- Email: [email protected]
Chapter 9 Some Generalizations
CHAPTER
9 Some Generulizutions
U p to this point all identities for special functions have been derived from a study of the representation theory of the Lie algebras Y(a, 6 ) and F6. However, the Lie theory approach to special functions is of much wider applicability than these examples may indicate. T o bolster this contention we use the results of Chapter 8 to discover some new Lie algebras which have realizations by generalized Lie derivatives in one and two complex variables. Corresponding to each of the new Lie algebras we will derive identities for special functions by relating these functions to the representation theory of the Lie algebra. T h e first five sections of this chapter are devoted to a brief examination of the Lie algebra X 5 .A study of X5 leads to new identities for the Hermite functions, the best known of which is Mehler’s theorem, (9.36). T h e special function theory of .X, presented here is by no means exhaustive, and the interested reader is invited to derive additional results. (Weisner [2] has obtained identities for the Hermite functions even more general than those derived here, by studying a 6-dimensional Lie algebra which contains X5as a subalgebra. However, his 6-dimensional algebra has no realizations by generalized Lie derivatives in one complex variable.) I n the last half of the chapter we introduce a family of 3-dimensional Lie algebras YP,*which forms a natural generalization of F3 91,1. T h e special functions associated with this family form a natural generalization of Bessel functions, and the identities obeyed by these functions are analogous to those derived for Bessel functions in Chapter 3 . T h e above examples are indicative of the use of Lie theory for the study of special functions. This theory can be employed both to obtain
9-1. THE LIE ALGEBRA .X5
299
properties of known special functions and to derive new functions which are interesting enough to deserve the label “special.” Additional examples of the applications of Lie theory to special functions are found in Vilenkin [2], [3], notably the special function theory of unitary representations of the orthogonal and Euclidean groups in n-space.
9-1
The Lie Algebra X5
X5is the 5-dimensional complex Lie algebra with basis and commutation relations
+&F*, [y3,91= 2 9 [f-,$+l = 8, [ f - ,91 = 2y+, [f+, 91 [&*, 81 = [ f 3 , 81 = [ 2 ,&] = 0. [23,2-*1
f*,
23,
&, 9
=
(9.1)
=0
Clearly, the 4-dimensional subalgebra of X5generated by f * , f 3 , d is isomorphic to 9 ( 0 , 1). T h e Lie algebra X, is of interest to us because it has realizations by differential operators in one complex variable. I n fact, for q = 2 in expression (8.28) one obtains the operators a1:
d dz’
-
z-
d
1, z, 9 ;
dz’
r = 5,
k
=
4,
s = 2,
which define an effective realization of X 5 .Every transitive effective realization of .X, by gd’s in one complex variable is an element of
QYQ.
-X, also has realizations in two complex variables. Using Lie’s tables (Lie [l], Vol. 111, pp. 71-73), and the methods of Chapter 8 one can show that every transitive effective realization of X5 by gd’s in two complex variables is an element of r71(pj), j = 1,..., 5, where
k=3,
r=5,
p2:
a
a
8%
az,
z2--z1-,
a
-
az,
a
a
+4 ;
tz,, - , 1, 22,8x2 az, r=5, k=3,
r=5,
s=l;
k=3,
s=l;
~ = 2 ;
3 00
9. SOME GENERALIZATIONS
9-2 The Lie Group K5
K5 is the 5-dimensional complex Lie group with elements
=g(q
+ e2-q‘, a + a’ + eTcb’,b + eTb’ + 2eZTcq‘,c + e-‘c‘, + 7
7’).
(9.3)
I n particular the identity element of K5 is g(0, O,O, 0, 0) and the inverse of g(q, a, b, c, T) is g ( - q r z T , -a
+ bc
-
2c2q, -be-7
+ 2cqecT, -ceT,
-7).
T h e associative law can be verified directly. This group has the 5 x 5 matrix realization ceT beer 2a - bc T g(q, a, b, c, T ) =
0 0
0 0
where now the group operation is matrix multiplication. I t is clear from (9.3) that the set of all group elements with q = 0 forms a subgroup of K5 isomorphic to G(0, 1) (compare with (4.11)). A simple computation using (9.3) or (9.4) shows that the Lie algebra of K5 is isomorphic to -X,. I n fact we can make the identification g ( q , a, b, c, 7) = e x p ( q 9 ) exp(a&) exp(b$+) exp(c$-) e x p ( ~ f ~ )(9.5)
x3,
where the elements 2*, &, B generate X5and satisfy the commutation relations (9.1). Equation (9.5) uniquely determines K5 as a local Lie group. Moreover, as a global group K5 is simply connected.
9-3. REPRESENTATIONS OF .X5
9-3
301
Representations of X5
Rather than try to give a complete analysis of the representation theory of .X, we restrict ourselves to irreducible representations which can be realized by elements of a(ci,)acting on spaces of analytic functions. For such representations it will be easy to compute the matrix elements. I n particular, we study the irreducible representations p of X, satisfying property A : p / 9 ( 0 , 1) ( p restricted to the subalgebra 9(0, 1)) is isomorphic to one of the irreducible representations R(w, m, , p) or To,p of 9(0,1). Such representations will yield additional information about special functions associated with 9 ( 0 ,I). Let p be an abstract irreducible representation of X5on a vector space V which satisfies the above requirement and let
p(Y*)
=
J*,
=
J3,
p ( 4 = E,
p(2) = Q
be operators on V . These operators clearly satisfy the commutation relations
[J3, 5’1 [J-, Q]
=
=
[J3, Ql
&J+, [J+, Q]
2J+,
=
[J-, J+l = E,
2Q,
=
[J*, El
0,
=
[J3, E]
=
[Q, El
= 0.
Theorem 9.1 Every irreducible representation p of X, satisfying property A is isomorphic to a representation in the following list: (1) R’(W m, , Y),
T h e spectrum of
uT.:
J3
W > P E @,
< Re m,
Y
f 0, w
+ m, not an integer.
is the set S = {m,
+ n: n an integer}.
0
(2)
, p E @,
w , m,
fL
< 1,
# 0.
T h e spectrum of J 3 is the set S = {-LO + n: n a nonnegative integer}. For each of the above classes the representation space V has a basis {fn,),m E S , such that JYm =
mfm
J’fm
9
Efm
= /+frn+l>
= pfm J-fm
Qfm
(m
1
= pfrn+z
+
(9-6)
Wlfm-1
for all m E S on the left-hand sides of these equations. All of the irreducible representations in classes ( 1 ) and (2) satisfy property A and no two of them are isomorphic.
9. SOME GENERALIZATIONS
302
T h e proof of this theorem is elementary and is left to the reader. We could of course find many other representations of X 5 ,but for purposes of illustration, the representations listed in Theorem 9. I will suffice.
9-4
The Representation K(w, m, , p)
We will find a realization of R’(w, m, , p ) given by some T‘ E @(a,) acting on a vector space Vi of analytic functions of z. According to the theory of Section 8-2, the space of cohomology classes of a ( E l ) is 2-dimensional. Moreover, every element of @(al) is cohomologous to a realization T’ of the form
T w o realizations of the form (9.7) are cohomologous if and only if they are identical. T o construct a realization of R’(w, m, , p ) in terms of the operators (9.7) let Yi be the space of all finite linear combinations of the functions &(z) = z m + w , defined for all m e S where S is the spectrum of the representation, and let c1 = - w , c2 = p. Then
for all m E S. Comparison of the3e expressions with (9.6) yields a realization of R’(w, m, , p ) in terms of linear differential operators. w is not an integer the functions fA(z)= zm+ware not Since m analytic and single-valued in a neighborhood of z = 0. This is very annoying for computational purposes. T o remedy the defect we map the space Yi onto the space “y; = cp-’(V;)where cp-l[ f ’ ] ( z ) = (p(z))-lf‘(z) E Yl , f ’ E V i , and p(z) = Then Yl has a basis of the form fm(z)= cp-“ fT3(,z)= zk where K = m - m, is an integer. cp induces a transformation T = ( p - 1 ~ ‘ mapping ~ a realization T‘ of .X5 by gd’s on Y ; into a realization T by gd’s on Yl ; see (8.3). Under this
+
Z~O+W.
9-4. THE REPRESENTATION R’(w, rn, , p)
transformation the operators (9.7) with c1 = into the operators
-w,
303
c2 = p are mapped
on Vl . T h e operators (9.9) and the basis functionsf,(z) = zk,m = m, + k, provide the desired realization of R‘(w, m,, p). (Note that with the exception of Q = pzz, these operators and basis functions are identical with those derived for a realization of the representation R ( w , m a ,p) of 3 ( 0 , 1) in Section 4-1.) Following our usual procedure we can extend this realization of R’(w, m a ,p) defined on Vl to a local multiplier representation of K5 on G l l , where GTl is the complex vector space of all functions of z analytic in some neighborhood of the point z = 1. Here Ul 3 “y; and Ul is invariant under the operators (9.9). T h e operators A(g), g = g(q, a , b, c , T) E K5 , defining the multiplier representation can be computed i n . a straightforward manner from expressions (9.5) and (9.9). T h e result is
we can show that every function h E yl C Ul has a unique Laurent expansion
which converges absolutely for all 1 z I > d, where 1 > dh 3 0. (Tl is invariant under A ( g ) , see Section 2-2.) Thus, the functions f,(z) = hk(z)= zk, m = m, + k, in Vl form an analytic basis for T h e matrix elements A , , ( g ) of the operators A ( g ) with respect to this basis are defined by
vl.
