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Certain classes of polynomial expansions and multiplication formulas
MATHEMATICAL AND COMPUTER MODELLING PERGAMON
Mathematical and Computer Modelling 37 (2003) 135-154 www.elsevier.com/locate/mcm
Certain
Classes of Polynomial Expansions and Multiplication Formulas KUNG-YU CHEN AND CHUAN-JEN CHYAN Department of Mathematics, Tamkang University Tamsui 25137, Taiwan, R.O.C.~math. t k a . edu. tw
H. M . S R I V A S T A V A Department of Mathematics and Statistics,University of Victoria Victoria, British Columbia V 8 W 3P4, Canada harimsriQmath, uric. ca
(Received May 2002; accepted June 2002) Abstract--The authors first present a class of expansions in a series of Bernoulli polynomials and then show how this general result can be applied to yield various (known or new) polynomial expansions. The corresponding expansion problem involving the Euler polynomials is then considered in an analogous manner. Several general multiplication formulas, involving (for example) certain families of generalized hypergeometric polynomials, are also investigated in the context especially of the classical Jacobi, Laguerre, and other related orthogonal polynomials. (~) 2003 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - P o l y n o m i a l expansions, Multiplication formulas, Bernoulli and Euler polynomials, Generalized hypergeometric polynomials, Generating functions, Dixon's summation theorem, Jacobi polynomials, Laguerre polynomials, Generalized Bessel polynomials, Hermite polynomials, Gegenbauer (or ultraspherical) polynomials, Generalized Rice polynomials, Chu-Vandermonde theorem, Legendre (or spherical) polynomials, Chebyshev polynomials.
1. I N T R O D U C T I O N
AND
PRELIMINARIES
For t h e g e n e r a l i z e d B e r n o u l l i p o l y n o m i a l s B (a) (x) ( a c C) defined by t h e g e n e r a t i n g f u n c t i o n
(cf., e.g., [1, p. 61, equation 1.6 (19)])
(
z
) ~ e xz
~ z~ = E BP)(x) ~.v
({z I < 27r; 1 a : = 1),
(1.1)
n=0
it is easily observed that
°+') (x + y) =
B(o) (x)
(y)
(1.2)
The present investigation was completed during the third-named author's visits to Tamka~g University at Tamsui in April 2002 and May 2002. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. 0895-7177/03/$ - see front matter © 2003 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(02)00328-X
Typeset by ~4A4S-TEX
136
K.-Y. CHENet al.
and
B~(~) (~ + 1)
B(~°) (~) . . . .
~_~
(x)
(n e No := {0,1,2,...}).
(1,3)
Furthermore, since B (°) (x)
=
(n ~ No),
x ~
(1.4)
upon setting/3 = 0 in (1.2), we obtain B(na) (x + y) =
k Bk
(1.5)
(x)
k=O
Now, by combining (1.3) and (1.5) (with y = 1), we find that
B(~"-I) (~) =
k=O
n] B (a) (x), k! (n -)¢ + 1)!
(1.6)
which, in the special case when a = 1, immediately yields the following expansion (cf., e.g., [2, p. 302, Exercise 1]): n[ xn = k=0 Z k! (n --k + 1)! Bk (x) (1.7) in series of the Bernoulli polynomials Bn(x) defined by Sn (x) := B(~1) (x)
(n e N0),
(1.8)
that is, by means of the generating function [3, Volume I, p. 36, equation 1.13 (2)] zeXZ
oo
ez - 1 = E
zn
Bn (x) ~.f
(Izl < 27r).
(1.9)
n=O
Motivated essentially by several recent works on the subject of polynomial expansions and multiplication formulas by (for example) Popov ([4,5]; see also [6,7]), we first apply (1.7) in order to develop a general class of polynomial expansions in series of the Bernoulli polynomials and investigate the corresponding expansion problem involving the Euler polynomials as well. We then consider several general multiplication formulas which are associated with (for example) certain families of generalized hypergeometric polynomials. We also indicate relevant connections of the polynomiM expansions and multiplication formulas, which are presented in this paper, with the corresponding (known or new) results involving many of the classical orthogonal polynomials.
2. C L A S S E S O F P O L Y N O M I A L EXPANSIONS INVOLVING BERNOULLI AND EULER POLYNOMIALS oo In terms of a suitably bounded double sequence {Am,n}m,n=O of essentially arbitrary (real or complex) parameters, we define the polynomials S g ( x ) by (cf. [8]; see also [9, p. 145, equation (3.1) with m = 0]) [~/gl n ) g k A ~,kX k (2.1) s~:N (~):= ~ ( - -~. k=O
(n e N o ; g e N := No \ {0}), where, as usual, [a] denotes the largest integer in a E R and (A)k is the Pochhammer symbol (or, more precisely, the shifted factorial, since (1)k = k!(k C No)) defined, in terms of Gamma functions, by F ( A + k ) _ [ 1,
(•)k:=
F(A)
~.A(;~+l)...(~+k-1),
( k = 0 ; A#.0),
(keN~eC).
(2.2)
Polynomial Expansions and Multiplication Formulas
137
By making use of the definition (2.1) in conjunction with the polynomial expansion (1.7), we have tn
$ ( g ; x , t ) : = E SNn (wX)-~. n=O oo tn [ ~ ]
(_n)Nj ~
J J
j=0
n=O
= ~
(-1)NJ
=
(--1)NJn!An+NjJwJtn+gJ'
n,j=o
n!J-------~"An+Nj'JJxJtn+NJ J sk (z) E k! ( j ~ k + 1)!
n,j=O =
~
(2.3)
k=0
(--1)N(j+k)
(X) o3J+kJ6n+N¢j+k)
n,j,k=o n! (j + 1)!k! An+N(j+k)d+kBk provided that the inversion of the order of summation is justified by absolute convergence of the series involved• If we now let n , n - N j and n, , n - Nk, successively, we find from (2.3) that [n/N]
S(N;x,t)=
\
9=0
n=0
tn E
k=0
(J + 1)!
(_l)Nk
(n-Nk)!k[
)
An3+kw3+k
(2.4) Bk (z).
Finally, upon equating the coefficients of t n from both sides of (2.4), we obtain the following general class of polynomial expansions in series of the Bernoulli polynomials:
In~N] ( n ) (Nk)! S N (cox) = E (--1)Nk N k k! k=0
(2.5) \
j=o
(J + 1)!
An3+kw J+k
Bk (x)
(n6N0; N6N). Next, in order to apply the general result (2.5) to a certain family of generalized hypergeometric polynomials, we set P
I1 (~J)k An,k = N -Nkj=l q
(A (L;An + #)) k
I] (~J)k j=l
(2.6)
(N, L E N ; n,k,p, qENo; .k,#,aj 6 C ( j = 1,...,p); /39 e C\Z o (j = 1,...,q; Z o := { 0 , - 1 , - 2 , . . . })), where A(N; A) abbreviates the array of N parameters A+N-I
A A+I
N'
N
''"'
N
(N6N)
K.-Y. CHEN et al.
138
and an empty product is interpreted (as usual) to be 1. Thus, for the generalized hypergeometric polynomials
.p(),,t,) ' n,N,L (x;p,q)
N+L+pFq
:=
[ A ( N ; - n ) ' A ( L ; A n + #)'c~I'''''O~p; ] X
(2.7) ,
Zl,. • ,Zq;
we find from the polynomial expansion (2.5) that
[n/N] ,D()~,t,) ( n ) (Nk)' • n,N,L(wx;p'q) = E (--Y)-lVk Nk k! k=O P
l-I (~5)k
5=1
(A (n; An + #))k Bk (x) w k
(2.8)
l-I (;35)~
j=l
A ( N ; - n + Nk),A(L;An + # + Lk),o1 + k,...,ap + k, 1;w-I • N+L+p+IFq+I
I
f~l+k, ..,f~q+k, 2; ] provided that each member of (2.8) exists. From among the classical orthogonal polynomials (see, for details, [10]), the Jacobi polynomials P(~'Z)(x), the Laguerre polynomials L(a)(x), and the Hermite polynomials Hn(x) can easily be shown to correspond (for example) to the special cases: pn(i,a+~+i) ( 1 - x
,1,1
•(0,tt) n,l,1 (X;0,2)
and
)
T;0,1
(Zl = a + i ) ,
(~1 ~-- O~-~ 1; ~2 = It)
p(o#,)( 1 ) ,2,1 -~-~; 0, 1
(~1
---- ].t),
respectively. Under these special cases, the hypergeometric polynomial expansion (2.8) readily yields p(a,f0 (1 - 2wx) =
.
+ _
(o~+ j3 + n + 1)k B k (X)
(2.9)
k=O
.3F2 (-n + k,o~ + fl + n + k + l,1;o~ + k + l,2;w) , L(a)(wx):~=o~(:+k)
a B k(x)
(2.10)
.2F2(-n + k, 1;a + k + l,2;w), and Hn ~
[,~/2] = x -cl/2)n E (--1)k 2k k=O
(2.11)
:3Fl ( A (2;-n T 2 k ) , l ; 2 ; - - ~ ) for the Jacobi, Laguerre, and Hermite polynomials, respectively. Obviously, since [10, p. 103, equation (5.3.4)]
L(a)(x)= lim ~ P ( a ' ~ ' ( 1 - 2 - ~ x ~ , If~l-*o¢L k P/)
(2.12)
Polynomial Expansions and Multiplication Formulas
139
the polynomial expansion (2.10) is a limit case of (2.9) when 03
w*
, --
and
IZ~l ~
~.
The polynomial expansion (2.10) (with w = 1) was given recently by Popov [5, p. 17, equation (7)] (see also [1, p. 69, Problem 15]). For the generalized Bessel polynomials yn(x; a, b) defined by (cf. [11,12]; see also [13, p. 75,
equation 1.9 (1)])
Yn (x;a,b):= ~ (nk)(a +n+k-k
2)k, (b) k
k=o
(2.13)
=2 Fo (-n,a + n - l;
;-b) '
a special case of the expansion formula (2.8). when N = L = 1,
p = q = 0,
and
x,
--
X /2
would yield the polynomial expansion Yn(Wx;(£-l)n+#+l,v)=~(k)(£n+#)kBk(x)(_w)
k
(2.14)
k=0
k, An+# + k, 1;2;w),
• 3F1 ( - n + which, for x*
,-~x
and
w~-~--
03
/2
assumes the form
y~ (wx; (;~ -
1) n + # + 1, u) =
(An + #)k Bk (x)
(2.15)
k=0
.3Fl (-n + k, An + # + k,1;2;-w) . In their
further
special cases when A = w = 1,
# = a-
1,
and
~ = b,
these last results (2.14) and (2.15) reduce at once to the forms
yn(x;a,b)=~-~(-1)k(nk)(a+n-1)kBk(-b) k=O
•3 F l ( - n + k , a + n + k
-
(2.16)
1,1;2;1)
and
(2.17)
k=0
.3Fl(-n+k,a+n+k-l,1;2;-b), respectively. The polynomial expansions (2.16) and (2.17) happen to be the by Popov [4, p. 10].
main
results in another paper
140
K.-Y. CItEN et al.
