Polynomials satisfying a binomial theorem

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

APPLICATIONS

32, 543-558 (1970)

Polynomials Satisfying a Binomial Theorem* B. C. CARLSON Departments of Mathematics and Physics, Iowa State University,

Ames, Iowa

Submitted by R. P. Boas, Jr.

1. INTRODUCTION In 1880 Appell [1] discussed sequences of polynomials p,(z), pr(.z), p,(z),... having the properties that p, is of exact degree n and

dPn

(?a= 1, 2, 3)...).

-&- = nPn-l>

Any such sequence is now called an Appell set ([5] Vol. 3, p. 235), the prototype being the sequence 1, .a, z2,... . Some authors omit the factor n on the right side of (1.1); although it is only a matter of normalization, we retain it so that p, will be an analog of zn rather than z”/n!. The members of an Appell set have the form p,(z)

= i am (i) m=o

2+-m,

(n = 0, 1,2 )... ),

(1.2)

where ($) is a binomial coefficient and allz = pm(O). Conversely, given any constant coefficients 01s, 01~, 01~,..., with 01s# 0, the polynomials (1.2) form an Appell set. More generally, for any choice of .zO, an Appell set satisfies

If h(t) is any formal power series in t such that h(O) # 0, the polynomials generated by h(t) et2 = go

PA4 $

(1.4)

* Work performed in the Ames Laboratory of the U. S. Atomic Energy Commission.

543

544

CARLSON

form an Appell set, as one readily verifies by differentiating with respect to z. Many of the classical sets of polynomials have these properties (see Section 3). An extensive bibliography is given by Scaravelli [IO]. The notion of an Appell set has been generalized in several directions. Multiple sequences of polynomials in two or more variables were discussed in Appell’s original paper. A class of generating relations more general than (1.4) has been considered extensively (see the bibliography in Boas and Buck [2]), and the resulting sets ( pJ are sometimes called generalized Appell sets. The extension of the differential recurrence relation (1 .l) to nonintegral values of 71has been studied by Truesdell [8], who gives references to earlier authors as well as historical notes in an appendix. In the present paper we shall consider a generalization to several variables which is quite distinct from the multiple sequences proposed by Appell. We emphasize the functional relation (1.3) and do not assume p, to be of degree n. After showing that Eqs. (1. I)-(1.3) have simple analogues in several variables, we shall prove an extension of (1.4) (see Theorem 3 and Corollary 4) which appears to be new even in the case of a single variable. Section 3 contains a list of examples of Appell sets, some well known and some not. Most are in one variable, but certain of these are illuminated by a relation between the case of one variable and the case of several. Examples 8, 9, 10, and 17 show that Rainville’s formula for Legendre polynomials and its various generalizations are equivalent to binomial theorems. The last section of the paper contains a composition theorem by which Appell sets can be combined to form new Appell sets in one or several variables.

2. THE CASE OF SEVERAL VARIABLES A way of defining an Appell set of polynomials in several variables is suggested by replacing x and x0 in (1.3) by z + h and z, respectively:

p,(a+ A)= & ( 1) P,(Z) xn-,

(n = 0, 1722-**1.

(2.1)

Since this reduces to the ordinary binomial theorem when p,(z) = zn, we shall call it a binomial theorem satisfied by the sequence { p,}.

p,>

DEFINITION 1. Let p = { = ( p, , p, , pa ,...} be a sequence of polynomials in the variables z1 , z2 ,..., zk, where k is any positive integer. The sequence is said to satisfy a binomial theorem if and only if the equation

POLYNOMIALS

SATISFYING

A BINOMIAL

545

THEOREM

holds for all complex values of .zl , z2 ,..., zlc, h and for every n = 0, 1,2 ,... . The collection of all sequences in k variables having this property will be denoted by A,. For brevity we shall write z + h = (zl + h, za + A,..., xk + h), so that (2.1) and (2.2) are henceforth the same. It should be noted that p,(z) is not required to be of exact degree n; for instance, (0, 1,2x, 3.z2,...}E A, although this sequence is not an Appell set according to usual terminology. In the sense that a polynomial in one variable is trivially a polynomial in two variables, A, C A, if K < m. If (pn} E A, and (qn} E A, , it is plain from (2.2) that A, contains also the sequence {cup, + /3qn}, where Q and fl are any complex numbers, and the sequence {8p,/&,} for each i = 1,2,..., K. THEOREM

1. The sequence { pn} is in A, if and only ;f

2 $$ = nP,-l,

(n = 0, 1,2,...),

(2.3)

where the right side is taken to be zero if n = 0. Proof.

Assume (2.3) to be true. Writing D = CFx, a/&

D”Pn =

(n Yrn!)