m
[A(g)htJ(z)
C
A&)hi(Z),
k
= 0 , fl, f2,-,
l=--m
(9.11)
9. SOME GENERALIZATIONS
304
Since the operators A(g) define a local multiplier representation of K5we obtain the addition theorem 5
Azdgigz)
=
C
I, k
Azi(gi)Ajlc(gB),
= 0,
i l , III~,...,
(9.12)
j=-m
valid for g , , g, in a sufficiently small neighborhood of the identity element. T o be precise we should determine the exact domain of validity of (9.12), just as in Section 4-1. However, it-will soon be evident that both sides of this equation are entire analytic functions of the group parameters of g , and g, . Thus, by analytic continuation equation (9.12) holds without restriction for all g, ,g , E K5 . T h e matrix elements A , , ( g ) can be evaluated directly from the generating function (9.11). We examine some special cases: (1) g ( 0 , a, b, c, T) ( q = 0 ) . I n this case (9.11) is identical with the generating function (4.8). Thus the matrix elements are proportional to the Laguerre functions,
+
where p = m, w is not an integer. ( 2 ) g(q, 0, b, 0,O). T h e generating function becomes exp[p(qz2
-+ bz)]zk =
m
Atk(g)z’. z=-5
Comparing this expression with the generating function (4.77) for the Hermite polynomials we find
Here, the matrix elements are entire functions of p and q. (3) g(q, 0, 0, c, 0). T h e generating function is
where R;+L(pq, c )
=
A,,(g). Thus, (9.15)
9-4. THE REPRESENTATION R’(w, m, , p)
305
where j ranges over all integral values such that the summand makes sense. There is no simple expression for the functions RP,in terms of functions of hypergeometric type. Various identities relating these matrix elements can be obtained from the addition theorem (9.12). (We have already derived identities for the matrix elements (9.13) in Section 4-1.) Thus, the relation g(q, 0, b, 0, 0 ) g(q’, 0, b‘, 0, 0 ) = g(4
+ 4’, 0, b + b’, 0, 0 )
implies (after some simplification) the identity (4
+4’Y2 n!
b
+ b‘ m=O
(9.16)
T h e relation
implies
There are many other identities contained in (9.12), but the reader can derive them for himself. I n analogy with Section 4-6 we can construct a realization of R‘(w, m, , p) by gd’s on a space of two complex variables where the action of -X, on this space is given by a member of @(fl1),(9.2). (The elements of a(&) when restricted to 9(0, 1) are identical with the type D’ operators.) From the theory of Section 8-2 it is easy to show that the space of is 2-dimensional and that every element of cohomology classes of a(&) this space is cohomologous to a realization of the form
Two realizations of this form are cohomologous if and only if they are identical. Note: T h e selection of a representative in each cohomology class given by (9.18) differs on the subalgebra 9 ( 0 , 1) from that chosen in Section 4-6. T h e representative selected here has been chosen for convenience in the computations to follow.
3 06
9. SOME GENERALIZATIONS
T o construct a realization of R’(w, m, , p ) using the operators (9.18) we must find nonzero functionsf,(x, y ) = Z,(x) emv such that Eqs. (9.6) are valid for all m E S = {m,+ n: n an integer}. I n terms of the functions Z,(x) these relations become (i) (ii)
(&d - P)z
m
= PZm+&43
d
-dx Z,(x) = ( m
+
W)
(9.19)
Zn-l(x),
Just as in Section 4-6 we can assume w = 0, p = 1 without any loss of generality for special function theory. Furthermore, a simple computation shows (i), (ii), and (iii) are compatible if and only if c = -1. Thus our problem reduces to finding a realization of R’(0, m, , 1) in terms of the operators (9.18) where c = -1. I n this case Eqs. (9.19) have the following linearly independent solutions for all m E S : (1) Z,(x) (2) Z,(x)
=
(-1)m-m
0
2-m/zH,(x/2/2)
=
(- l)m-mo exp(x2/4)D,(x),
+ 1 ) 2(m+l)/zH-,-1(42/2) = exp(x2/4)e--iw(m+1)/2T(m + 1) D-,-l(ix). =
exp(x2/2)e--i7i(m+1)/21‘(m
(9.20)
T h e H,(x) are Hermite functions defined in terms of parabolic cylinder functions by H,(x) = 2mIz exp x2/2D,(d/zx). When m is a positive integer, which is not the case here, H,(x) is a Hermite polynomial. T h e fact that expressions (9.20) satisfy the recursion relations (i) and (ii) follows easily from (4.66) and (4.67). Relation (iii) can be derived from (i) and (ii). These solutions are entire functions of x. Clearly, if the functions 2, , m E S , are given by either (1) or (2), then the functionsf,(x, y ) = Z,(x) emu form an analytic basis for a realization of the representation R’(0,m, , 1) of X, . As usual this representation of S, can be extended to a local multiplier representation of K , by operators T(g),g E K , , on the space 9of all functions analytic in a neighborhood of the point (xo, yo) = (0, 0). A straightforward computation using (9.5) and (9.18) yields
~ E F (9.21) ,
where t
=
ev.
9-4. THE REPRESENTATION R'(w, m, , p)
307
T h e matrix elements of the operators T(g) with respect to a basis fm(x, t ) = ZJx) tm satisfying (9.19) are given by the functions A,,(g), (9.11), where w = 0, p = 1. Thus we obtain the relations
Corresponding to the basis
for all integers k, this equation can be written in the form x2t29 - txb
-
bV/2
x
+ bt
-
ct-1
-
2ct q
For q = 0, (9.22) is identical with (4.70) so we will omit this case. If a = c = T = 0, q = -$, (9.22) reduces (after some simplification) to the identity
(9.23)
valid for all m E @ not an integer. (In the next section we will show that (9.23) is also valid for m a nonnegative integer.) If a = b = T = 0, q = - $, (9.22) simplifies to
m
where h:(c)
=
(-
d/2)-n I?;( -4, z/%)
has the generating function
9. SOME GENERALIZATIONS
308
Here m is not an integer. However, in the next section we will show that this equation remains true for m a nonnegative integer. Similarly, the second set of basis vectors (9.20) can be used to derive identities for the Hermite functions. For example the case a = c = T = 0, q = -4, yields the identity
a generalization of (9.16). By substituting appropriate choices for the group parameter into (9.22) the reader can derive additional identities obeyed by the Hermite functions.
9-5
The Representation
f’w.p
I n analogy with the procedure of the last section we look for a realization of the representation t’u,pgiven by some T E 6Y(El) acting on a vector space of analytic functions of z. A comparison of expressions (4.17) and (9.7) shows how to proceed. Namely, in (9.7) we set c1 = - w , c2 = p to obtain J3
=
d
--
dz
w,
d
J- = dz,
J+ = pz,
E
= p,
Q
= pz2.
(9.24)
Designate by V2the space of all finite linear combinations of the functions zk, k = 0, 1, 2, ..., and define the basis vectors fm of V2by fm(x) = h k ( 2 ) = zk where m = -w k, k 3 0. Clearly, hk(z) =
+
JY, = (--w J-f,
+ z dzd l
zk =
mj,,
J+L
= (pz) z k =
+
,
Efm = /+fin ,
d dz
= - zk = ( m
Qfm
=
-w)fm-l
(P’)zk = d m + z
c~fn+~,
(9.25)
-
‘These relations define a realization of Tb,, on Vz. Moreover, this realization can easily be extended to a local multiplier representation B of K5on the space 6Y2 consisting of entire functions of z. T h e operators B(g),g = g(p, a, b, c, T) E K 5 , defining this representation are easily computed: [ B ( g ) f ] ( z= ) exp[p(p2
+ bz + a )
-
-w7]f(eTz
+ eTc), f~ rY2.
(9.26)
309
9-5. THE REPRESENTATION
Furthermore it is straightforward to show that the operators B(g) form a global representation of K 5 .T h e matrix elements B,,(g) of B(g) with respect to the basis (f, = hk) are given by m
LB(g) h k l ( Z )
=
Blk(g) hl(z),
= 0,
1, &***,
l=O
or
T h e addition theorem for the matrix elements is (9.28)
valid for all g, ,g, E K5 . Corresponding to some special choices of the group parameters the matrix elements have the following explicit expressions: (1) g(0, a b, c, T). For q = 0, (9.27) becomes identical with the generating function (4.22). The matrix elements are thus proportional to associated Laguerre polynomials.
(2) g(q, 0, b, 0, 0). Exactly as in (9.14) we find
(3) g(q, 0, 0, c, 0). In this case (9.27) reduces to (9.30)
where R;-,(pq, c)
=
Blk(g).Thus, (9.31)
31 0
9. SOME GENERALIZATIONS
where j ranges over all integral values such that the summand is defined. Comparing (9.30) with the generating function (4.77) for the Hermite polynomials we find (9.32)
Just as in the last section, the addition theorem (9.28) can be used to derive identities obeyed by the matrix elements. This task will be left to the reader. we now Armed with a knowledge of the matrix elements of procede to construct a realization of this representation by gd’s on a space of two complex variables, where the action of X5is given by a member of a(&), (9.2). I n particular we will use the operators (9.18) with c = -p. T o construct a realization of using these operators it is necessary to find nonzero functions f,(x,y) = Z,(x) emu, m = - w , -a 1,..., such that Eqs. (9.6) are valid with
+
Just as in the derivation of Eqs. (9.19), it is easy to show that without loss of generality for special function theory we can assume w = 0, p = 1. Then, expressed in terms of the functions Zm(x),Eqs. (9.6) become the recursion relations
(9.33)
where m takes the values 0, 1, 2, ... . The functions Z,(x) are determined to within an arbitrary constant by (9.33). I n fact relation (ii) implies Z,(x) = c for some constant c. If we set c = 1, the remaining solutions Z,(x) are uniquely determined: Z,(x)
=
(-1)m
2-m/2Hm(x/z/Z)
=
(-1)m
exp(x2/4) Dm(x),
(9.34)
9-5. T H E REPRESENTATION
f'u,p
311
where the H,(x) are Hermite polynomials. Thus the vectors f , ( x , y ) = Z,(x) emy, m = 0, 1, 2 ,..., where Z,(x) is given by (9.34), form a basis for a realization of the representation of Z5. This infinitesimal representation of .X5can be extended to a local multiplier representation of K5 by operators T(g),g E K 5 ,on the space 9of all functions analytic in a neighborhood of the point (9, y o ) = (0,o). Indeed we have already computed the operators T(g) on 9, Eq. (9.21). T h e matrix elements of T(g) with respect to an analytic basis (fm(x, y ) ) satisfying (9.33) are the functions B,,(g) given by (9.27) ( w = 0, p = 1). Consequently,
c BZk(g)fZ(x,y), ' m
[ T ( g ) f k l ( x , y== )
=
O, 2 , * * * ~
1=0
or
(9.35)
0 this expression is identical with (4.76). For a
= c = T = 0, it reduces to (9.23) where now m is a nonnegative integer. I n the special case n = 0, (9.23) becomes
For q
q
=
=
-B,
(9.36)
which is Mehler's theorem, (4.158). Finally, if a = b = T = 0, q = -*, (9.35) reduces to the identity
W
=
C h$-,(c) H,(x) tZ-'-,
(9.37)
1=0
where hF-E--k(c)= (- 2/Z)k-zRf-k(-4, 4 Z c ) is given by the generating function m
exp(--z2/4)(z
-
2c)k
=
1 hl'_k(c)
21.