We remark in passing that, by employing a slightly different notation, the generalized Bessel polynomials
(
x)
y n ( x ; ( a - 1 ) n + ~ + 2 , 2 ) : = 2F0 -n, a n + ~ + l ; =:
;--~
(2.18)
Z e c)
were considered by Khan and Ahmad [14] as a further generalization of the Bessel polynomials. While each of the polynomial expansions (2.14) and (2.15) can easily be restated in the Y-notation of (2.18), such families of generalized Bessel polynomials as those occurring in (2.14), (2.15), and (2.18) were actually investigated, over two decades ago, by Srivastava [15] in the context of generating functions (see also Srivastava and Manocha [13, p. 92, Problem 18; pp. 386-388, Problems 9 and 10]). We turn our attention now to the expansion problem, corresponding to (2.5) and (2.8), in series of the Euler polynomials En (x) defined by means of the generating function [3, Volume I, p. 40, equation 1.14 (2)] oo Zn 2eXZ - Z E , (x) (Izl < ~) (2.19)
eZ+l
~.
rt~O
•
The following relationships between the Bernoulli and Euler polynomials follow easily from the definitions (1.9) and (2.19):
(2.20)
2
(n E No).
For the Euler polynomials En (x), it is known also that (cf., e.g., [1, pp. 63-64]) E,~ (x + 1) + En (x) = 2x n
(n c N0)
(2.21)
and k=O
Obviously, therefore, by combining (2.21) and (2.22) (with y = 1), we obtain the following analogue of (1.7) for the Euler polynomials:
=5
En(z)+
Ek(z)
(2.23)
.
k=0
Making use of (2.23) in place of (1.7), our derivation of the polynomial expansion (2.5) can now be applied mutatis mutandis in order to obtain the following general class of polynomial expansions in series of the Euler polynomials:
1 [n/N]
SN ( W X ) = 5 ~
(--l)Nk
( n ) (Nk)! Yk k[
k=O
• (A,~,k+
Nk)gj An,j+kJ.) wkEk (x) j!
(2.24)
[ ( n - N k ) / g ] ( - - n "4-
Z j=O
(n~No; N~N).
Polynomial Expansions and Multiplication Formulas
141
Fhrthermore, for the generalized hypergeometric polynomials defined by (2.7), we find from (2.24) with the coefficients An,k chosen as in (2.6) that ~,(),,~) 1 r'n'N'L(wx;p'q)=7 E
(--N)-Nk
n Nk
k!
k=O
P 1-I (<~J)k j=l -q (A (L; An + U))k Ek (x) wk
(2.25)
17I (ZJ)k
j=l
.(
FqpN+L
[A(N;-n+Nk),A(L;An+#+Lk),al+k .... ,ap+k; co , 13a + k,...,/3q + k;
provided that each member of (2.25) exists. Just as we applied (2.8) above, we can easily deduce the following consequences of (2.25) for the Jacobi, Laguerre, Hermite, and Bessel polynomials: P(~'Z) (1 - 2wx) =
~
(c~ +
f7 + n + 1)k Ek (x)
(2.26)
•{l÷ 2Fl(-n÷k,,:l÷13÷n÷k÷l;o~,÷k+l;w)} or, equivalently,
P(~'Z)(1--2wX)=-~k~=0
k!
(el+•+n÷
l)kEk(x)
(2.27)
.~(n,÷a~ "i':,~-k p(~+k,n+k) t \n - a/
n (~) (wx) = -~ k=o
(1 -- 2w) } •
"
a
(2.28)
• {1+ 1 F l ( - n + k ; o ~ + k + l ; w ) } or, equivalently,
L(#) (~:)
{
• 1+2F0
(
=
1 n -7 ~
(~Ek
.
(x)
(2.29)
k=0
A(2;-n+2k);
;-~
')}
(2.3o)
or, equivalently,
Hn
( w ) ~
=
1
I,,/2i
x -(l/2)n E
/'n ~ (2k)!~
(-1)k \ 2 k ) --~--! r'k (x)
k=0
• {(2Lo)n-2k +Hn-2k(O))};
(2.311
K.-Y. CHEN et al.
142
(An + #)k Ek (x)
y,~ (wx; (A - 1) n + # + 1, v) = ~
(2.32)
k=O o2
•{l+2Fo(-n+k, An+#+k;
;-~)}
or, equivalently,
Yn (wx; (A - 1) n + # + 1, u) = ~
(An + #)k Ek (x)
(2.33)
k=O
• {1 + Y~-k (w; ( A - 1)n + # + 2k + 1, v)}. Numerous further applications of (2.8) and (2.25), associated with other families of hypergeometric polynomials, can be presented similarly. 3. M U L T I P L I C A T I O N FORMULAS INVOLVING HYPERGEOMETRIC POLYNOMIALS We begin by considering a special case of the polynomials SN(x) defined by (2.1) when An,k = (A + n)g k Ck
(n,k E N0; N E N; A E C),
(3.1)
where {C~}n~__0 is a suitably bounded sequence of essentially arbitrary (real or complex) parameters. In this special case, the definition (2.1) can be rewritten in the form [n/N]
TN ( x ; A ) : :
(--n)gk(A+n)g kckxk
E
(3.2)
k!
k=O
(hE N0; N E N ; A c C ) . Then, in terms also of a sequence {D ,~}n=O,we can derive the following general class of multiplication formulas: oo
ECIDNI
~_~
(xtN)Z l!
l=O
--
z.-. k=O
(--t)k
k!(-A-~'k)k
oa
D k +l
TN (x; A) E k
l-~O
(A+2k+
tl
1), l!
(3.3)
(A e C\Zo; N e N), provided that each member of (3.3) exists. Although the multiplication formula (3.3) can be deduced from several known multivariable expansion formulas (cf., e.g., [16, p. 300, equation (1.4)]; see also [13, p. 277, Problem 25 (i)] and the references cited therein), we choose to present here the outline of a direct proof of (3.3) for the sake of completeness. If, for convenience, we denote the right-hand side of (3.3) by T(N; z, t), and apply the definition (3.2), we obtain
T(N;x,t):=
k[ k=0
" TN (=; A) (A + 2k + 1)t/! ( + )k z=o
= k,l=O j=O E (k - Nj)! (A)2k (A + c¢ =
l,j=O
tl+Nj l! (A + 2 N j +
]2k+l
CjDk+t
Xa..
l! (3.4)
1)1CjD +Nj 2
• 3F2 [ 1 A + N j , A + 2 N j + I + I ;
J"
1 ,
143
Polynomial Expansions and Multiplication Formulas
which, in view of Dixon's summation theorem [17, p. 243, Entry (III.8)], leads us to the left-hand side of (3.3) by observing the well-poised hypergeometric 3F2 series in (3.4) to be equal to the Kronecker delta gl,0 (1 • No). The above method of derivation of (3.3) can be applied mutatis mutandis in order to derive the following confluent case of (3.3) when X i
x ) -~,
t ~---~At,
and
( N • N) :
}AI--*oo oo
E
(xtN)ll! - ~ ~ (-t)k Ugk (x) E Dk+t~.t~
/=0
k=0
(3.5)
/=0
(N • N), provided that each member of (3.5) exists,
ug(x)
being defined by
:=
~--- k--0
(3.6)
"~"
(n•No; N•N). Upon setting Ck = N - 2 N k j = l
j=l
q
j=l
(k•N0; N • N )
(3.7)
(k • No),
(3.8)
j=l
and
1~ (pJ)k I] (~J)k j=l Dk = j=l j=l
j=l
if we use such contracted notations as (for example) (ap) for the array of p parameters Otl, • . . , ~ p ,
A[N; (ap)] for the array of Np parameters aA a 3 + l N' N ""
a3+N-1 N
(j=l,...,p;
N•N),
and so on, we find from the multiplication formulas (3.3) and (3.5) that [ (ap), A IN; (p~)] ;
]
p+N.Fq+Nv L (Zq), " [N; (o-v)] ; xtNN(u-v-2)N (3.9) = E
j=l
j=l
j=l
(--t) k
j=l
144
K.-Y.
[
CHEN
et al.
(~/r)+k'(flu)+k;
]
"r+uFs+v+l
t A+2k+l,(Ss)+k,(av)+k;
A (N;-k),A
+ k),(ap),A IN; (Ss)] ;
(N;A
"p+N(s+2)Fq+Nr
]
xN (s-r)N
(3.9)(cont.)
(flq), A [N; (~r)] ;
p-q
r+u
p-q=N(v-u)+l
and
IxtN[ < N(V-u+2)N; r + u = s + v + 2 and Itl.< 1; N E N ) , and
p+NuFq+Nv [ (OIp) ' A [N; (pu)] ; xtN N(u-v-1)N 1 (flq), A [N; (a.)] ;
Zj=I
=
j=l
(-t)
j=l
[
(7~) + k, (p~) + k; ]
• r+uFs+v
t
(3.10)
(Ss) + k, (Or) + k;
•p+N(s+l)Fq+Nr
[ A (N;-k),(ap) ,A [N; (Ss)] ;
xN (s-r)g
J
(flq), A IN; (~r)] ;
p-q
l, r +u < s + v + l; p - q = N ( v - u ) +
l and
Ixtg[ < N(V-u+l)N; r + u = s + v + 1 and It I < 1; N E N ) , respectively. With a view to applying (3.9) and (3.10) to some of the aforementioned classical orthogonal polynomials, we set N = 1 and
u,
~u + 1
(Pu+l --- - n ; n c No),
so that (3.9) simplifies to the finite form
pTu+lFqTv
'
'
(~), (~o);
xt =
tk
j=l
j=l
~o
(~ + k)~ fl (~J)k fl (~J)~ j=l
j=l
-~+k,(w)+k,(p~)+k; ] " r+u+lFs+v+l
t A + 2k + 1, (Ss) + k, (Zv) + k;
,÷~÷2F~+~[ -k'~ + a'(~')'(~2 (~), ('~);; ] ~
,
(3.11)
Polynomial Expansions and Multiplication Formulas
145
while (3.10) yields
l~I (~J)kl] (p~)k
[-n,(~p),(p~); ] p+u+ l tq+v
xt
(G), (~)
;
~:0
fl (ej)~ fl (oj)~ j=l
j=l
-n + k, ('M + k, (p~)+ k; ]
• r+u+lFswv
(~s) +
k,
(o~) +
[-k,(~p),(~); ] • p-Fs+ltq+r
t k;
(3.12)
J
X
L
(z~),('~,);
which obviously is a confluent case of (3.9) when X x, , ~,
t~--,At,
and
IAI--*oe.
By appealing to the multiplication formula (3.11), S£nchez-Ruiz [181 derived several interesting linearization and connection formulas involving squares of the Gegenbauer (or ultraspherical) polynomials C,~(xt). These classical orthogonal polynomials are known to have many different hypergeometric representations including (for example) the following relatively more familiar one (cf. [2, p. 280, equation 144 (20)]):
"-1n-In+l; x2 ] 2 '
C~(x)= ( 2 v +zn~- 21 F) l n
-!
1 u+~;
J
'
(3.13)
which, in view of the Pfaff-Kummer transformation (cf., e.g., [3, Volume I, p. 64, equation 2.1.4 (22)])
2Fl (a,b;c;z) = (1- z)-a 2Fi (a,c-b;C; z_ ) (larg(1-z)l~-e;
0
(3.14)
c~Zo),
assumes the equivalent form 1 1 C~ (x) = (2u + n - 1 ) 2F1 I --~n,v +-~n; n 1
1 -- X 2
]
.
(3.15)
u+ 2 , Making use of this last hypergeometric representation (3.15) in conjunction with Clausen's identity [3, Volume I, p. 185, equation 4.3 (1)] {
1)} 2= 3F2 ( 2a,2b,a+b;2a+2b, a+b+-~;z 1 ),
(
2F1 a,b;a+b+-~;z
(3.16)
it is easily seen that
I
{C~ (x)}2 = (2u + n - 1 ) 3F2 --n, 2p + n, 1~; 1 - x 21 . 2v, p + ~;
(3.17)
146
K.-Y. CHEN
et al.
Thus, upon setting p=q=2
(~1 = 2 v + n ;
a 2 = v ; ~l=2V; ~ 2 = v + ~ ) ,
( 71 = 2 v + n ; 72 = v; 73 = A+ 2' 1 61
r-l=s=2
u = v = O,
x t
~ bx,
2v; 62
v+ 1)
a
and
t=~,
the multiplication formula (3.11) yields the following linearization formula for the square of the Gegenbauer (or ultraspherical) polynomials:
= (n!)------'Tk=O
• 4F3
(A)2;¢ (u + 1/2)k
- n + k,u + k,2u + n + k,A + k + -~; 1 A + 2 k + 1,2u + k,u + k + 2;
(4-b)
C~2k "
In precisely the same manner, but with different choices for the various parameters involved, we find from (3.11) that
(2 )2o C~n ( ~ )
= (2n)!
2(:)
k=0
(k,) 2(u + {(A)k} 2 (u + 1/2)k
-n+k,u+n+k,2A+k,A+k+-~;
• 4F3
C~ ~
2
1 2A + 2k + 1,A+ k , u + k + 2;
and
{c: • 5F4
}2 (n!)2 k=0
{(A)k} 2 (u + 1/2)k -4-b-
- n + k,2u + n + k,u + k,2A + k,A + k + -~; 1 2A + 2k + l,A + k, 2u + k,u + k + -~;
~
2 '
For a = b = 1 and x ~ 1 - x 2, (3.18)-(3.20) would obviously correspond to the main results of S~nchez-Ruiz [18, p. 265, Theorems 1 and 2; p. 226, Theorem 3]. Both (3.11) and (3.12) appeared recently in the work of S£nchez-Ruiz and Dehesa [19, p. 580, Lemmas 1.1 and 1.2], who applied each of these results to several families of generalized hypergeometric polynomials (see also the various related works cited in [19]). A special case of (3.12) when p -- q = 0 was applied earlier by Srivastava [20, p. A663, equation (4.1)] in order to derive the corresponding multiplication formulas for the classical Jacobi and Laguerre polynomials. For the sake of completeness, we choose to recall here Srivastava's main applications of the multiplication formula (3.12), involving the Jacobi and Laguerre polynomials, as follows (cf. [20, p. A663]; see also [13, pp. 260-262, Problems 3 and 4]): P(~") ( 1 - ( 1 - x)y) =
£(:::)(
k=O
a + fl + n +
)(
A+#+n+k
[ - n + k , a + /~ + n + k + l,A + k + l; •P(k~'u+'~-k) (x) 3F2 A+#+n+k+l,a+k+l;
] y ,
yk (3.21)
Polynomial Expansions and Multiplication Formulas
147
which, in its special case when p = a + j3 - A, yields
_
k
(x)
k=O
-n+k,A+k+l;
• 2F1
L(a) (xy) =
I
] yJ ;
(3.22)
a+k+l;
~,n- k) y
~k (x) 2F1
y ,
k=0
(3.23)
a+k+l;
which, for/~ = a, reduces at once to the following well-known (rather classical) result attributed to Feldheim (1912-1944) [3, Volume II, p. 192, equation 10.12 (40)]:
k=0
In light of some other developments by, for example, [6,7], here we only apply (3.11) to the Jacobi and Bessel polynomials, and also to the generalized Rice polynomials H~(~'z) [4, P, x] defined by (cf., e.g., [13, p. 140, equation 2.6 (13)])
H(c''z)[{,p,x] :=
3F2 ( - n , a + 3 + n + 1,~;a + 1,p;x),
(3.25)
which, for a = /3 = 0, were considered by Rice (1907-1986) [13, p. 140, equation 2.6(14)] and which, for ~ = p, yield the Jacobi polynomials
(3.26)
p(a,e) (x) = H (a'z) [¢, ¢, ~l ( 1 - x ) ] First of all, upon setting
p=q=O,
u=v=l X e----~ ~1
(l-x)
(pl = a + $ + n + l ; 1
and
r-1
al = ~ + 1 ) , =s=0
A ~--* A + t z + 1,
(Ti=A+l)
(3.11) yields the following multiplication formula for the J acobi polynomials:
k=O
[-n+k,A+k+l,a+j3+n+k+l; "~]¢ P(~'")
(x) t k 3F2
a+k+l,A+#+2k+2;
]
(3.27)
$
which, for t = 1, was given also in the aforementioned work of S£nchez-Ruiz and Dehesa [19, p. 581, equation (8)]. Similarly, for the generalized Bessel polynomials defined by (2.13), we obtain
Yn(xt;a,/3) =k=o ~ (nk) (a÷n÷k-2)k
( A + 2 k - 2 ) k -1 ( _ ~ ) k
(3.28)
•yk(x;A,p) 2Fl(-n+k,a+n+k-1;A+2k;#---;) •
148
K.-Y. CHEN et aL
In the case of the generalized Rice polynomials defined by (3.25), (3.11) readily yields the following generalization of the multiplication formula (3.27):
H (a'z) [~,p, xt]= ~ ( : + - k ) ( a + ~ + n + k ) ( A + # + 2 k ) - i k
k=O
M(~,~) [rj,a,x] 5F4
tk
(~)k (~)k
k
(p)k
[-n+k,,~+k+l,a+fl+n+k+l,~+k,a+k;
]
(3.29)
t a + k + l, )~+ # + 2k + 2, rl + k, p + k;
which obviously corresponds to (3.27) when = p,
7? = a,
and
x,
, 1 (1 - z ) .
In view of (3.26), this last multiplication formula (3.29) can be specialized to derive each of the following connection formulas between Jacobi polynomials and the generalized Rice polynomials:
\n - k]
k
-n+k,A+k+l,a+fl+n+k+l,a+k;
]
•Hk(~'") [77,a, x] 4F3
(3.30)
t
a+k+l,A+#+2k+2,~+k; and = k=0
k
k
(-~k-
[-n+k,.X+k+ l , a + / 3 + n + k + l,~+k; ] P(~'") (1 - 2x) 4F3 "~ k
(3.31)
t
a+k+l,A+#+2k+2,p+k;
Analogous connection formulas between Jacobi polynomials and the generalized Bessel polynomials would follow directly from (3.11), and we thus obtain
= k=o k k ] •yk(x;X,#) 2 F 2 ( - n + k , a + ~ + n + k + l ; a + k + l , A + 2 k ; t )
-~.
(3.32)
and
k=0
k
"
"
(3.33)
"'~(~'") (l + ~-~) 3Fl (-n + k,A + k + l,a + n + k-1; A + # + 2k + Some obvious further special cases of these last relationships (3.32) and (3.33) were proven earlier by Popov [6]. More recently, by using a slightly different notation for the generalized Bessel polynomials, the multiplication formula (3.28) was derived in another paper by Popov [7, p. 20, equation (1)]. Since 2F1 (-n + k,a + n + k - 1; ~ + 2k; 1) = ( - 1 ) n-~ (a - )~),~-k ()t)2k (A),~+k (3.34)
(0
Polynomial Expansions and Multiplication Formulas
149
by the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], a special case of (3.28) when t= #
and
x ~-~ ~x
yields
k
k=O •yk ( ~ x ; ) , ,
(A+k-1)n+l
(3.35)
~),
which, for f~ = #, was stated incorrectly by Popov [7, p. 21, equation (3)]. The various multiplication formulas considered in this section can indeed be proven directly by means of such known expansions as (for example) the following familiar analogue of (1.7) and (2.23) [2, p. 262, equation 136 (2)]: n
x,~=n!E(_l)k(n+a ~ a+~+2k+l \ n - k / ( a + ~ + k + 1)~+~
k=O
(1 - 2z) P•("'•)
(8.36)
in the case of the Jacobi polynomials. In the cases of the classical Laguerre and Hermite polynomials, we have [2, p. 207, equation 118 (2)]
k (x)
(3.37)
[n/2] /, n ~ (2k)! H (2xF = Z \ 2 k ) T ~-2~ (x), k=O
(8.38)
xn = n ! E ( - 1 ) k \ n _ k ]
k=O and [2, p. 194, equation 110 (5)]
respectively. More generally, by equating the coefficients of t Nn (n E No; N E N) on both sides of the expansion formulas (3.9) and (3.10), we can show that
"=
]=1
Xn --
7 (~))~ FI (7))N~
\j=l
_
_
(Nn)! E (-1)k k=O
(A Jr- ~)Nn+l (3.39)
j=l
A (N;-k),A
(N;A + k), (ap),A [N; (ha)I;
• p+N(~+2)8+N.
xN(S-r+2) N]
(&), A IN; (~)] ;
and
-
\j=l
,=1 7 (Zj)~ FI (TJ)N,,
j=l
~ : ~ ~ {_l)k (Nn!) k=o
[ A (N; - k ) , (ap), A IN; (6s)] ; xN (s-r+l)N m
p+N(s+l)Fq+Nr
L
(Zq), A [N; (~,)] ;
(3.40)
150
K.-Y. CHEN et aL
respectively. For N = 1 (and r = s = 0), (3.39) and (3.40) immediately yield the following known results recorded by (for example) Prudnikov et al. [21, Volume 2, p. 389, Entry 5.3.2.2; p. 388, Entry 5.3.1.4]:
- k , A + k, (ap) ; =
zn = (3j)
(A + k)n+l p+2Fq
(-1)k
x
(3.41)
k=o
\j=l and =
x~ = E
(--1)k
p+lFq
x
,
(3.42)
k=0
\j=l respectively. These last results (3.41) and (3.42), as well as their q-extensions, were applied recently by Area et al. [22] to various hypergeometric and q-hypergeometric polynomials. 4. F U R T H E R
REMARKS
AND
OBSERVATIONS
For the classical Laguerre polynomials L(~)(x), (3.12) with p=q=O,
r-l=s=0
('h=l+fl),
x,
, bx,
and
u=v-l=0 a t=b
(al = 1 + c ~ ) ,
immediately yields the following multiplication formula (cf. [13, p. 261, Problem 3 Off)]):
L (~) (ax) =
+
L(k~) (bx)
2F1
(4.1)
k=O
a+k+l;
which may be compared also with (3.23) above. As a matter of fact, in view of the limit relationship (2.12), (4.1) can easily be deduced as a confluent case of (3.27) by first setting x,
~1---
2bx
a#
and
/Z
t = =-=,
and then letting m i n {[fl[, I/z[} --* c~.