1=0
31 2
9. SOME GENERALIZATIONS
This completes our analysis of the special function theory of X 5 . Note that the identities for Hermite functions derived here have been obtained from a study of the operator realizations a1 and P I , (8.28) and (9.2). We could also have used p5 , but this would not have led to any new results. T h e operators p 2 , p 3 , p4, on the other hand, do not yield but rather lead realizations of the representations R’(w, m, , p ) or T;+, to realizations of representations of X5which we have not classified here. I n particular, p2 can be used to realize certain reducible representations in such a way that the basis vectors turn out to be proportional t o generalized Laguerre functions. Th u s a study of p2 leads to new identities for Laguerre functions.
9-6 The Lie Algebra 5??p,q Corresponding to a pair of positive integers ( p , q) let gPrq be the f + , 3- and commucomplex 3-dimensional Lie algebra with basis tation relations
x3,
is isomorphic to Y3. I n addition the following isomorphisms Clearly, are easily established: Lemma 9.1
for all positive integers gP,*E 5??q,P; 9’np,np gP,*;
P, 9, n. According to this lemma every Lie algebra 5??p,,,, is isomorphic to a Lie algebra gP,*such that (1) p and q are relatively prime positive integers; (2) p is odd; and (3) if q is odd then p > q. Consequently, from now on the pair ( p , q) will be assumed to satisfy properties (1)-(3). It is , 5??P‘,q’ with subscripts satisfying obvious that two Lie algebras gp,, these properties, are isomorphic if and only if p = p‘ and q = 4’. Applying the techniques developed in Chapter 8 we can determine all by gd’s in one or two of the transitive effective realizations of Yp,, complex variables. Thus, from (8.26) there follows the realization 71
:
a -
aY ’
epy,
e-qy;
r
=
3, k
=
2, s
= 0.
(9.39)
Every transitive effective realization of gp,,by gd’s in one complex variable is an element of OZ(ql). Similarly, by making use of Lie’s tables (Lie [I], Vol. 111, pp. 71-73),
9-7.THE LIE GROUP G,,Q
31 3
it can be shown that every transitive effective realization of 9p,q by gd’s in two complex variables is an element of 0l(sj), j = 1, 2, 3, where r=3, 6, :
--PZ,
a a az, + 4% az, ’
a -
Y
az, ’
=
3. k
r=3,
9-7
k=l, s=O; =
1,
k = l ,
s = 0;
(9.40)
s=O.
The Lie Group Gp,q
Corresponding to a pair of relatively prime positive integers ( p , q) such that p is odd, denote by Gp,Qthe 3-dimensional complex Lie group with elements
g ( h c, 7)’
b, c,
E
I%,
and group multiplication
It is easy to check that the multiplication is associative. Furthermore, e = g(0, 0, 0) is the identity element and g(--e-PTb, -@c, -T) is the unique inverse of the group element g(b, c, T). GP,*has a 4 x 4 matrix realization
(9.42)
where matrix multiplication corresponds to the group operation. Comparing (9.42) with (1.35) we see that Gl,l is isomorphic to T , . T h e Lie algebra of G,,* can be computed from either of the expressions (9.41) or (9.42), and is easily recognized to be isomorphic to Y P , ,. Indeed we can make the identification
where f * ,
y 3generate 9fp,qand satisfy the commutation relations (9.38).
31 4
9-8
9. SOME GENERALIZATIONS
Representations of gp,q
Let p be a representation of g’p,q on an abstract vector space V and let = J*, p(Y3)= J3, be operators on V . Clearly,
p(f*)
Note that the operator C p * q = (J+)*(J-)” = (J-)”( J+)q commutes , on V . For p irreducible we would with all operators p(01), 01 E gP,, expect CP,* to be a multiple of I. I n strict analogy with our study of the representation theory of F3 we will classify all representations p satisfying the properties: (i) p is irreducible (ii) Each eigenvalue of J 3 has multiplicity equal to one. There is a countable basis for V consisting of eigenvectors of J3.
(9.45)
We give without proof the results of the straightforward classification. satisfying (9.45) and Theorem 9.2 Every representation p of gP,, 0, on V , is isomorphic to a representation of the form for which Cp’q QP,*(w,m,) defined for w , m, E @ such that w # 0 and 0 Rc m, < 1. T h e spectrum of J3 corresponding to QP,*(w,m,) is given by 5’ = (m, + n: n an integer) and there is a basis i f m } , m E S , f0r.V such that
+
<
J+fm
= mfm
J3fm
= wfm+v,
(9.46)
J-fm
= wfm-,
,
C”.ym= (I+)*( J - ) ” f i n
.
= wp+,fm
QpPQ(w‘, mi) if and only if There exist isomorphisms QP,,(w, m,) m, = mi , w = w‘d, where d is a ( p g)th root of unity. I n accordance with the usual procedure, we can use the operators to construct a realization of the representation QP,*(m,m,). This can be done by introducing the change of variable z = e” in (9.39) and considering the operators
+
J3
=z
d
+ c1 ,
J+
=C~ZP,
J-
= c~z-‘J
(9.47)
in Ol(fl). Thus, let ”y; be the complex vector space of all finite linear combinations of the functions h,(x) = zn, n = 0, * l, &2, ..., and define operators J*, J 3 on rY; by (9.47) where c1 = m, , c2 = w . Define
9-8. REPRESENTATIONS OF gP,@
the basis vectorsf, of Vl byf,(z) runs over the integers. Then
J+fm
= ( u z p ) ~ "= wfm+n
1
=
h,(z) where m
31 5 =
m,
+ n and n
= ( w ~ - Z" ~ )= wfm-,
J-fm
,
(9.48)
c.ym = w ~ + " f m . . These equations agree with (9.46) and yield a realization of Qp,p(w,m,). T h e differential operators (9.47) (cl = m, , c2 = w ) induce a multiplier representation A of G,,* on the space GZl consisting of those functions f ( z ) which are analytic and single-valued for all x f 0. This multiplier representation is defined by operators A(g),g = g(6, c, T ) E Gp,* : [A(g)f](z) = exp[w(bzP
+ cz-@)+ r n o ~ ] f ( e T z ) ,
f~ & .
(9.49)
Clearly, GZl is invariant under the operators A(g) and the group property A(g1g2) = A(g,) A(g2) is valid for all g, g, E G,,* . T h e matrix elements A,,(g) of A(g) with respect to the analytic basis {f, = h,) of GZl are defined by 9
c m
[A(g) h k l b )
=
M g ) hdz),
g E GD,*>
k
= 0 , il, *2,...,
(9.50)
I=-m
or exp[w(bzP
+ c x q ) + (m,+ k )
a,
T]
zk =
A,,(g) zz
(9.51)
I=--m
where g
= g(6, c,
T).
Explicitly, (9.52)
(9.53)
T h e nonnegative integers sl ,n l are uniquely determined by the properties:
(1) 1 = n,p - siq, (2) If I = zip - sip where n ; , s; are nonnegative integers, then n;
+ si 3 n, +
s,.
Since p and q are relatively prime, the integers n, , s l can easily be shown to exist for all 1. For example, if p = q = 1 then s, = 0, n, = I for I 3 0 and s, = -1, n, = 0 for 1 < 0.
31 6
9. SOME GENERALIZATIONS
If bc # 0 we can introduce new group parameters r, v defined by b = r v p / ( p + p), c = r v - Q / ( p 9). I n terms of these new parameters the matrix elements are
+
(9.54)
(9.55)
It follows from the ratio test that 17,q(r)is an entire function of r for all integers 1. Here I;,’(r) = (-i)zJz(ir), 13 0, is the ordinary “modified Bessel function” (see ErdClyi et al. [l], Vol. 11). T h u s we can consider the functions If’*(r) to be a group-theoretic generalization of Bessel functions. Substitute (9.54) into (9.51) to obtain the simple generating function Y
(z”
+ z-.)I
f
=
IpqY)
2.
(9.56)
Z=-m
T h e group property of the operators A(g) implies the addition theorem
c Azj(g1) A&!Z), m
Azk(glg2) =
(9.57)
j=-W
valid for all g, ,g,
E
Gp,q. This leads immediately to the identity m
FP.*(bl
+ b, , + cZ) = 2 F;:q(b, ~1
, c ~ ) F ; ” (, c2). ~~
(9.58)
j=-m
We could also use the addition theorem to derive identities involving the functions Iy3q(r).However, this will not be done here as most of the results so obtained are special cases of identities which will be derived in Section 9-10.
9-9
Recursion Relations for the Matrix Elements
Denote by GZ(Gp,J the space consisting of all entire functions on the group Gp,*, i.e., of all entire functions of the parameters b, c, T. Clearly, the matrix elements A,,(g) are elements of 6Y(Gp,J. T h e representation P of Gp,qon LT(Gp,J defined by [P(g’)fl(g) = f(gg’),
f
.
U(GP,*)Y
(9.59)
317
9-9. RECURSION RELATIONS FOR MATRIX ELEMENTS for all g, g‘ E Gp,qsatisfies the relation
and determines Gp,qas a transformation group. According to (9.57) the action of P on a matrix element Ajk is given by
c m
[P(g‘)Ajkl(g) = Ajk(gg’) =
Adg’) M g ) .