By appropriately modifying the above limit process, we can obtain connection formulas between Jacobi and Laguerre polynomials as follows. First of all, in (3.27) we let x,
2bx
a#
) 1 - - -/z,
t =Y'
and
Iiz] --~ cxD,
so that we have
k=O
k (4.2)
"(b) k [ - n + k ' A + k + l ' a + f l + n + k + l ; b ] a3 F+1 k + 1;
Polynomial Expansions and Multiplication Formulas
And, upon letting
a
x ~ ~ 1 - 2bx,
t = -~,
and
151
Ifll -~ oc,
(3.27) also yields the connection formula
k=0
a + k + l,A + # + 2k + 2; Each of the results, which we have presented in this paper for the Jacobi polynomials P~(~'z) (x), can indeed be applied in order to derive the corresponding result involving such simpler orthogonal polynomials as theGegenbauer (or ultraspherical) polynomials [cf. equation (3.13) and (3.15)]
c: (x)
/
(\u + nn-
2u +nn - 1 p(u_l/2,u_l/2){x. ~ j,
(4.4)
the Legendre (or ultraspherical) polynomials Pn (x) -- C~/2 (x) = p(O,O) (x) ,
(4.5)
the Chebyshev polynomials of the first kind Tn (x) ---- n
-- ~
P(-1/2'-l/2)(x) = -~nC~ (x),
(4.6)
n
where C ° (x) := lira { v - l c ~ (x)}, lul-~0
(4.7)
and the Chebyshev polynomials of the second kind P(1/~'l/2)(x) u~(x) = 5
= C~ (x).
(4.8)
n+
Next, with a view to applying the multiplication formula (4.1) in the evaluation of integrals involving products of Laguerre polynomials, we recall the familiar orthogonality property [10, p. 100, equation (5.1.1)] fo ° x ~ e - X L ~ ) (x) L (~) (x) dx - F ((~ + n + 1) 5m n n!
(4.9)
(9~(a) > -1; m , n E No), where 6m,n denotes the Kronecker delta. Thus, making use of (4.9) in conjunction with the multiplication formula (4.1), we readily obtain the integral formula o ° z~ e-bX L ~ ) (az) L (~~) (bx) dx =
(~-I-:) _
(fl:n)anF
(fl+l) ( bfl+n+l 2F1 - m + n ,
fl+n+l;t~+n+l;-~
(9~(fl) > - 1 ; m > - n > - O (m, n E N 0 ) ; iR(b) > 0 ) .
a)
(4.10)
K.-Y. CHEN et al.
152
In spite of the fact that much more general integral formulas than (4.10), involving products of two or more Laguerre polynomials, have been considered in the mathematical literature rather extensively (see, for details, [23]; see also equations (4.16) and (4.17) below), we do find it to be worthwhile to mention here the following three special cases of the integral formula (4.10). First, (4.10) with m = n reduces immediately to the form
/o c~ x~e-b~L (~) (ax) n (~) (bz) dx = a'~F (~ + n + 1) n!b~+n+ 1 (~(/~) > -1; n 6 N0; ~ ( b ) > 0).
(4.11)
Second, by means of the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], already referred to in connection with (3.34), a special case of (4.10) when b = a yields the integral formula
/o c~ xZe-aXL~ ) (az) L (~) (az) dx F ( / ~ + n + 1) ( a - f l ) m _ n -
aZ+l
(4.12)
(m - n)!n!
(~R (/3) > -1; m >- n >- 0 (m,n C No); !R(a) > 0). Third, in its special case when/3 = a, (4.10) readily yields the integral formula
/o °~ x~e-b~L~ ) (az) n (~) (bx) dz = r (~ + m + 1) (b - a ) m - ~ a ~
ba+m+l
(4.1a)
(m - n)! n!
(~R(c~)>-l; m > _ n > O (m, n E N 0 ) ; ! R ( b ) > 0 ) . Both (4.11) (for a = b = 1) and (4.12) (for a = 1 and n = m - 1) appeared quite recently in the work of Mavromatis and Alassar [24, p. 904, equation (6); p. 905, equation (11)], using a slightly different notation for the Laguerre polynomials. The integral formula (4.13) (with, of course, c~ -- A, b = c, and a , , b) provides the corrected version of another result recorded by Prudnikov et al. [25, Volume 3, p. 478, Entry 2.19.14.10]. Yet another interesting application of the multiplication formula (4.1) in evaluating integrals involving products of two Laguerre polynomials can easily be presented here. Indeed, by writing (4.1) in the forms
j=o \ m - j ]
¢
2F1
(4.14) a+j+l;
and k=0
-k)
k (cx)
2F1
,
(4.15)
~+k+l; and appealing to the orthogonality property (4.9) once again, we readily arrive at the following integral formula (cf., e.g., [23, p. 312, equation (25)]):
o ° x~e-CXL (a) (ax) L (z) (bx) dx
-- C7+1P(~'-~-l)min(m'n)E(:-~k)(:'~-~k) ("f;k) ~ab~ k k=0 • 2F1
-
\ c'2/] 2F1
a+k+l; ~+k+l; (!R(v) > -1; m , n 6 N0; ~R(c) > 0).
(4.16)
PolynomialExpansionsand MultiplicationFormulas
153
By means of the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], a special case of (4.16) when c = b leads us to (cf. [25, Volume 3, p. 478, Entry 2.19.14.8]; see also [23, p. 307, equation (8)])
~oc~ x'le-b~L(ma) --
r(~q-1) rain(re'n)(: b'r+l
Z
+
-
.2F1 ( - m + k , ~ + k
(ax) n (~) (bx) dx
k)(
V
; )
k ( f l - V)n-k
k
(41 )
°)
+ l ; a + k + l; ~
(!R(y) > -1; m , n e No; !}l(b) > 0), which, for 7 =/3, immediately yields (4.10). Furthermore, upon setting a = b = c in the integral formula (4.16), we similarly obtain the following mild generalization of (4.12):
fo ~ z're-a~L(~ ) (ax) L ~ ) (ax) dx +k k ) (~ - ~ ) ~ _ ~ ( z - ~)~_~
F (m~ i++1 nl () m ' ~ )~( a ~
(4.18)
(fit (~,) > -1; m , n E No; !R(a) > 0), which, for V = fl, reduces at once to (4.12). Finally, in view of the orthogonality property for the Jacobi polynomials [10, p. 68, equation (4.3.3)]:
L1
1 (1 - z) ~ (1 +
=
X)~ Ptm~'~) (x) p(~,Z) (z) dx
2~+Z+IF (a + n + 1) F (/3 + n + 1) n! (a +/3 + 2n + 1 ) r ( a +/3 + n + 1) & ' "
(4.19)
(min{ER(a),9t(/3)} > -1; m , n E No), the above method of derivation of the integral formula (4.16) can be applied mutatis mutandis in order to find from the multiplication formula (3.27) that
_1 (1 -
X) A
(1 + x)" P(ma'~) (1 - (1 - x) u) P(~P'°) (1 - (1 - x) v) dx
1 ----- 2A+/~+1
min(ra'n~,rr)(?T~' ~, ---I-O:~ ~] (:-F-- ~) (°~-t-/3-F~'rl'-Fk) k (P-t-G-F~-Fk)F~-)( k=O
-m+k,A+k+l,a+/3+m+k+l; . (uv) k B ()~ + k + 1, p + k + 1) 3/?2
u
(4.20)
a+k+l,A+#+2k+l;
•3F2
v p+k+l,X+.+2k+l;
(min{!tl()~),91(#)} > -1; m , n e No), where B (c~,/3) denotes the familiar Beta function defined by B (~,/3) . - r ( ~ ) r ( / 3 ) - : B ( / 3 , ~ ) . r (~ +/3)
(42~)
In fact, by virtue of the limit relationship (2.12), it is not difficult to deduce the integral formula (4.16) as a confluent case of (4.20).
154
K.-Y. CHEN et al.