(9.60)
l=-m
Comparing this expression with (9.50) we find that for fixed j the functions (Aik},K = 0, f l , f 2 , ..., form a basis for a realization of the representation Qp,,(w, m,) of Gp,qunder the action of the operators P(g).Therefore, the Lie derivatives J*, J 3 defined on a ( G p , Jby
(9.61)
JYk) =
d
I
[ P ( ~ x~$~)f’l(g) P
.r=O
f~ @(Gp,Q),
satisfy the relations J3A9&)
=
(%
4- k) A,&), J-Ajdg)
(J’)“ (J-)” A,,(g)
CP*QAJk(g)=
=
=
J+Aidg)
WAili+pk),
“A,,-&),
wYfyAjk(g)>
(9.62)
i,k
=
0, & I >
*’>***
*
Equations (9.62) yield recursion relations and differential equations for the matrix elements A j , . We will use these equations to derive recursion relations for the generalized Bessel functions I ~ , * ( I Thus, ). we assign to a group element g = g(b, c, T ) , bc # 0, the local coordinates [ I , u, T], where b = rvp/(p q), c = ~v-q/i(p 9). Then if g‘ = g(b‘, c’, T‘), a straightforward computation shows the local coordinates of gg‘ are
+
[Y(
1
v(1
+ ( p + q)
r-lv-PeW)q/(P+y)
+
(1 + ( p f q ) y-laG‘e-YTC’
+ ( p + q ) r-lv-PePTb’)ll(P+y)(1
i
Y/(P+Y) 1
7
( p + q ) Y-~vQ~-RTc’)~/(P+*), T + T’]
for 1 b’ 1, 1 c‘ I sufficiently small. It follows from the definition (9.61) that the Lie derivatives J*, J 3 are
(9.63)
31 8
9. SOME GENERALIZATIONS
Substituting (9.54) and (9.63) into (9.62) and factoring out the dependence on v and T, we obtain the equations
= I$*(r),
m
=
0, i l , *2
,... .
(9.64)
I n particular, the functions Ig*((r)are solutions of an ordinary differential q. equation of order p
+
9-10
Realizations in Two Variables
T h e operators 6, and 6, , (9.40) can be used to construct realizations of
Qp,,(w, m,) on spaces of functions of two complex variables. However, the basis functions so obtained are elementary and of little interest. On the other hand a realization of Qp,q(w,m,) by 6, does lead to new results. T o see this we introduce new variables x , y , where z1= xePY/(p q), in the expression for 6, :
z2 = xe-""/(p
+
+ q),
T o construct a realization of Qp,q(w,m,) using the operators (9.65) we must find nonzero functions &(x, y ) = Z,(x) e c r n y ; m = m, n, n = 0, f l , 5 2 ,..., such that expressions (9.46) are valid. These expressions are equivalent to the equations
+
d
) ... (q dx X
dx
m-qp X
)(p
d
m-(p-l)q
z
+
X
1
(9.67)
where without loss of generality for special function theory, we have set w = 1. Thus, the functions Z,,L(x)are solutions of differential equations of order p + q.
9-1 0. REALIZATIONS IN TWO VARIABLES
319
I n general it is possible to findp + q linearly independent solutions of Eqs. (9.66) and (9.67). However, we will be content with a single solution. It is a consequence of (9.64) that if m, = 0, the functions Z,(x) = I ? ) ~ ( Qare ( x )solutions of these equations. Following a standard procedure in the theory of special functions we shall use analytic ( Xm) an arbitrary complex number and continuation to define I ; L ~for show that the functions Z,(x) = Z?)$(x) satisfy the required relations (9.66) and (9.67) even if m, f 0. An application of the Cauchy residue theorem to (9.56) yields 27riI:q(r)
=
s,
z-”-l exp
[ z (z” + 2-91 dz, P+4 m
= 0,f l ,
*2 ,...,
where C is a simple closed contour surrounding the origin in the z-plane. Without changing the value of the integral, the contour C can be deformed into a loop D which starts at infinity on the negative real z-axis, encircles the origin counterclockwise and returns to its starting point. ( T he proof of this statement is almost word for word the same as the well-known proof of the analogous statement in the special case p = q = 1, see Whittaker and Watson [I], Chapter 17). Since p is odd, the integrand is bounded on D for Re Y > 0. Thus, we have
s‘”’
where --m denotes integration along D. Since the integrand is now single valued on D even for m complex, we can use (9.68) to define the function IPg(r) for arbitrary values of m and Re Y > 0. Moreover, by differentiating under the integral sign in (9.68) we can verify that Eqs. (9.64) remain valid for arbitrary m. T h u s the functions Z,,(x) = IP;;(x), m = m, n, define a realization of the representation Q p , q ( m o , 1) of %I, ., We can obtain more information about the generalized Bessel functions *;:Z by mimicking the standard treatment of ordinary Bessel functions in terms of contour integrals as given, for example, by Whittaker and Watson [I], Chapter 17. If r > 0, introduction of the new variable w = ( ~ / ( p+ q ) ) l / p z in (9.68) leads to the expression
+
T h e right-hand side of this equation can be analytically continued to define I:iq(y) as an analytic, but not single valued, function of r for all
320
9. SOME GENERALIZATIONS
r # 0. I n particular r-m/pI$g(r) is an entire function of r ( P + q ) / P . T o obtain a series expansion for Zgp(r) we use
in (9.69) and integrate term by term. From the theory of the gamma function (Whittaker and Watson [l], Chapter 12),
Thus,
For p = 1 this expression simplifies to
We could construct additional solutions of equations (9.64), linearly independent of the functions IEQ(r),by choosing a different path of integration in (9.68) (see Khriptun [I]). However, for purposes of illustration we will be content with the solutions ZZ*(r).
9-1 I
Generating Functions for Generalized Bessel Functions
+
As shown above, the functions fm(x, y ) = I%(.) e c r n y , m = m, n, n an integer, form a basis for a realization of the representationQ,,q( 1, m,) of gP,,. Following our usual procedure we will extend this representation to a local multiplier representation of Gp,q in the space F of functions analytic i n ' a neighborhood of (xo,y o ) = (1, 0). T h e operators T(g),g = g(b, c , T ) E Gp,p defining this multiplier representation can easily be computed from (9.65) and (9.43). T h e result is
fE F ,
(9.72)
9-11. GENERATING FUNCTIONS
321
+
+
valid for j b(p q)/xtp I < 1, 1 c(p q) tq/x I < 1, where t = e”. The matrix elements of T(g) with respect to the analytic basis (fmfx, t ) = P Z ( x ) t-”), are the functions A,,(g) given by (9.52) and (9.54). Thus, m
[T(g)frn,,+kI(x,t )
C
=
= 0,
Azk(g)fm,+Z(X, t ) ,
i l , +2,-*.,
Z=-CC
or b(p m
+ q))Q:(P+Q)
(1
+
c(p
+ q) tQ)n.(P+Q)] X
c F!;Q(b, m
=
c ) I2$x)
tl,
Z=--a,
I b(p for all m
E
$?.If c “ : 1
[X
+ q ) W I < 1,
=
0, t
(1
+ y-b(p )Qf + q )
=
I .(p
t q) tQ/X I
< 1,
(9.73)
1, this expression simplifies to / ( P + d ](1
+ b(p + q )
)rn/(P+Q)
X
while for b
=
0, t
I:.[+
=
1 it becomes
+
c(p
+ ,I))”’(”+Q)] (
+
c(p
+ q))-”L;(”+Q) X
+
If b = c = ri(p Q ) # 0 the matrix elements can be expressed in terms of generalized Bessel functions and (9.73) becomes y
e Q I X ( l
Q/(P+Q)
+,t”)
) (1, +
rtg
pl(P+g)
I(
x
+ rt-”
x f rtQ
~/(P+Q!
1
For p = Q = 1, Eq. (9.73) is equivalent to the Bessel function identity (3.29).
322
9. SOME GENERALIZATIONS
Similarly, in analogy with Weisner’s treatment of Bessel functions as given in Section 3-4, we can use the operators
expressions (9.65), to derive generating functions which are not related to the matrix elements A,,(g). T h e basic observation to be made is: If f ( x , t ) is a solution of the equation C p , v = ( J + ) q ( J - ) p f = wf then the function T(g)f given by the formal expression (9.72) is also a solution of this equation. Moreover, iff is a solution of the equation (xIJ+
+ xzJ- + x,J’)lf(x,
t ) = xf(x, t )
for constants x1 , x2 , x3 , A, then T(g)f is a solution of
As an illustration of these remarks, consider the function f ( x , t ) = I z * ( x )tm, m E @. In this case C p , q f = f, JY = -mf, so the function T(g)f satisfies
where g = g(b, c, T) E G p , q .If c in the form
=T =
0, b
Since x - ” / p I E , q ( x ) is an entire function of expansion in t P about t = 0:
c-
=
1, T(g)f can be written
x(p+q)lp,
m
h(x, t )
=
n=-m
h,(x) tn”,
I X t Y I
h has a Laurent
+ q.
323
9-11. GENERATING FUNCTIONS
Substituting this expansion into the first equation (9.74), we find Z,,(x) = h-,(x) is a solution of the generalized Bessel equation (9.67) for n = 0, 4 1 , 4 2 , ... . At this point we use the fact: For m an integer the only solutions of (9.67) which are regular at x = 0 are cI?:(x), c a constant. This statement can be verified by computing the indicia1 equation of (9.67) (see Ince [I], Chapter 7). Since h(x, t ) is bounded for x = 0, we have h,(x) = c,I&'(x), c, E @. Hence, m
h(x, t )
=
c
C " l g ( x ) t".
n=-m
+
T h e equation (-p J+ J3) h(x, t ) = -mh(x, t ) implies = ( m - np) c, . T h e constant co can be determined by setting x = 0 in h(x, t ) and using (9.70). Consequently, co =
T ( - m / p ) sin[r(m Pr
+ 1)1 ,
cn =
(-1)" T[(.P
-
mYP1 sin[r(m
+ 111
Pn
and we obtain the identity
139"
When
p
=
Ic p
+ q.
q = 1 this expression is equivalent to (3.38).