REFERENCES 1. H.M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic, Dordrecht, (2001). 2. E.D. Rainville, Special Functions, Macmillan, New York, (1960); Reprinted by Chelsea Publishing, Bronx, New York, (1971). 3. A. Erd61yi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Volumes I and II, McGraw-Hill, New York, (1953). 4. B.S. Popov, Some relations between Bessel and Bernoulli polynomials, Ann. Inst. Math. (Skopje) 38, 7-10, (1997). 5. B.S. Popov, Expressions of Laguerre polynomials through Bernoulli polynomials, Mat. Bilten 22 (48), 15-18, (1998). 6. B.S. Popov, Some relations between Jacobi and Bessel polynomials, Mat. Bilten 23 (49), 5-10, (1999). 7. B.S. Popov, Some relations of Bessel polynomials, Mat. Bilten 24 (50), 19-22, (2000). 8. H.M. Srivastava, A contour integral involving Fox's H-function, Indian J. Math. 14, 1-6, (1972). 9. B. Gonz~lez, J. Matera and H.M. Srivastava, Some q-generating functions and associated generalized hypergeometric polynomials, Mathl. Comput. Modelling 34 (1/2), 133-175, (2001). 10. G. SzegS, Orthogonal Polynomials, Volume 23 in the Series on Colloquium Publications, Fourth Edition, American Mathematical Society, Providence, RI, (1975). 11. E. Grosswald, Bessel polynomials, Lecture Notes in Mathematics, Volume 698, Springer-Verlag, Berlin, (1978). 12. H.L. Krall and O. Frink, A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65, 100-115, (1949). 13. H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, (1984). 14. M.A. Khan and K. Ahmad, On a generalization of Bessel polynomials suggested by the polynomials L~'f~(x) of Prabhakar and Rekha, J. Analysis 6, 151-161, (1998). 15. H.M. Srivastava, Some generating functions for Laguerre and Bessel polynomials, Bull. Inst. Math. Acad. Sinica 8, 571-579, (1980). 16. H.M. Srivastava, Some polynomial expansions for functions of several variables, IMA J. Appl. Math. 27, 299-306, (1981). 17. L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, (1966). 18. J. S~nchez-Ruiz, Linearization and connection formulae involving squares of Gegenbauer polynomials, Appl. Math. Lett. 14 (3), 261-267, (2001). 19. J. S~nchez-Ruiz and J.S. Dehesa, Some connection and linearization problems for polynomials in and beyond the Askey scheme, J. Comput. Appl. Math. 133, 579-591, (2001). 20. H.M. Srivastava, Sur quelques formules de multiplication concernant les polynSmes hyperg~om~triques g~n~ral -is&s, C.R. Acad. Sci. Paris Sdr. A-B 272, A661-A664, (1971). 21. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series, Volume 2, Special Functions, (Translated from the Russian by N.M. Queen), Gordon and Breach Science, New York, (1988). 22. I. Area, E. Godoy, A. Ronveaux and A. Zarzo, Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas, J. Comput. Appl. Math. 133, 151-162, (2001). 23. P.-A. Lee, S.-H. Ong and H.M. Srivastava, Some integrals of the products of Laguerre polynomials, Internat. J. Comput. Math. 78, 303-321, (2001). 24. H.A. Mavromatis and R.S. Alassar, Two new associated Laguerre integral results, Appl. Math. Lett. 14 (7), 903-905, (2001). 25. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series, Volume 3, More Special Functions, Second Edition (Translated from the Russian by G.G. Gould), Gordon and Breach Science, New York, (1990).
Mathematical and Computer Modelling 37 (2003) 135-154 www.elsevier.com/locate/mcm
Certain
Classes of Polynomial Expansions and Multiplication Formulas KUNG-YU CHEN AND CHUAN-JEN CHYAN Department of Mathematics, Tamkang University Tamsui 25137, Taiwan, R.O.C.
H. M . S R I V A S T A V A Department of Mathematics and Statistics,University of Victoria Victoria, British Columbia V 8 W 3P4, Canada harimsriQmath, uric. ca
(Received May 2002; accepted June 2002) Abstract--The authors first present a class of expansions in a series of Bernoulli polynomials and then show how this general result can be applied to yield various (known or new) polynomial expansions. The corresponding expansion problem involving the Euler polynomials is then considered in an analogous manner. Several general multiplication formulas, involving (for example) certain families of generalized hypergeometric polynomials, are also investigated in the context especially of the classical Jacobi, Laguerre, and other related orthogonal polynomials. (~) 2003 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - P o l y n o m i a l expansions, Multiplication formulas, Bernoulli and Euler polynomials, Generalized hypergeometric polynomials, Generating functions, Dixon's summation theorem, Jacobi polynomials, Laguerre polynomials, Generalized Bessel polynomials, Hermite polynomials, Gegenbauer (or ultraspherical) polynomials, Generalized Rice polynomials, Chu-Vandermonde theorem, Legendre (or spherical) polynomials, Chebyshev polynomials.
1. I N T R O D U C T I O N
AND
PRELIMINARIES
For t h e g e n e r a l i z e d B e r n o u l l i p o l y n o m i a l s B (a) (x) ( a c C) defined by t h e g e n e r a t i n g f u n c t i o n
(cf., e.g., [1, p. 61, equation 1.6 (19)])
(
z
) ~ e xz
~ z~ = E BP)(x) ~.v
({z I < 27r; 1 a : = 1),
(1.1)
n=0
it is easily observed that
°+') (x + y) =
B(o) (x)
(y)
(1.2)
The present investigation was completed during the third-named author's visits to Tamka~g University at Tamsui in April 2002 and May 2002. This work was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. 0895-7177/03/$ - see front matter © 2003 Elsevier Science Ltd. All rights reserved. PII: S0895-7177(02)00328-X
Typeset by ~4A4S-TEX
136
K.-Y. CHENet al.
and
B~(~) (~ + 1)
B(~°) (~) . . . .
~_~
(x)
(n e No := {0,1,2,...}).
(1,3)
Furthermore, since B (°) (x)
=
(n ~ No),
x ~
(1.4)
upon setting/3 = 0 in (1.2), we obtain B(na) (x + y) =
k Bk
(1.5)
(x)
k=O
Now, by combining (1.3) and (1.5) (with y = 1), we find that
B(~"-I) (~) =
k=O
n] B (a) (x), k! (n -)¢ + 1)!
(1.6)
which, in the special case when a = 1, immediately yields the following expansion (cf., e.g., [2, p. 302, Exercise 1]): n[ xn = k=0 Z k! (n --k + 1)! Bk (x) (1.7) in series of the Bernoulli polynomials Bn(x) defined by Sn (x) := B(~1) (x)
(n e N0),
(1.8)
that is, by means of the generating function [3, Volume I, p. 36, equation 1.13 (2)] zeXZ
oo
ez - 1 = E
zn
Bn (x) ~.f
(Izl < 27r).
(1.9)
n=O
Motivated essentially by several recent works on the subject of polynomial expansions and multiplication formulas by (for example) Popov ([4,5]; see also [6,7]), we first apply (1.7) in order to develop a general class of polynomial expansions in series of the Bernoulli polynomials and investigate the corresponding expansion problem involving the Euler polynomials as well. We then consider several general multiplication formulas which are associated with (for example) certain families of generalized hypergeometric polynomials. We also indicate relevant connections of the polynomiM expansions and multiplication formulas, which are presented in this paper, with the corresponding (known or new) results involving many of the classical orthogonal polynomials.
2. C L A S S E S O F P O L Y N O M I A L EXPANSIONS INVOLVING BERNOULLI AND EULER POLYNOMIALS oo In terms of a suitably bounded double sequence {Am,n}m,n=O of essentially arbitrary (real or complex) parameters, we define the polynomials S g ( x ) by (cf. [8]; see also [9, p. 145, equation (3.1) with m = 0]) [~/gl n ) g k A ~,kX k (2.1) s~:N (~):= ~ ( - -~. k=O
(n e N o ; g e N := No \ {0}), where, as usual, [a] denotes the largest integer in a E R and (A)k is the Pochhammer symbol (or, more precisely, the shifted factorial, since (1)k = k!(k C No)) defined, in terms of Gamma functions, by F ( A + k ) _ [ 1,
(•)k:=
F(A)
~.A(;~+l)...(~+k-1),
( k = 0 ; A#.0),
(keN~eC).
(2.2)
Polynomial Expansions and Multiplication Formulas
137
By making use of the definition (2.1) in conjunction with the polynomial expansion (1.7), we have tn
$ ( g ; x , t ) : = E SNn (wX)-~. n=O oo tn [ ~ ]
(_n)Nj ~
J J
j=0
n=O
= ~
(-1)NJ
=
(--1)NJn!An+NjJwJtn+gJ'
n,j=o
n!J-------~"An+Nj'JJxJtn+NJ J sk (z) E k! ( j ~ k + 1)!
n,j=O =
~
(2.3)
k=0
(--1)N(j+k)
(X) o3J+kJ6n+N¢j+k)
n,j,k=o n! (j + 1)!k! An+N(j+k)d+kBk provided that the inversion of the order of summation is justified by absolute convergence of the series involved• If we now let n , n - N j and n, , n - Nk, successively, we find from (2.3) that [n/N]
S(N;x,t)=
\
9=0
n=0
tn E
k=0
(J + 1)!
(_l)Nk
(n-Nk)!k[
)
An3+kw3+k
(2.4) Bk (z).
Finally, upon equating the coefficients of t n from both sides of (2.4), we obtain the following general class of polynomial expansions in series of the Bernoulli polynomials:
In~N] ( n ) (Nk)! S N (cox) = E (--1)Nk N k k! k=0
(2.5) \
j=o
(J + 1)!
An3+kw J+k
Bk (x)
(n6N0; N6N). Next, in order to apply the general result (2.5) to a certain family of generalized hypergeometric polynomials, we set P
I1 (~J)k An,k = N -Nkj=l q
(A (L;An + #)) k
I] (~J)k j=l
(2.6)
(N, L E N ; n,k,p, qENo; .k,#,aj 6 C ( j = 1,...,p); /39 e C\Z o (j = 1,...,q; Z o := { 0 , - 1 , - 2 , . . . })), where A(N; A) abbreviates the array of N parameters A+N-I
A A+I
N'
N
''"'
N
(N6N)
K.-Y. CHEN et al.
138
and an empty product is interpreted (as usual) to be 1. Thus, for the generalized hypergeometric polynomials
.p(),,t,) ' n,N,L (x;p,q)
N+L+pFq
:=
[ A ( N ; - n ) ' A ( L ; A n + #)'c~I'''''O~p; ] X
(2.7) ,
Zl,. • ,Zq;
we find from the polynomial expansion (2.5) that
[n/N] ,D()~,t,) ( n ) (Nk)' • n,N,L(wx;p'q) = E (--Y)-lVk Nk k! k=O P
l-I (~5)k
5=1
(A (n; An + #))k Bk (x) w k
(2.8)
l-I (;35)~
j=l
A ( N ; - n + Nk),A(L;An + # + Lk),o1 + k,...,ap + k, 1;w-I • N+L+p+IFq+I
I
f~l+k, ..,f~q+k, 2; ] provided that each member of (2.8) exists. From among the classical orthogonal polynomials (see, for details, [10]), the Jacobi polynomials P(~'Z)(x), the Laguerre polynomials L(a)(x), and the Hermite polynomials Hn(x) can easily be shown to correspond (for example) to the special cases: pn(i,a+~+i) ( 1 - x
,1,1
•(0,tt) n,l,1 (X;0,2)
and
)
T;0,1
(Zl = a + i ) ,
(~1 ~-- O~-~ 1; ~2 = It)
p(o#,)( 1 ) ,2,1 -~-~; 0, 1
(~1
---- ].t),
respectively. Under these special cases, the hypergeometric polynomial expansion (2.8) readily yields p(a,f0 (1 - 2wx) =
.
+ _
(o~+ j3 + n + 1)k B k (X)
(2.9)
k=O
.3F2 (-n + k,o~ + fl + n + k + l,1;o~ + k + l,2;w) , L(a)(wx):~=o~(:+k)
a B k(x)
(2.10)
.2F2(-n + k, 1;a + k + l,2;w), and Hn ~
[,~/2] = x -cl/2)n E (--1)k 2k k=O
(2.11)
:3Fl ( A (2;-n T 2 k ) , l ; 2 ; - - ~ ) for the Jacobi, Laguerre, and Hermite polynomials, respectively. Obviously, since [10, p. 103, equation (5.3.4)]
L(a)(x)= lim ~ P ( a ' ~ ' ( 1 - 2 - ~ x ~ , If~l-*o¢L k P/)
(2.12)
Polynomial Expansions and Multiplication Formulas
139
the polynomial expansion (2.10) is a limit case of (2.9) when 03
w*
, --
and
IZ~l ~
~.