(9.76)
9 Some Generulizutions
U p to this point all identities for special functions have been derived from a study of the representation theory of the Lie algebras Y(a, 6 ) and F6. However, the Lie theory approach to special functions is of much wider applicability than these examples may indicate. T o bolster this contention we use the results of Chapter 8 to discover some new Lie algebras which have realizations by generalized Lie derivatives in one and two complex variables. Corresponding to each of the new Lie algebras we will derive identities for special functions by relating these functions to the representation theory of the Lie algebra. T h e first five sections of this chapter are devoted to a brief examination of the Lie algebra X 5 .A study of X5 leads to new identities for the Hermite functions, the best known of which is Mehler’s theorem, (9.36). T h e special function theory of .X, presented here is by no means exhaustive, and the interested reader is invited to derive additional results. (Weisner [2] has obtained identities for the Hermite functions even more general than those derived here, by studying a 6-dimensional Lie algebra which contains X5as a subalgebra. However, his 6-dimensional algebra has no realizations by generalized Lie derivatives in one complex variable.) I n the last half of the chapter we introduce a family of 3-dimensional Lie algebras YP,*which forms a natural generalization of F3 91,1. T h e special functions associated with this family form a natural generalization of Bessel functions, and the identities obeyed by these functions are analogous to those derived for Bessel functions in Chapter 3 . T h e above examples are indicative of the use of Lie theory for the study of special functions. This theory can be employed both to obtain
9-1. THE LIE ALGEBRA .X5
299
properties of known special functions and to derive new functions which are interesting enough to deserve the label “special.” Additional examples of the applications of Lie theory to special functions are found in Vilenkin [2], [3], notably the special function theory of unitary representations of the orthogonal and Euclidean groups in n-space.
9-1
The Lie Algebra X5
X5is the 5-dimensional complex Lie algebra with basis and commutation relations
+&F*, [y3,91= 2 9 [f-,$+l = 8, [ f - ,91 = 2y+, [f+, 91 [&*, 81 = [ f 3 , 81 = [ 2 ,&] = 0. [23,2-*1
f*,
23,
&, 9
=
(9.1)
=0
Clearly, the 4-dimensional subalgebra of X5generated by f * , f 3 , d is isomorphic to 9 ( 0 , 1). T h e Lie algebra X, is of interest to us because it has realizations by differential operators in one complex variable. I n fact, for q = 2 in expression (8.28) one obtains the operators a1:
d dz’
-
z-
d
1, z, 9 ;
dz’
r = 5,
k
=
4,
s = 2,
which define an effective realization of X 5 .Every transitive effective realization of .X, by gd’s in one complex variable is an element of
QYQ.
-X, also has realizations in two complex variables. Using Lie’s tables (Lie [l], Vol. 111, pp. 71-73), and the methods of Chapter 8 one can show that every transitive effective realization of X5 by gd’s in two complex variables is an element of r71(pj), j = 1,..., 5, where
k=3,
r=5,
p2:
a
a
8%
az,
z2--z1-,
a
-
az,
a
a
+4 ;
tz,, - , 1, 22,8x2 az, r=5, k=3,
r=5,
s=l;
k=3,
s=l;
~ = 2 ;
3 00
9. SOME GENERALIZATIONS
9-2 The Lie Group K5
K5 is the 5-dimensional complex Lie group with elements
=g(q
+ e2-q‘, a + a’ + eTcb’,b + eTb’ + 2eZTcq‘,c + e-‘c‘, + 7
7’).
(9.3)
I n particular the identity element of K5 is g(0, O,O, 0, 0) and the inverse of g(q, a, b, c, T) is g ( - q r z T , -a
+ bc
-
2c2q, -be-7
+ 2cqecT, -ceT,
-7).
T h e associative law can be verified directly. This group has the 5 x 5 matrix realization ceT beer 2a - bc T g(q, a, b, c, T ) =
0 0
0 0
where now the group operation is matrix multiplication. I t is clear from (9.3) that the set of all group elements with q = 0 forms a subgroup of K5 isomorphic to G(0, 1) (compare with (4.11)). A simple computation using (9.3) or (9.4) shows that the Lie algebra of K5 is isomorphic to -X,. I n fact we can make the identification g ( q , a, b, c, 7) = e x p ( q 9 ) exp(a&) exp(b$+) exp(c$-) e x p ( ~ f ~ )(9.5)
x3,
where the elements 2*, &, B generate X5and satisfy the commutation relations (9.1). Equation (9.5) uniquely determines K5 as a local Lie group. Moreover, as a global group K5 is simply connected.
9-3. REPRESENTATIONS OF .X5
9-3
301
Representations of X5
Rather than try to give a complete analysis of the representation theory of .X, we restrict ourselves to irreducible representations which can be realized by elements of a(ci,)acting on spaces of analytic functions. For such representations it will be easy to compute the matrix elements. I n particular, we study the irreducible representations p of X, satisfying property A : p / 9 ( 0 , 1) ( p restricted to the subalgebra 9(0, 1)) is isomorphic to one of the irreducible representations R(w, m, , p) or To,p of 9(0,1). Such representations will yield additional information about special functions associated with 9 ( 0 ,I). Let p be an abstract irreducible representation of X5on a vector space V which satisfies the above requirement and let
p(Y*)
=
J*,
=
J3,
p ( 4 = E,
p(2) = Q
be operators on V . These operators clearly satisfy the commutation relations
[J3, 5’1 [J-, Q]
=
=
[J3, Ql
&J+, [J+, Q]
2J+,
=
[J-, J+l = E,
2Q,
=
[J*, El
0,
=
[J3, E]
=
[Q, El
= 0.
Theorem 9.1 Every irreducible representation p of X, satisfying property A is isomorphic to a representation in the following list: (1) R’(W m, , Y),
T h e spectrum of
uT.:
J3
W > P E @,
< Re m,
Y
f 0, w
+ m, not an integer.
is the set S = {m,
+ n: n an integer}.
0
(2)
, p E @,
w , m,
fL
< 1,
# 0.
T h e spectrum of J 3 is the set S = {-LO + n: n a nonnegative integer}. For each of the above classes the representation space V has a basis {fn,),m E S , such that JYm =
mfm
J’fm
9
Efm
= /+frn+l>
= pfm J-fm
Qfm
(m
1
= pfrn+z
+
(9-6)
Wlfm-1
for all m E S on the left-hand sides of these equations. All of the irreducible representations in classes ( 1 ) and (2) satisfy property A and no two of them are isomorphic.
9. SOME GENERALIZATIONS
302
T h e proof of this theorem is elementary and is left to the reader. We could of course find many other representations of X 5 ,but for purposes of illustration, the representations listed in Theorem 9. I will suffice.
9-4
The Representation K(w, m, , p)
We will find a realization of R’(w, m, , p ) given by some T‘ E @(a,) acting on a vector space Vi of analytic functions of z. According to the theory of Section 8-2, the space of cohomology classes of a ( E l ) is 2-dimensional. Moreover, every element of @(al) is cohomologous to a realization T’ of the form
T w o realizations of the form (9.7) are cohomologous if and only if they are identical. T o construct a realization of R’(w, m, , p ) in terms of the operators (9.7) let Yi be the space of all finite linear combinations of the functions &(z) = z m + w , defined for all m e S where S is the spectrum of the representation, and let c1 = - w , c2 = p. Then
for all m E S. Comparison of the3e expressions with (9.6) yields a realization of R’(w, m, , p ) in terms of linear differential operators. w is not an integer the functions fA(z)= zm+ware not Since m analytic and single-valued in a neighborhood of z = 0. This is very annoying for computational purposes. T o remedy the defect we map the space Yi onto the space “y; = cp-’(V;)where cp-l[ f ’ ] ( z ) = (p(z))-lf‘(z) E Yl , f ’ E V i , and p(z) = Then Yl has a basis of the form fm(z)= cp-“ fT3(,z)= zk where K = m - m, is an integer. cp induces a transformation T = ( p - 1 ~ ‘ mapping ~ a realization T‘ of .X5 by gd’s on Y ; into a realization T by gd’s on Yl ; see (8.3). Under this
+
Z~O+W.
9-4. THE REPRESENTATION R’(w, rn, , p)
transformation the operators (9.7) with c1 = into the operators
-w,
303
c2 = p are mapped
on Vl . T h e operators (9.9) and the basis functionsf,(z) = zk,m = m, + k, provide the desired realization of R‘(w, m,, p). (Note that with the exception of Q = pzz, these operators and basis functions are identical with those derived for a realization of the representation R ( w , m a ,p) of 3 ( 0 , 1) in Section 4-1.) Following our usual procedure we can extend this realization of R’(w, m a ,p) defined on Vl to a local multiplier representation of K5 on G l l , where GTl is the complex vector space of all functions of z analytic in some neighborhood of the point z = 1. Here Ul 3 “y; and Ul is invariant under the operators (9.9). T h e operators A(g), g = g(q, a , b, c , T) E K5 , defining the multiplier representation can be computed i n . a straightforward manner from expressions (9.5) and (9.9). T h e result is
we can show that every function h E yl C Ul has a unique Laurent expansion
which converges absolutely for all 1 z I > d, where 1 > dh 3 0. (Tl is invariant under A ( g ) , see Section 2-2.) Thus, the functions f,(z) = hk(z)= zk, m = m, + k, in Vl form an analytic basis for T h e matrix elements A , , ( g ) of the operators A ( g ) with respect to this basis are defined by
vl.
m
[A(g)htJ(z)
C
A&)hi(Z),
k
= 0 , fl, f2,-,
l=--m
(9.11)
9. SOME GENERALIZATIONS
304
Since the operators A(g) define a local multiplier representation of K5we obtain the addition theorem 5
Azdgigz)
=
C
I, k
Azi(gi)Ajlc(gB),
= 0,
i l , III~,...,
(9.12)
j=-m
valid for g , , g, in a sufficiently small neighborhood of the identity element. T o be precise we should determine the exact domain of validity of (9.12), just as in Section 4-1. However, it-will soon be evident that both sides of this equation are entire analytic functions of the group parameters of g , and g, . Thus, by analytic continuation equation (9.12) holds without restriction for all g, ,g , E K5 . T h e matrix elements A , , ( g ) can be evaluated directly from the generating function (9.11). We examine some special cases: (1) g ( 0 , a, b, c, T) ( q = 0 ) . I n this case (9.11) is identical with the generating function (4.8). Thus the matrix elements are proportional to the Laguerre functions,
+
where p = m, w is not an integer. ( 2 ) g(q, 0, b, 0,O). T h e generating function becomes exp[p(qz2
-+ bz)]zk =
m
Atk(g)z’. z=-5
Comparing this expression with the generating function (4.77) for the Hermite polynomials we find
Here, the matrix elements are entire functions of p and q. (3) g(q, 0, 0, c, 0). T h e generating function is
where R;+L(pq, c )
=
A,,(g). Thus, (9.15)
9-4. THE REPRESENTATION R’(w, m, , p)
305
where j ranges over all integral values such that the summand makes sense. There is no simple expression for the functions RP,in terms of functions of hypergeometric type. Various identities relating these matrix elements can be obtained from the addition theorem (9.12). (We have already derived identities for the matrix elements (9.13) in Section 4-1.) Thus, the relation g(q, 0, b, 0, 0 ) g(q’, 0, b‘, 0, 0 ) = g(4
+ 4’, 0, b + b’, 0, 0 )
implies (after some simplification) the identity (4
+4’Y2 n!
b
+ b‘ m=O
(9.16)
T h e relation
implies
There are many other identities contained in (9.12), but the reader can derive them for himself. I n analogy with Section 4-6 we can construct a realization of R‘(w, m, , p) by gd’s on a space of two complex variables where the action of -X, on this space is given by a member of @(fl1),(9.2). (The elements of a(&) when restricted to 9(0, 1) are identical with the type D’ operators.) From the theory of Section 8-2 it is easy to show that the space of is 2-dimensional and that every element of cohomology classes of a(&) this space is cohomologous to a realization of the form
Two realizations of this form are cohomologous if and only if they are identical. Note: T h e selection of a representative in each cohomology class given by (9.18) differs on the subalgebra 9 ( 0 , 1) from that chosen in Section 4-6. T h e representative selected here has been chosen for convenience in the computations to follow.
3 06
9. SOME GENERALIZATIONS
T o construct a realization of R’(w, m, , p ) using the operators (9.18) we must find nonzero functionsf,(x, y ) = Z,(x) emv such that Eqs. (9.6) are valid for all m E S = {m,+ n: n an integer}. I n terms of the functions Z,(x) these relations become (i) (ii)
(&d - P)z
m
= PZm+&43
d
-dx Z,(x) = ( m
+
W)
(9.19)
Zn-l(x),
Just as in Section 4-6 we can assume w = 0, p = 1 without any loss of generality for special function theory. Furthermore, a simple computation shows (i), (ii), and (iii) are compatible if and only if c = -1. Thus our problem reduces to finding a realization of R’(0, m, , 1) in terms of the operators (9.18) where c = -1. I n this case Eqs. (9.19) have the following linearly independent solutions for all m E S : (1) Z,(x) (2) Z,(x)
=
(-1)m-m
0
2-m/zH,(x/2/2)
=
(- l)m-mo exp(x2/4)D,(x),
+ 1 ) 2(m+l)/zH-,-1(42/2) = exp(x2/4)e--iw(m+1)/2T(m + 1) D-,-l(ix). =
exp(x2/2)e--i7i(m+1)/21‘(m
(9.20)
T h e H,(x) are Hermite functions defined in terms of parabolic cylinder functions by H,(x) = 2mIz exp x2/2D,(d/zx). When m is a positive integer, which is not the case here, H,(x) is a Hermite polynomial. T h e fact that expressions (9.20) satisfy the recursion relations (i) and (ii) follows easily from (4.66) and (4.67). Relation (iii) can be derived from (i) and (ii). These solutions are entire functions of x. Clearly, if the functions 2, , m E S , are given by either (1) or (2), then the functionsf,(x, y ) = Z,(x) emu form an analytic basis for a realization of the representation R’(0,m, , 1) of X, . As usual this representation of S, can be extended to a local multiplier representation of K , by operators T(g),g E K , , on the space 9of all functions analytic in a neighborhood of the point (xo, yo) = (0, 0). A straightforward computation using (9.5) and (9.18) yields
~ E F (9.21) ,
where t
=
ev.
9-4. THE REPRESENTATION R'(w, m, , p)
307
T h e matrix elements of the operators T(g) with respect to a basis fm(x, t ) = ZJx) tm satisfying (9.19) are given by the functions A,,(g), (9.11), where w = 0, p = 1. Thus we obtain the relations
Corresponding to the basis
for all integers k, this equation can be written in the form x2t29 - txb
-
bV/2
x
+ bt
-
ct-1
-
2ct q
For q = 0, (9.22) is identical with (4.70) so we will omit this case. If a = c = T = 0, q = -$, (9.22) reduces (after some simplification) to the identity
(9.23)
valid for all m E @ not an integer. (In the next section we will show that (9.23) is also valid for m a nonnegative integer.) If a = b = T = 0, q = - $, (9.22) simplifies to
m
where h:(c)
=
(-
d/2)-n I?;( -4, z/%)
has the generating function
9. SOME GENERALIZATIONS
308
Here m is not an integer. However, in the next section we will show that this equation remains true for m a nonnegative integer. Similarly, the second set of basis vectors (9.20) can be used to derive identities for the Hermite functions. For example the case a = c = T = 0, q = -4, yields the identity
a generalization of (9.16). By substituting appropriate choices for the group parameter into (9.22) the reader can derive additional identities obeyed by the Hermite functions.
9-5
The Representation
f’w.p
I n analogy with the procedure of the last section we look for a realization of the representation t’u,pgiven by some T E 6Y(El) acting on a vector space of analytic functions of z. A comparison of expressions (4.17) and (9.7) shows how to proceed. Namely, in (9.7) we set c1 = - w , c2 = p to obtain J3
=
d
--
dz
w,
d
J- = dz,
J+ = pz,
E
= p,
Q
= pz2.
(9.24)
Designate by V2the space of all finite linear combinations of the functions zk, k = 0, 1, 2, ..., and define the basis vectors fm of V2by fm(x) = h k ( 2 ) = zk where m = -w k, k 3 0. Clearly, hk(z) =
+
JY, = (--w J-f,
+ z dzd l
zk =
mj,,
J+L
= (pz) z k =
+
,
Efm = /+fin ,
d dz
= - zk = ( m
Qfm
=
-w)fm-l
(P’)zk = d m + z
c~fn+~,
(9.25)
-
‘These relations define a realization of Tb,, on Vz. Moreover, this realization can easily be extended to a local multiplier representation B of K5on the space 6Y2 consisting of entire functions of z. T h e operators B(g),g = g(p, a, b, c, T) E K 5 , defining this representation are easily computed: [ B ( g ) f ] ( z= ) exp[p(p2
+ bz + a )
-
-w7]f(eTz
+ eTc), f~ rY2.
(9.26)
309
9-5. THE REPRESENTATION
Furthermore it is straightforward to show that the operators B(g) form a global representation of K 5 .T h e matrix elements B,,(g) of B(g) with respect to the basis (f, = hk) are given by m
LB(g) h k l ( Z )
=
Blk(g) hl(z),
= 0,
1, &***,
l=O
or
T h e addition theorem for the matrix elements is (9.28)
valid for all g, ,g, E K5 . Corresponding to some special choices of the group parameters the matrix elements have the following explicit expressions: (1) g(0, a b, c, T). For q = 0, (9.27) becomes identical with the generating function (4.22). The matrix elements are thus proportional to associated Laguerre polynomials.
(2) g(q, 0, b, 0, 0). Exactly as in (9.14) we find
(3) g(q, 0, 0, c, 0). In this case (9.27) reduces to (9.30)
where R;-,(pq, c)
=
Blk(g).Thus, (9.31)
31 0
9. SOME GENERALIZATIONS
where j ranges over all integral values such that the summand is defined. Comparing (9.30) with the generating function (4.77) for the Hermite polynomials we find (9.32)
Just as in the last section, the addition theorem (9.28) can be used to derive identities obeyed by the matrix elements. This task will be left to the reader. we now Armed with a knowledge of the matrix elements of procede to construct a realization of this representation by gd’s on a space of two complex variables, where the action of X5is given by a member of a(&), (9.2). I n particular we will use the operators (9.18) with c = -p. T o construct a realization of using these operators it is necessary to find nonzero functions f,(x,y) = Z,(x) emu, m = - w , -a 1,..., such that Eqs. (9.6) are valid with
+
Just as in the derivation of Eqs. (9.19), it is easy to show that without loss of generality for special function theory we can assume w = 0, p = 1. Then, expressed in terms of the functions Zm(x),Eqs. (9.6) become the recursion relations
(9.33)
where m takes the values 0, 1, 2, ... . The functions Z,(x) are determined to within an arbitrary constant by (9.33). I n fact relation (ii) implies Z,(x) = c for some constant c. If we set c = 1, the remaining solutions Z,(x) are uniquely determined: Z,(x)
=
(-1)m
2-m/2Hm(x/z/Z)
=
(-1)m
exp(x2/4) Dm(x),
(9.34)
9-5. T H E REPRESENTATION
f'u,p
311
where the H,(x) are Hermite polynomials. Thus the vectors f , ( x , y ) = Z,(x) emy, m = 0, 1, 2 ,..., where Z,(x) is given by (9.34), form a basis for a realization of the representation of Z5. This infinitesimal representation of .X5can be extended to a local multiplier representation of K5 by operators T(g),g E K 5 ,on the space 9of all functions analytic in a neighborhood of the point (9, y o ) = (0,o). Indeed we have already computed the operators T(g) on 9, Eq. (9.21). T h e matrix elements of T(g) with respect to an analytic basis (fm(x, y ) ) satisfying (9.33) are the functions B,,(g) given by (9.27) ( w = 0, p = 1). Consequently,
c BZk(g)fZ(x,y), ' m
[ T ( g ) f k l ( x , y== )
=
O, 2 , * * * ~
1=0
or
(9.35)
0 this expression is identical with (4.76). For a
= c = T = 0, it reduces to (9.23) where now m is a nonnegative integer. I n the special case n = 0, (9.23) becomes
For q
q
=
=
-B,
(9.36)
which is Mehler's theorem, (4.158). Finally, if a = b = T = 0, q = -*, (9.35) reduces to the identity
W
=
C h$-,(c) H,(x) tZ-'-,
(9.37)
1=0
where hF-E--k(c)= (- 2/Z)k-zRf-k(-4, 4 Z c ) is given by the generating function m
exp(--z2/4)(z
-
2c)k
=
1 hl'_k(c)
21.