The polynomial expansion (2.10) (with w = 1) was given recently by Popov [5, p. 17, equation (7)] (see also [1, p. 69, Problem 15]). For the generalized Bessel polynomials yn(x; a, b) defined by (cf. [11,12]; see also [13, p. 75,
equation 1.9 (1)])
Yn (x;a,b):= ~ (nk)(a +n+k-k
2)k, (b) k
k=o
(2.13)
=2 Fo (-n,a + n - l;
;-b) '
a special case of the expansion formula (2.8). when N = L = 1,
p = q = 0,
and
x,
--
X /2
would yield the polynomial expansion Yn(Wx;(£-l)n+#+l,v)=~(k)(£n+#)kBk(x)(_w)
k
(2.14)
k=0
k, An+# + k, 1;2;w),
• 3F1 ( - n + which, for x*
,-~x
and
w~-~--
03
/2
assumes the form
y~ (wx; (;~ -
1) n + # + 1, u) =
(An + #)k Bk (x)
(2.15)
k=0
.3Fl (-n + k, An + # + k,1;2;-w) . In their
further
special cases when A = w = 1,
# = a-
1,
and
~ = b,
these last results (2.14) and (2.15) reduce at once to the forms
yn(x;a,b)=~-~(-1)k(nk)(a+n-1)kBk(-b) k=O
•3 F l ( - n + k , a + n + k
-
(2.16)
1,1;2;1)
and
(2.17)
k=0
.3Fl(-n+k,a+n+k-l,1;2;-b), respectively. The polynomial expansions (2.16) and (2.17) happen to be the by Popov [4, p. 10].
main
results in another paper
140
K.-Y. CItEN et al.
We remark in passing that, by employing a slightly different notation, the generalized Bessel polynomials
(
x)
y n ( x ; ( a - 1 ) n + ~ + 2 , 2 ) : = 2F0 -n, a n + ~ + l ; =:
;--~
(2.18)
Z e c)
were considered by Khan and Ahmad [14] as a further generalization of the Bessel polynomials. While each of the polynomial expansions (2.14) and (2.15) can easily be restated in the Y-notation of (2.18), such families of generalized Bessel polynomials as those occurring in (2.14), (2.15), and (2.18) were actually investigated, over two decades ago, by Srivastava [15] in the context of generating functions (see also Srivastava and Manocha [13, p. 92, Problem 18; pp. 386-388, Problems 9 and 10]). We turn our attention now to the expansion problem, corresponding to (2.5) and (2.8), in series of the Euler polynomials En (x) defined by means of the generating function [3, Volume I, p. 40, equation 1.14 (2)] oo Zn 2eXZ - Z E , (x) (Izl < ~) (2.19)
eZ+l
~.
rt~O
•
The following relationships between the Bernoulli and Euler polynomials follow easily from the definitions (1.9) and (2.19):
(2.20)
2
(n E No).
For the Euler polynomials En (x), it is known also that (cf., e.g., [1, pp. 63-64]) E,~ (x + 1) + En (x) = 2x n
(n c N0)
(2.21)
and k=O
Obviously, therefore, by combining (2.21) and (2.22) (with y = 1), we obtain the following analogue of (1.7) for the Euler polynomials:
=5
En(z)+
Ek(z)
(2.23)
.
k=0
Making use of (2.23) in place of (1.7), our derivation of the polynomial expansion (2.5) can now be applied mutatis mutandis in order to obtain the following general class of polynomial expansions in series of the Euler polynomials:
1 [n/N]
SN ( W X ) = 5 ~
(--l)Nk
( n ) (Nk)! Yk k[
k=O
• (A,~,k+
Nk)gj An,j+kJ.) wkEk (x) j!
(2.24)
[ ( n - N k ) / g ] ( - - n "4-
Z j=O
(n~No; N~N).
Polynomial Expansions and Multiplication Formulas
141
Fhrthermore, for the generalized hypergeometric polynomials defined by (2.7), we find from (2.24) with the coefficients An,k chosen as in (2.6) that ~,(),,~) 1 r'n'N'L(wx;p'q)=7 E
(--N)-Nk
n Nk
k!
k=O
P 1-I (<~J)k j=l -q (A (L; An + U))k Ek (x) wk
(2.25)
17I (ZJ)k
j=l
.(
FqpN+L
[A(N;-n+Nk),A(L;An+#+Lk),al+k .... ,ap+k; co , 13a + k,...,/3q + k;
provided that each member of (2.25) exists. Just as we applied (2.8) above, we can easily deduce the following consequences of (2.25) for the Jacobi, Laguerre, Hermite, and Bessel polynomials: P(~'Z) (1 - 2wx) =
~
(c~ +
f7 + n + 1)k Ek (x)
(2.26)
•{l÷ 2Fl(-n÷k,,:l÷13÷n÷k÷l;o~,÷k+l;w)} or, equivalently,
P(~'Z)(1--2wX)=-~k~=0
k!
(el+•+n÷
l)kEk(x)
(2.27)
.~(n,÷a~ "i':,~-k p(~+k,n+k) t \n - a/
n (~) (wx) = -~ k=o
(1 -- 2w) } •
"
a
(2.28)
• {1+ 1 F l ( - n + k ; o ~ + k + l ; w ) } or, equivalently,
L(#) (~:)
{
• 1+2F0
(
=
1 n -7 ~
(~Ek
.
(x)
(2.29)
k=0
A(2;-n+2k);
;-~
')}
(2.3o)
or, equivalently,
Hn
( w ) ~
=
1
I,,/2i
x -(l/2)n E
/'n ~ (2k)!~
(-1)k \ 2 k ) --~--! r'k (x)
k=0
• {(2Lo)n-2k +Hn-2k(O))};
(2.311
K.-Y. CHEN et al.
142
(An + #)k Ek (x)
y,~ (wx; (A - 1) n + # + 1, v) = ~
(2.32)
k=O o2
•{l+2Fo(-n+k, An+#+k;
;-~)}
or, equivalently,
Yn (wx; (A - 1) n + # + 1, u) = ~
(An + #)k Ek (x)
(2.33)
k=O
• {1 + Y~-k (w; ( A - 1)n + # + 2k + 1, v)}. Numerous further applications of (2.8) and (2.25), associated with other families of hypergeometric polynomials, can be presented similarly. 3. M U L T I P L I C A T I O N FORMULAS INVOLVING HYPERGEOMETRIC POLYNOMIALS We begin by considering a special case of the polynomials SN(x) defined by (2.1) when An,k = (A + n)g k Ck
(n,k E N0; N E N; A E C),
(3.1)
where {C~}n~__0 is a suitably bounded sequence of essentially arbitrary (real or complex) parameters. In this special case, the definition (2.1) can be rewritten in the form [n/N]
TN ( x ; A ) : :
(--n)gk(A+n)g kckxk
E
(3.2)
k!
k=O
(hE N0; N E N ; A c C ) . Then, in terms also of a sequence {D ,~}n=O,we can derive the following general class of multiplication formulas: oo
ECIDNI
~_~
(xtN)Z l!
l=O
--
z.-. k=O
(--t)k
k!(-A-~'k)k
oa
D k +l
TN (x; A) E k
l-~O
(A+2k+
tl
1), l!
(3.3)
(A e C\Zo; N e N), provided that each member of (3.3) exists. Although the multiplication formula (3.3) can be deduced from several known multivariable expansion formulas (cf., e.g., [16, p. 300, equation (1.4)]; see also [13, p. 277, Problem 25 (i)] and the references cited therein), we choose to present here the outline of a direct proof of (3.3) for the sake of completeness. If, for convenience, we denote the right-hand side of (3.3) by T(N; z, t), and apply the definition (3.2), we obtain
T(N;x,t):=
k[ k=0
" TN (=; A) (A + 2k + 1)t/! ( + )k z=o
= k,l=O j=O E (k - Nj)! (A)2k (A + c¢ =
l,j=O
tl+Nj l! (A + 2 N j +
]2k+l
CjDk+t
Xa..
l! (3.4)
1)1CjD +Nj 2
• 3F2 [ 1 A + N j , A + 2 N j + I + I ;
J"
1 ,
143
Polynomial Expansions and Multiplication Formulas
which, in view of Dixon's summation theorem [17, p. 243, Entry (III.8)], leads us to the left-hand side of (3.3) by observing the well-poised hypergeometric 3F2 series in (3.4) to be equal to the Kronecker delta gl,0 (1 • No). The above method of derivation of (3.3) can be applied mutatis mutandis in order to derive the following confluent case of (3.3) when X i
x ) -~,
t ~---~At,
and
( N • N) :
}AI--*oo oo
E
(xtN)ll! - ~ ~ (-t)k Ugk (x) E Dk+t~.t~
/=0
k=0
(3.5)
/=0
(N • N), provided that each member of (3.5) exists,
ug(x)
being defined by
:=
~--- k--0
(3.6)
"~"
(n•No; N•N). Upon setting Ck = N - 2 N k j = l
j=l
q
j=l
(k•N0; N • N )
(3.7)
(k • No),
(3.8)
j=l
and
1~ (pJ)k I] (~J)k j=l Dk = j=l j=l
j=l
if we use such contracted notations as (for example) (ap) for the array of p parameters Otl, • . . , ~ p ,
A[N; (ap)] for the array of Np parameters aA a 3 + l N' N ""
a3+N-1 N
(j=l,...,p;
N•N),
and so on, we find from the multiplication formulas (3.3) and (3.5) that [ (ap), A IN; (p~)] ;
]
p+N.Fq+Nv L (Zq), " [N; (o-v)] ; xtNN(u-v-2)N (3.9) = E
j=l
j=l
j=l
(--t) k
j=l
144
K.-Y.
[
CHEN
et al.
(~/r)+k'(flu)+k;
]
"r+uFs+v+l
t A+2k+l,(Ss)+k,(av)+k;
A (N;-k),A
+ k),(ap),A IN; (Ss)] ;
(N;A
"p+N(s+2)Fq+Nr
]
xN (s-r)N
(3.9)(cont.)
(flq), A [N; (~r)] ;
p-q
r+u
p-q=N(v-u)+l
and
IxtN[ < N(V-u+2)N; r + u = s + v + 2 and Itl.< 1; N E N ) , and
p+NuFq+Nv [ (OIp) ' A [N; (pu)] ; xtN N(u-v-1)N 1 (flq), A [N; (a.)] ;
Zj=I
=
j=l
(-t)
j=l
[
(7~) + k, (p~) + k; ]
• r+uFs+v
t
(3.10)
(Ss) + k, (Or) + k;
•p+N(s+l)Fq+Nr
[ A (N;-k),(ap) ,A [N; (Ss)] ;
xN (s-r)g
J
(flq), A IN; (~r)] ;
p-q
l, r +u < s + v + l; p - q = N ( v - u ) +
l and
Ixtg[ < N(V-u+l)N; r + u = s + v + 1 and It I < 1; N E N ) , respectively. With a view to applying (3.9) and (3.10) to some of the aforementioned classical orthogonal polynomials, we set N = 1 and
u,
~u + 1
(Pu+l --- - n ; n c No),
so that (3.9) simplifies to the finite form
pTu+lFqTv
'
'
(~), (~o);
xt =
tk
j=l
j=l
~o
(~ + k)~ fl (~J)k fl (~J)~ j=l
j=l
-~+k,(w)+k,(p~)+k; ] " r+u+lFs+v+l
t A + 2k + 1, (Ss) + k, (Zv) + k;
,÷~÷2F~+~[ -k'~ + a'(~')'(~2 (~), ('~);; ] ~
,
(3.11)
Polynomial Expansions and Multiplication Formulas
145
while (3.10) yields
l~I (~J)kl] (p~)k
[-n,(~p),(p~); ] p+u+ l tq+v
xt
(G), (~)
;
~:0
fl (ej)~ fl (oj)~ j=l
j=l
-n + k, ('M + k, (p~)+ k; ]
• r+u+lFswv
(~s) +
k,
(o~) +
[-k,(~p),(~); ] • p-Fs+ltq+r
t k;
(3.12)
J
X
L
(z~),('~,);
which obviously is a confluent case of (3.9) when X x, , ~,
t~--,At,
and
IAI--*oe.
By appealing to the multiplication formula (3.11), S£nchez-Ruiz [181 derived several interesting linearization and connection formulas involving squares of the Gegenbauer (or ultraspherical) polynomials C,~(xt). These classical orthogonal polynomials are known to have many different hypergeometric representations including (for example) the following relatively more familiar one (cf. [2, p. 280, equation 144 (20)]):
"-1n-In+l; x2 ] 2 '
C~(x)= ( 2 v +zn~- 21 F) l n
-!
1 u+~;
J
'
(3.13)
which, in view of the Pfaff-Kummer transformation (cf., e.g., [3, Volume I, p. 64, equation 2.1.4 (22)])
2Fl (a,b;c;z) = (1- z)-a 2Fi (a,c-b;C; z_ ) (larg(1-z)l~-e;
0
(3.14)
c~Zo),
assumes the equivalent form 1 1 C~ (x) = (2u + n - 1 ) 2F1 I --~n,v +-~n; n 1
1 -- X 2
]
.
(3.15)
u+ 2 , Making use of this last hypergeometric representation (3.15) in conjunction with Clausen's identity [3, Volume I, p. 185, equation 4.3 (1)] {
1)} 2= 3F2 ( 2a,2b,a+b;2a+2b, a+b+-~;z 1 ),
(
2F1 a,b;a+b+-~;z
(3.16)
it is easily seen that
I
{C~ (x)}2 = (2u + n - 1 ) 3F2 --n, 2p + n, 1~; 1 - x 21 . 2v, p + ~;
(3.17)
146
K.-Y. CHEN
et al.
Thus, upon setting p=q=2
(~1 = 2 v + n ;
a 2 = v ; ~l=2V; ~ 2 = v + ~ ) ,
( 71 = 2 v + n ; 72 = v; 73 = A+ 2' 1 61
r-l=s=2
u = v = O,
x t
~ bx,
2v; 62
v+ 1)
a
and
t=~,
the multiplication formula (3.11) yields the following linearization formula for the square of the Gegenbauer (or ultraspherical) polynomials:
= (n!)------'Tk=O
• 4F3
(A)2;¢ (u + 1/2)k
- n + k,u + k,2u + n + k,A + k + -~; 1 A + 2 k + 1,2u + k,u + k + 2;
(4-b)
C~2k "
In precisely the same manner, but with different choices for the various parameters involved, we find from (3.11) that
(2 )2o C~n ( ~ )
= (2n)!
2(:)
k=0
(k,) 2(u + {(A)k} 2 (u + 1/2)k
-n+k,u+n+k,2A+k,A+k+-~;
• 4F3
C~ ~
2
1 2A + 2k + 1,A+ k , u + k + 2;
and
{c: • 5F4
}2 (n!)2 k=0
{(A)k} 2 (u + 1/2)k -4-b-
- n + k,2u + n + k,u + k,2A + k,A + k + -~; 1 2A + 2k + l,A + k, 2u + k,u + k + -~;
~
2 '
For a = b = 1 and x ~ 1 - x 2, (3.18)-(3.20) would obviously correspond to the main results of S~nchez-Ruiz [18, p. 265, Theorems 1 and 2; p. 226, Theorem 3]. Both (3.11) and (3.12) appeared recently in the work of S£nchez-Ruiz and Dehesa [19, p. 580, Lemmas 1.1 and 1.2], who applied each of these results to several families of generalized hypergeometric polynomials (see also the various related works cited in [19]). A special case of (3.12) when p -- q = 0 was applied earlier by Srivastava [20, p. A663, equation (4.1)] in order to derive the corresponding multiplication formulas for the classical Jacobi and Laguerre polynomials. For the sake of completeness, we choose to recall here Srivastava's main applications of the multiplication formula (3.12), involving the Jacobi and Laguerre polynomials, as follows (cf. [20, p. A663]; see also [13, pp. 260-262, Problems 3 and 4]): P(~") ( 1 - ( 1 - x)y) =
£(:::)(
k=O
a + fl + n +
)(
A+#+n+k
[ - n + k , a + /~ + n + k + l,A + k + l; •P(k~'u+'~-k) (x) 3F2 A+#+n+k+l,a+k+l;
] y ,
yk (3.21)
Polynomial Expansions and Multiplication Formulas
147
which, in its special case when p = a + j3 - A, yields
_
k
(x)
k=O
-n+k,A+k+l;
• 2F1
L(a) (xy) =
I
] yJ ;
(3.22)
a+k+l;
~,n- k) y
~k (x) 2F1
y ,
k=0
(3.23)
a+k+l;
which, for/~ = a, reduces at once to the following well-known (rather classical) result attributed to Feldheim (1912-1944) [3, Volume II, p. 192, equation 10.12 (40)]:
k=0
In light of some other developments by, for example, [6,7], here we only apply (3.11) to the Jacobi and Bessel polynomials, and also to the generalized Rice polynomials H~(~'z) [4, P, x] defined by (cf., e.g., [13, p. 140, equation 2.6 (13)])
H(c''z)[{,p,x] :=
3F2 ( - n , a + 3 + n + 1,~;a + 1,p;x),
(3.25)
which, for a = /3 = 0, were considered by Rice (1907-1986) [13, p. 140, equation 2.6(14)] and which, for ~ = p, yield the Jacobi polynomials
(3.26)
p(a,e) (x) = H (a'z) [¢, ¢, ~l ( 1 - x ) ] First of all, upon setting
p=q=O,
u=v=l X e----~ ~1
(l-x)
(pl = a + $ + n + l ; 1
and
r-1
al = ~ + 1 ) , =s=0
A ~--* A + t z + 1,
(Ti=A+l)
(3.11) yields the following multiplication formula for the J acobi polynomials:
k=O
[-n+k,A+k+l,a+j3+n+k+l; "~]¢ P(~'")
(x) t k 3F2
a+k+l,A+#+2k+2;
]
(3.27)
$
which, for t = 1, was given also in the aforementioned work of S£nchez-Ruiz and Dehesa [19, p. 581, equation (8)]. Similarly, for the generalized Bessel polynomials defined by (2.13), we obtain
Yn(xt;a,/3) =k=o ~ (nk) (a÷n÷k-2)k
( A + 2 k - 2 ) k -1 ( _ ~ ) k
(3.28)
•yk(x;A,p) 2Fl(-n+k,a+n+k-1;A+2k;#---;) •
148
K.-Y. CHEN et aL
In the case of the generalized Rice polynomials defined by (3.25), (3.11) readily yields the following generalization of the multiplication formula (3.27):
H (a'z) [~,p, xt]= ~ ( : + - k ) ( a + ~ + n + k ) ( A + # + 2 k ) - i k
k=O
M(~,~) [rj,a,x] 5F4
tk
(~)k (~)k
k
(p)k
[-n+k,,~+k+l,a+fl+n+k+l,~+k,a+k;
]
(3.29)
t a + k + l, )~+ # + 2k + 2, rl + k, p + k;
which obviously corresponds to (3.27) when = p,
7? = a,
and
x,
, 1 (1 - z ) .
In view of (3.26), this last multiplication formula (3.29) can be specialized to derive each of the following connection formulas between Jacobi polynomials and the generalized Rice polynomials:
\n - k]
k
-n+k,A+k+l,a+fl+n+k+l,a+k;
]
•Hk(~'") [77,a, x] 4F3
(3.30)
t
a+k+l,A+#+2k+2,~+k; and = k=0
k
k
(-~k-
[-n+k,.X+k+ l , a + / 3 + n + k + l,~+k; ] P(~'") (1 - 2x) 4F3 "~ k
(3.31)
t
a+k+l,A+#+2k+2,p+k;
Analogous connection formulas between Jacobi polynomials and the generalized Bessel polynomials would follow directly from (3.11), and we thus obtain
= k=o k k ] •yk(x;X,#) 2 F 2 ( - n + k , a + ~ + n + k + l ; a + k + l , A + 2 k ; t )
-~.