1=0
31 2
9. SOME GENERALIZATIONS
This completes our analysis of the special function theory of X 5 . Note that the identities for Hermite functions derived here have been obtained from a study of the operator realizations a1 and P I , (8.28) and (9.2). We could also have used p5 , but this would not have led to any new results. T h e operators p 2 , p 3 , p4, on the other hand, do not yield but rather lead realizations of the representations R’(w, m, , p ) or T;+, to realizations of representations of X5which we have not classified here. I n particular, p2 can be used to realize certain reducible representations in such a way that the basis vectors turn out to be proportional t o generalized Laguerre functions. Th u s a study of p2 leads to new identities for Laguerre functions.
9-6 The Lie Algebra 5??p,q Corresponding to a pair of positive integers ( p , q) let gPrq be the f + , 3- and commucomplex 3-dimensional Lie algebra with basis tation relations
x3,
is isomorphic to Y3. I n addition the following isomorphisms Clearly, are easily established: Lemma 9.1
for all positive integers gP,*E 5??q,P; 9’np,np gP,*;
P, 9, n. According to this lemma every Lie algebra 5??p,,,, is isomorphic to a Lie algebra gP,*such that (1) p and q are relatively prime positive integers; (2) p is odd; and (3) if q is odd then p > q. Consequently, from now on the pair ( p , q) will be assumed to satisfy properties (1)-(3). It is , 5??P‘,q’ with subscripts satisfying obvious that two Lie algebras gp,, these properties, are isomorphic if and only if p = p‘ and q = 4’. Applying the techniques developed in Chapter 8 we can determine all by gd’s in one or two of the transitive effective realizations of Yp,, complex variables. Thus, from (8.26) there follows the realization 71
:
a -
aY ’
epy,
e-qy;
r
=
3, k
=
2, s
= 0.
(9.39)
Every transitive effective realization of gp,,by gd’s in one complex variable is an element of OZ(ql). Similarly, by making use of Lie’s tables (Lie [I], Vol. 111, pp. 71-73),
9-7.THE LIE GROUP G,,Q
31 3
it can be shown that every transitive effective realization of 9p,q by gd’s in two complex variables is an element of 0l(sj), j = 1, 2, 3, where r=3, 6, :
--PZ,
a a az, + 4% az, ’
a -
Y
az, ’
=
3. k
r=3,
9-7
k=l, s=O; =
1,
k = l ,
s = 0;
(9.40)
s=O.
The Lie Group Gp,q
Corresponding to a pair of relatively prime positive integers ( p , q) such that p is odd, denote by Gp,Qthe 3-dimensional complex Lie group with elements
g ( h c, 7)’
b, c,
E
I%,
and group multiplication
It is easy to check that the multiplication is associative. Furthermore, e = g(0, 0, 0) is the identity element and g(--e-PTb, -@c, -T) is the unique inverse of the group element g(b, c, T). GP,*has a 4 x 4 matrix realization
(9.42)
where matrix multiplication corresponds to the group operation. Comparing (9.42) with (1.35) we see that Gl,l is isomorphic to T , . T h e Lie algebra of G,,* can be computed from either of the expressions (9.41) or (9.42), and is easily recognized to be isomorphic to Y P , ,. Indeed we can make the identification
where f * ,
y 3generate 9fp,qand satisfy the commutation relations (9.38).
31 4
9-8
9. SOME GENERALIZATIONS
Representations of gp,q
Let p be a representation of g’p,q on an abstract vector space V and let = J*, p(Y3)= J3, be operators on V . Clearly,
p(f*)
Note that the operator C p * q = (J+)*(J-)” = (J-)”( J+)q commutes , on V . For p irreducible we would with all operators p(01), 01 E gP,, expect CP,* to be a multiple of I. I n strict analogy with our study of the representation theory of F3 we will classify all representations p satisfying the properties: (i) p is irreducible (ii) Each eigenvalue of J 3 has multiplicity equal to one. There is a countable basis for V consisting of eigenvectors of J3.
(9.45)
We give without proof the results of the straightforward classification. satisfying (9.45) and Theorem 9.2 Every representation p of gP,, 0, on V , is isomorphic to a representation of the form for which Cp’q QP,*(w,m,) defined for w , m, E @ such that w # 0 and 0 Rc m, < 1. T h e spectrum of J3 corresponding to QP,*(w,m,) is given by 5’ = (m, + n: n an integer) and there is a basis i f m } , m E S , f0r.V such that
+
<
J+fm
= mfm
J3fm
= wfm+v,
(9.46)
J-fm
= wfm-,
,
C”.ym= (I+)*( J - ) ” f i n
.
= wp+,fm
QpPQ(w‘, mi) if and only if There exist isomorphisms QP,,(w, m,) m, = mi , w = w‘d, where d is a ( p g)th root of unity. I n accordance with the usual procedure, we can use the operators to construct a realization of the representation QP,*(m,m,). This can be done by introducing the change of variable z = e” in (9.39) and considering the operators
+
J3
=z
d
+ c1 ,
J+
=C~ZP,
J-
= c~z-‘J
(9.47)
in Ol(fl). Thus, let ”y; be the complex vector space of all finite linear combinations of the functions h,(x) = zn, n = 0, * l, &2, ..., and define operators J*, J 3 on rY; by (9.47) where c1 = m, , c2 = w . Define
9-8. REPRESENTATIONS OF gP,@
the basis vectorsf, of Vl byf,(z) runs over the integers. Then
J+fm
= ( u z p ) ~ "= wfm+n
1
=
h,(z) where m
31 5 =
m,
+ n and n
= ( w ~ - Z" ~ )= wfm-,
J-fm
,
(9.48)
c.ym = w ~ + " f m . . These equations agree with (9.46) and yield a realization of Qp,p(w,m,). T h e differential operators (9.47) (cl = m, , c2 = w ) induce a multiplier representation A of G,,* on the space GZl consisting of those functions f ( z ) which are analytic and single-valued for all x f 0. This multiplier representation is defined by operators A(g),g = g(6, c, T ) E Gp,* : [A(g)f](z) = exp[w(bzP
+ cz-@)+ r n o ~ ] f ( e T z ) ,
f~ & .
(9.49)
Clearly, GZl is invariant under the operators A(g) and the group property A(g1g2) = A(g,) A(g2) is valid for all g, g, E G,,* . T h e matrix elements A,,(g) of A(g) with respect to the analytic basis {f, = h,) of GZl are defined by 9
c m
[A(g) h k l b )
=
M g ) hdz),
g E GD,*>
k
= 0 , il, *2,...,
(9.50)
I=-m
or exp[w(bzP
+ c x q ) + (m,+ k )
a,
T]
zk =
A,,(g) zz
(9.51)
I=--m
where g
= g(6, c,
T).
Explicitly, (9.52)
(9.53)
T h e nonnegative integers sl ,n l are uniquely determined by the properties:
(1) 1 = n,p - siq, (2) If I = zip - sip where n ; , s; are nonnegative integers, then n;
+ si 3 n, +
s,.
Since p and q are relatively prime, the integers n, , s l can easily be shown to exist for all 1. For example, if p = q = 1 then s, = 0, n, = I for I 3 0 and s, = -1, n, = 0 for 1 < 0.
31 6
9. SOME GENERALIZATIONS
If bc # 0 we can introduce new group parameters r, v defined by b = r v p / ( p + p), c = r v - Q / ( p 9). I n terms of these new parameters the matrix elements are
+
(9.54)
(9.55)
It follows from the ratio test that 17,q(r)is an entire function of r for all integers 1. Here I;,’(r) = (-i)zJz(ir), 13 0, is the ordinary “modified Bessel function” (see ErdClyi et al. [l], Vol. 11). T h u s we can consider the functions If’*(r) to be a group-theoretic generalization of Bessel functions. Substitute (9.54) into (9.51) to obtain the simple generating function Y
(z”
+ z-.)I
f
=
IpqY)
2.
(9.56)
Z=-m
T h e group property of the operators A(g) implies the addition theorem
c Azj(g1) A&!Z), m
Azk(glg2) =
(9.57)
j=-W
valid for all g, ,g,
E
Gp,q. This leads immediately to the identity m
FP.*(bl
+ b, , + cZ) = 2 F;:q(b, ~1
, c ~ ) F ; ” (, c2). ~~
(9.58)
j=-m
We could also use the addition theorem to derive identities involving the functions Iy3q(r).However, this will not be done here as most of the results so obtained are special cases of identities which will be derived in Section 9-10.
9-9
Recursion Relations for the Matrix Elements
Denote by GZ(Gp,J the space consisting of all entire functions on the group Gp,*, i.e., of all entire functions of the parameters b, c, T. Clearly, the matrix elements A,,(g) are elements of 6Y(Gp,J. T h e representation P of Gp,qon LT(Gp,J defined by [P(g’)fl(g) = f(gg’),
f
.
U(GP,*)Y
(9.59)
317
9-9. RECURSION RELATIONS FOR MATRIX ELEMENTS for all g, g‘ E Gp,qsatisfies the relation
and determines Gp,qas a transformation group. According to (9.57) the action of P on a matrix element Ajk is given by
c m
[P(g‘)Ajkl(g) = Ajk(gg’) =
Adg’) M g ) .