(3.32)
and
k=0
k
"
"
(3.33)
"'~(~'") (l + ~-~) 3Fl (-n + k,A + k + l,a + n + k-1; A + # + 2k + Some obvious further special cases of these last relationships (3.32) and (3.33) were proven earlier by Popov [6]. More recently, by using a slightly different notation for the generalized Bessel polynomials, the multiplication formula (3.28) was derived in another paper by Popov [7, p. 20, equation (1)]. Since 2F1 (-n + k,a + n + k - 1; ~ + 2k; 1) = ( - 1 ) n-~ (a - )~),~-k ()t)2k (A),~+k (3.34)
(0
Polynomial Expansions and Multiplication Formulas
149
by the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], a special case of (3.28) when t= #
and
x ~-~ ~x
yields
k
k=O •yk ( ~ x ; ) , ,
(A+k-1)n+l
(3.35)
~),
which, for f~ = #, was stated incorrectly by Popov [7, p. 21, equation (3)]. The various multiplication formulas considered in this section can indeed be proven directly by means of such known expansions as (for example) the following familiar analogue of (1.7) and (2.23) [2, p. 262, equation 136 (2)]: n
x,~=n!E(_l)k(n+a ~ a+~+2k+l \ n - k / ( a + ~ + k + 1)~+~
k=O
(1 - 2z) P•("'•)
(8.36)
in the case of the Jacobi polynomials. In the cases of the classical Laguerre and Hermite polynomials, we have [2, p. 207, equation 118 (2)]
k (x)
(3.37)
[n/2] /, n ~ (2k)! H (2xF = Z \ 2 k ) T ~-2~ (x), k=O
(8.38)
xn = n ! E ( - 1 ) k \ n _ k ]
k=O and [2, p. 194, equation 110 (5)]
respectively. More generally, by equating the coefficients of t Nn (n E No; N E N) on both sides of the expansion formulas (3.9) and (3.10), we can show that
"=
]=1
Xn --
7 (~))~ FI (7))N~
\j=l
_
_
(Nn)! E (-1)k k=O
(A Jr- ~)Nn+l (3.39)
j=l
A (N;-k),A
(N;A + k), (ap),A [N; (ha)I;
• p+N(~+2)8+N.
xN(S-r+2) N]
(&), A IN; (~)] ;
and
-
\j=l
,=1 7 (Zj)~ FI (TJ)N,,
j=l
~ : ~ ~ {_l)k (Nn!) k=o
[ A (N; - k ) , (ap), A IN; (6s)] ; xN (s-r+l)N m
p+N(s+l)Fq+Nr
L
(Zq), A [N; (~,)] ;
(3.40)
150
K.-Y. CHEN et aL
respectively. For N = 1 (and r = s = 0), (3.39) and (3.40) immediately yield the following known results recorded by (for example) Prudnikov et al. [21, Volume 2, p. 389, Entry 5.3.2.2; p. 388, Entry 5.3.1.4]:
- k , A + k, (ap) ; =
zn = (3j)
(A + k)n+l p+2Fq
(-1)k
x
(3.41)
k=o
\j=l and =
x~ = E
(--1)k
p+lFq
x
,
(3.42)
k=0
\j=l respectively. These last results (3.41) and (3.42), as well as their q-extensions, were applied recently by Area et al. [22] to various hypergeometric and q-hypergeometric polynomials. 4. F U R T H E R
REMARKS
AND
OBSERVATIONS
For the classical Laguerre polynomials L(~)(x), (3.12) with p=q=O,
r-l=s=0
('h=l+fl),
x,
, bx,
and
u=v-l=0 a t=b
(al = 1 + c ~ ) ,
immediately yields the following multiplication formula (cf. [13, p. 261, Problem 3 Off)]):
L (~) (ax) =
+
L(k~) (bx)
2F1
(4.1)
k=O
a+k+l;
which may be compared also with (3.23) above. As a matter of fact, in view of the limit relationship (2.12), (4.1) can easily be deduced as a confluent case of (3.27) by first setting x,
~1---
2bx
a#
and
/Z
t = =-=,
and then letting m i n {[fl[, I/z[} --* c~.
By appropriately modifying the above limit process, we can obtain connection formulas between Jacobi and Laguerre polynomials as follows. First of all, in (3.27) we let x,
2bx
a#
) 1 - - -/z,
t =Y'
and
Iiz] --~ cxD,
so that we have
k=O
k (4.2)
"(b) k [ - n + k ' A + k + l ' a + f l + n + k + l ; b ] a3 F+1 k + 1;
Polynomial Expansions and Multiplication Formulas
And, upon letting
a
x ~ ~ 1 - 2bx,
t = -~,
and
151
Ifll -~ oc,
(3.27) also yields the connection formula
k=0
a + k + l,A + # + 2k + 2; Each of the results, which we have presented in this paper for the Jacobi polynomials P~(~'z) (x), can indeed be applied in order to derive the corresponding result involving such simpler orthogonal polynomials as theGegenbauer (or ultraspherical) polynomials [cf. equation (3.13) and (3.15)]
c: (x)
/
(\u + nn-
2u +nn - 1 p(u_l/2,u_l/2){x. ~ j,
(4.4)
the Legendre (or ultraspherical) polynomials Pn (x) -- C~/2 (x) = p(O,O) (x) ,
(4.5)
the Chebyshev polynomials of the first kind Tn (x) ---- n
-- ~
P(-1/2'-l/2)(x) = -~nC~ (x),
(4.6)
n
where C ° (x) := lira { v - l c ~ (x)}, lul-~0
(4.7)
and the Chebyshev polynomials of the second kind P(1/~'l/2)(x) u~(x) = 5
= C~ (x).
(4.8)
n+
Next, with a view to applying the multiplication formula (4.1) in the evaluation of integrals involving products of Laguerre polynomials, we recall the familiar orthogonality property [10, p. 100, equation (5.1.1)] fo ° x ~ e - X L ~ ) (x) L (~) (x) dx - F ((~ + n + 1) 5m n n!
(4.9)
(9~(a) > -1; m , n E No), where 6m,n denotes the Kronecker delta. Thus, making use of (4.9) in conjunction with the multiplication formula (4.1), we readily obtain the integral formula o ° z~ e-bX L ~ ) (az) L (~~) (bx) dx =
(~-I-:) _
(fl:n)anF
(fl+l) ( bfl+n+l 2F1 - m + n ,
fl+n+l;t~+n+l;-~
(9~(fl) > - 1 ; m > - n > - O (m, n E N 0 ) ; iR(b) > 0 ) .
a)
(4.10)
K.-Y. CHEN et al.
152
In spite of the fact that much more general integral formulas than (4.10), involving products of two or more Laguerre polynomials, have been considered in the mathematical literature rather extensively (see, for details, [23]; see also equations (4.16) and (4.17) below), we do find it to be worthwhile to mention here the following three special cases of the integral formula (4.10). First, (4.10) with m = n reduces immediately to the form
/o c~ x~e-b~L (~) (ax) n (~) (bz) dx = a'~F (~ + n + 1) n!b~+n+ 1 (~(/~) > -1; n 6 N0; ~ ( b ) > 0).
(4.11)
Second, by means of the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], already referred to in connection with (3.34), a special case of (4.10) when b = a yields the integral formula
/o c~ xZe-aXL~ ) (az) L (~) (az) dx F ( / ~ + n + 1) ( a - f l ) m _ n -
aZ+l
(4.12)
(m - n)!n!
(~R (/3) > -1; m >- n >- 0 (m,n C No); !R(a) > 0). Third, in its special case when/3 = a, (4.10) readily yields the integral formula
/o °~ x~e-b~L~ ) (az) n (~) (bx) dz = r (~ + m + 1) (b - a ) m - ~ a ~
ba+m+l
(4.1a)
(m - n)! n!
(~R(c~)>-l; m > _ n > O (m, n E N 0 ) ; ! R ( b ) > 0 ) . Both (4.11) (for a = b = 1) and (4.12) (for a = 1 and n = m - 1) appeared quite recently in the work of Mavromatis and Alassar [24, p. 904, equation (6); p. 905, equation (11)], using a slightly different notation for the Laguerre polynomials. The integral formula (4.13) (with, of course, c~ -- A, b = c, and a , , b) provides the corrected version of another result recorded by Prudnikov et al. [25, Volume 3, p. 478, Entry 2.19.14.10]. Yet another interesting application of the multiplication formula (4.1) in evaluating integrals involving products of two Laguerre polynomials can easily be presented here. Indeed, by writing (4.1) in the forms
j=o \ m - j ]
¢
2F1
(4.14) a+j+l;
and k=0
-k)
k (cx)
2F1
,
(4.15)
~+k+l; and appealing to the orthogonality property (4.9) once again, we readily arrive at the following integral formula (cf., e.g., [23, p. 312, equation (25)]):
o ° x~e-CXL (a) (ax) L (z) (bx) dx
-- C7+1P(~'-~-l)min(m'n)E(:-~k)(:'~-~k) ("f;k) ~ab~ k k=0 • 2F1
-
\ c'2/] 2F1
a+k+l; ~+k+l; (!R(v) > -1; m , n 6 N0; ~R(c) > 0).
(4.16)
PolynomialExpansionsand MultiplicationFormulas
153
By means of the Chu-Vandermonde theorem [17, p. 243, Entry (III.4)], a special case of (4.16) when c = b leads us to (cf. [25, Volume 3, p. 478, Entry 2.19.14.8]; see also [23, p. 307, equation (8)])
~oc~ x'le-b~L(ma) --
r(~q-1) rain(re'n)(: b'r+l
Z
+
-
.2F1 ( - m + k , ~ + k
(ax) n (~) (bx) dx
k)(
V
; )
k ( f l - V)n-k
k
(41 )
°)
+ l ; a + k + l; ~
(!R(y) > -1; m , n e No; !}l(b) > 0), which, for 7 =/3, immediately yields (4.10). Furthermore, upon setting a = b = c in the integral formula (4.16), we similarly obtain the following mild generalization of (4.12):
fo ~ z're-a~L(~ ) (ax) L ~ ) (ax) dx +k k ) (~ - ~ ) ~ _ ~ ( z - ~)~_~
F (m~ i++1 nl () m ' ~ )~( a ~
(4.18)
(fit (~,) > -1; m , n E No; !R(a) > 0), which, for V = fl, reduces at once to (4.12). Finally, in view of the orthogonality property for the Jacobi polynomials [10, p. 68, equation (4.3.3)]:
L1
1 (1 - z) ~ (1 +
=
X)~ Ptm~'~) (x) p(~,Z) (z) dx
2~+Z+IF (a + n + 1) F (/3 + n + 1) n! (a +/3 + 2n + 1 ) r ( a +/3 + n + 1) & ' "
(4.19)
(min{ER(a),9t(/3)} > -1; m , n E No), the above method of derivation of the integral formula (4.16) can be applied mutatis mutandis in order to find from the multiplication formula (3.27) that
_1 (1 -
X) A
(1 + x)" P(ma'~) (1 - (1 - x) u) P(~P'°) (1 - (1 - x) v) dx
1 ----- 2A+/~+1
min(ra'n~,rr)(?T~' ~, ---I-O:~ ~] (:-F-- ~) (°~-t-/3-F~'rl'-Fk) k (P-t-G-F~-Fk)F~-)( k=O
-m+k,A+k+l,a+/3+m+k+l; . (uv) k B ()~ + k + 1, p + k + 1) 3/?2
u
(4.20)
a+k+l,A+#+2k+l;
•3F2
v p+k+l,X+.+2k+l;
(min{!tl()~),91(#)} > -1; m , n e No), where B (c~,/3) denotes the familiar Beta function defined by B (~,/3) . - r ( ~ ) r ( / 3 ) - : B ( / 3 , ~ ) . r (~ +/3)
(42~)
In fact, by virtue of the limit relationship (2.12), it is not difficult to deduce the integral formula (4.16) as a confluent case of (4.20).
154
K.-Y. CHEN et al.
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