(9.60)
l=-m
Comparing this expression with (9.50) we find that for fixed j the functions (Aik},K = 0, f l , f 2 , ..., form a basis for a realization of the representation Qp,,(w, m,) of Gp,qunder the action of the operators P(g).Therefore, the Lie derivatives J*, J 3 defined on a ( G p , Jby
(9.61)
JYk) =
d
I
[ P ( ~ x~$~)f’l(g) P
.r=O
f~ @(Gp,Q),
satisfy the relations J3A9&)
=
(%
4- k) A,&), J-Ajdg)
(J’)“ (J-)” A,,(g)
CP*QAJk(g)=
=
=
J+Aidg)
WAili+pk),
“A,,-&),
wYfyAjk(g)>
(9.62)
i,k
=
0, & I >
*’>***
*
Equations (9.62) yield recursion relations and differential equations for the matrix elements A j , . We will use these equations to derive recursion relations for the generalized Bessel functions I ~ , * ( I Thus, ). we assign to a group element g = g(b, c, T ) , bc # 0, the local coordinates [ I , u, T], where b = rvp/(p q), c = ~v-q/i(p 9). Then if g‘ = g(b‘, c’, T‘), a straightforward computation shows the local coordinates of gg‘ are
+
[Y(
1
v(1
+ ( p + q)
r-lv-PeW)q/(P+y)
+
(1 + ( p f q ) y-laG‘e-YTC’
+ ( p + q ) r-lv-PePTb’)ll(P+y)(1
i
Y/(P+Y) 1
7
( p + q ) Y-~vQ~-RTc’)~/(P+*), T + T’]
for 1 b’ 1, 1 c‘ I sufficiently small. It follows from the definition (9.61) that the Lie derivatives J*, J 3 are
(9.63)
31 8
9. SOME GENERALIZATIONS
Substituting (9.54) and (9.63) into (9.62) and factoring out the dependence on v and T, we obtain the equations
= I$*(r),
m
=
0, i l , *2
,... .
(9.64)
I n particular, the functions Ig*((r)are solutions of an ordinary differential q. equation of order p
+
9-10
Realizations in Two Variables
T h e operators 6, and 6, , (9.40) can be used to construct realizations of
Qp,,(w, m,) on spaces of functions of two complex variables. However, the basis functions so obtained are elementary and of little interest. On the other hand a realization of Qp,q(w,m,) by 6, does lead to new results. T o see this we introduce new variables x , y , where z1= xePY/(p q), in the expression for 6, :
z2 = xe-""/(p
+
+ q),
T o construct a realization of Qp,q(w,m,) using the operators (9.65) we must find nonzero functions &(x, y ) = Z,(x) e c r n y ; m = m, n, n = 0, f l , 5 2 ,..., such that expressions (9.46) are valid. These expressions are equivalent to the equations
+
d
) ... (q dx X
dx
m-qp X
)(p
d
m-(p-l)q
z
+
X
1
(9.67)
where without loss of generality for special function theory, we have set w = 1. Thus, the functions Z,,L(x)are solutions of differential equations of order p + q.
9-1 0. REALIZATIONS IN TWO VARIABLES
319
I n general it is possible to findp + q linearly independent solutions of Eqs. (9.66) and (9.67). However, we will be content with a single solution. It is a consequence of (9.64) that if m, = 0, the functions Z,(x) = I ? ) ~ ( Qare ( x )solutions of these equations. Following a standard procedure in the theory of special functions we shall use analytic ( Xm) an arbitrary complex number and continuation to define I ; L ~for show that the functions Z,(x) = Z?)$(x) satisfy the required relations (9.66) and (9.67) even if m, f 0. An application of the Cauchy residue theorem to (9.56) yields 27riI:q(r)
=
s,
z-”-l exp
[ z (z” + 2-91 dz, P+4 m
= 0,f l ,
*2 ,...,
where C is a simple closed contour surrounding the origin in the z-plane. Without changing the value of the integral, the contour C can be deformed into a loop D which starts at infinity on the negative real z-axis, encircles the origin counterclockwise and returns to its starting point. ( T he proof of this statement is almost word for word the same as the well-known proof of the analogous statement in the special case p = q = 1, see Whittaker and Watson [I], Chapter 17). Since p is odd, the integrand is bounded on D for Re Y > 0. Thus, we have
s‘”’
where --m denotes integration along D. Since the integrand is now single valued on D even for m complex, we can use (9.68) to define the function IPg(r) for arbitrary values of m and Re Y > 0. Moreover, by differentiating under the integral sign in (9.68) we can verify that Eqs. (9.64) remain valid for arbitrary m. T h u s the functions Z,,(x) = IP;;(x), m = m, n, define a realization of the representation Q p , q ( m o , 1) of %I, ., We can obtain more information about the generalized Bessel functions *;:Z by mimicking the standard treatment of ordinary Bessel functions in terms of contour integrals as given, for example, by Whittaker and Watson [I], Chapter 17. If r > 0, introduction of the new variable w = ( ~ / ( p+ q ) ) l / p z in (9.68) leads to the expression
+
T h e right-hand side of this equation can be analytically continued to define I:iq(y) as an analytic, but not single valued, function of r for all
320
9. SOME GENERALIZATIONS
r # 0. I n particular r-m/pI$g(r) is an entire function of r ( P + q ) / P . T o obtain a series expansion for Zgp(r) we use
in (9.69) and integrate term by term. From the theory of the gamma function (Whittaker and Watson [l], Chapter 12),
Thus,
For p = 1 this expression simplifies to
We could construct additional solutions of equations (9.64), linearly independent of the functions IEQ(r),by choosing a different path of integration in (9.68) (see Khriptun [I]). However, for purposes of illustration we will be content with the solutions ZZ*(r).
9-1 I
Generating Functions for Generalized Bessel Functions
+
As shown above, the functions fm(x, y ) = I%(.) e c r n y , m = m, n, n an integer, form a basis for a realization of the representationQ,,q( 1, m,) of gP,,. Following our usual procedure we will extend this representation to a local multiplier representation of Gp,q in the space F of functions analytic i n ' a neighborhood of (xo,y o ) = (1, 0). T h e operators T(g),g = g(b, c , T ) E Gp,p defining this multiplier representation can easily be computed from (9.65) and (9.43). T h e result is
fE F ,
(9.72)
9-11. GENERATING FUNCTIONS
321
+
+
valid for j b(p q)/xtp I < 1, 1 c(p q) tq/x I < 1, where t = e”. The matrix elements of T(g) with respect to the analytic basis (fmfx, t ) = P Z ( x ) t-”), are the functions A,,(g) given by (9.52) and (9.54). Thus, m
[T(g)frn,,+kI(x,t )
C
=
= 0,
Azk(g)fm,+Z(X, t ) ,
i l , +2,-*.,
Z=-CC
or b(p m
+ q))Q:(P+Q)
(1
+
c(p
+ q) tQ)n.(P+Q)] X
c F!;Q(b, m
=
c ) I2$x)
tl,
Z=--a,
I b(p for all m
E
$?.If c “ : 1
[X
+ q ) W I < 1,
=
0, t
(1
+ y-b(p )Qf + q )
=
I .(p
t q) tQ/X I
< 1,
(9.73)
1, this expression simplifies to / ( P + d ](1
+ b(p + q )
)rn/(P+Q)
X
while for b
=
0, t
I:.[+
=
1 it becomes
+
c(p
+ ,I))”’(”+Q)] (
+
c(p
+ q))-”L;(”+Q) X
+
If b = c = ri(p Q ) # 0 the matrix elements can be expressed in terms of generalized Bessel functions and (9.73) becomes y
e Q I X ( l
Q/(P+Q)
+,t”)
) (1, +
rtg
pl(P+g)
I(
x
+ rt-”
x f rtQ
~/(P+Q!
1
For p = Q = 1, Eq. (9.73) is equivalent to the Bessel function identity (3.29).
322
9. SOME GENERALIZATIONS
Similarly, in analogy with Weisner’s treatment of Bessel functions as given in Section 3-4, we can use the operators
expressions (9.65), to derive generating functions which are not related to the matrix elements A,,(g). T h e basic observation to be made is: If f ( x , t ) is a solution of the equation C p , v = ( J + ) q ( J - ) p f = wf then the function T(g)f given by the formal expression (9.72) is also a solution of this equation. Moreover, iff is a solution of the equation (xIJ+
+ xzJ- + x,J’)lf(x,
t ) = xf(x, t )
for constants x1 , x2 , x3 , A, then T(g)f is a solution of
As an illustration of these remarks, consider the function f ( x , t ) = I z * ( x )tm, m E @. In this case C p , q f = f, JY = -mf, so the function T(g)f satisfies
where g = g(b, c, T) E G p , q .If c in the form
=T =
0, b
Since x - ” / p I E , q ( x ) is an entire function of expansion in t P about t = 0:
c-
=
1, T(g)f can be written
x(p+q)lp,
m
h(x, t )
=
n=-m
h,(x) tn”,
I X t Y I
h has a Laurent
+ q.
323
9-11. GENERATING FUNCTIONS
Substituting this expansion into the first equation (9.74), we find Z,,(x) = h-,(x) is a solution of the generalized Bessel equation (9.67) for n = 0, 4 1 , 4 2 , ... . At this point we use the fact: For m an integer the only solutions of (9.67) which are regular at x = 0 are cI?:(x), c a constant. This statement can be verified by computing the indicia1 equation of (9.67) (see Ince [I], Chapter 7). Since h(x, t ) is bounded for x = 0, we have h,(x) = c,I&'(x), c, E @. Hence, m
h(x, t )
=
c
C " l g ( x ) t".
n=-m
+
T h e equation (-p J+ J3) h(x, t ) = -mh(x, t ) implies = ( m - np) c, . T h e constant co can be determined by setting x = 0 in h(x, t ) and using (9.70). Consequently, co =
T ( - m / p ) sin[r(m Pr
+ 1)1 ,
cn =
(-1)" T[(.P
-
mYP1 sin[r(m
+ 111
Pn
and we obtain the identity
139"
When
p
=
Ic p
+ q.
q = 1 this expression is equivalent to (3.38).
(9.76)