P,-nz 3

, we have

(m, n = 0, 1, 2 ,... ),

where the right-hand side is taken to be zero if m > n. If we define g,(h) = p,(z + /\) for fixed z, then ~dnzgn(4 = Dmp,(z + A). d/l”’

Hence the expansion of g, in powers of X is

g,(4 = i Drn~nC4$ >

(n = 0, 1,2 ,... ),

m=O

or

P,(z +4=mio (E)P,-&) A”,(n=0,1,

L.),

which differs from (2.1) only by a change of summation index. Conversely, if the last equation is valid, we differentiate it once with respect to h and put X = 0 to obtain (2.3).

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Theorem 1 shows that the degree of p, cannot exceed n if {pa} E A, , for in this casep, must be a constant; if p, = 0 the degree of p, is less than n. In the case of two or more variables, the degree of p, is unrestricted; an example in which it exceeds n is furnished by the sequence W”(x

- YXX + Y)“) = {P&Y YN E A2 .

COROLLARY 1. (9 If CPJ E Ak , then {Y,> 6 Ak , where sd.4 = a-" P,(W) and wi = ol.zi + pi, (i = 1, 2 ,..., k). The constants ac, & , /$ ,..., Brc may be any complex numbers provided o( # 0. (ii) If { pn} E A, , then ((k) p,-,} E A, for each m = 0, 1, 2 ,.,., where (z) p,-, is taken to be zero if n < m. Proof.

(i) DcI~(Z) = O1l--nnp,-l(w) = q,-,(z). (ii) The sequence

is seen to be in A, by differentiating (2.2) m times. The following theorem will be useful because it allows one to conclude that a given sequence is in A, if it satisfies a binomial theorem for a single value of zi . If k > 1 any one of the x’s would serve equally well in place of z1 . THEOREM 2. If (2.2) holdsfor some value of z, , all values of x2 , xS ,..., xk , h, and every n = 0, I,2 ,..., then { pn} E A, . Proof. We assume that (2.2) is valid for every n = 0, 1, 2 ,..., if z1 = 5, where 5 is some fixed number:

p,(t + A, z2 + L,

zk

+

h)

=

m$o

(i)

Prn(t,

X2 ,...,

xk)

h”-“.

(2.4)

Since xs , z, ,..., xk , h are unrestricted, we may replace zj by xj - z1 + 5, (j = 2, 3,..., n), and h by A + z1 - 5 for any value of z1 . Then

for all values of z1 , z2 ,..., zk , A. Substituting

547

POLYNOMIALS SATISFYING A BINOMIAL THEOREM

and rearranging the order of summation, we have

p,(x + A)= i (p

i

.S=O

(;)P,(L%

-

21 +

L...,-%

-.

.u-

5)

nZ=O

- (x1 - ,),-? By (2.4) the inner sum is p,(.z, , z2 ,..., z,), and hence

P&+4=to(:)P&4 x-3

(n = 0, 1,2 )...),

for all x1 , z2 ,..., zk , A. COROLLARY 2.

If

the sequence { pn} satisfies

for some value of 5, all values of z1 , x2 ,..., zk and every n = 0, 1, 2 ,..., then { ~4 E 4. Proof. In (2.5) we replace x1 by 5 + h and zi by xj -+ A, (j = 2, 3,,.., k), and apply Theorem 2. Choosing 5 = 0, for example, we deduce from Corollary 2 that the validity of

= 5 ( ;) ~~(0, x2 - x1 ,..., xk - x1) x;-“, m=o

(n = 0, 1,2 ,... ), (2.6)

for all values of zr , z2 ,..., xk is a necessary and sufficient condition that 1. When K = 1, this result implies a one-to-one correspondence between the sequences 144 = i P&N in 4 and the sequences {an) of complex numbers. For if we specify&(O) = 01,, (n = 0, I,2 ,... ), an element of A, is uniquely determined by { PJ E A, 3 necessity being an immediate consequence of Definition

p,(x)

=

i

p,(O)(E)

xn-m,

(n = 0, 1,2 ,... ).

m=0

Examples of the correspondence p(O) f-f p(x) are (1, 0, 0,O ,... } f-) (1, z, z2, z3,...} = e(l)(z), (0, 1, 0,O ,...} t) (0, 1,22, 3z2,... } = et2)(z),

(2.7)

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CARLSON

and more generally ecs)(0) = {a,,} f+ I( :) z?-~/ = e(@(z),

(s = 0, 1, 2 ,... ),

(2.8)

where a,, is the Kronecker delta and (y) is taken to be zero if s > n. The sequences ecS)furnish a basis in the linear vector space A,, and p t A, if and only if PC4 = ,fo Pm(O)@Y4,

(2.9)

for (2.9) is equivalent to the system (2.7). If K > 1 there is exactly one element of Ak for each sequence of arbitrarily chosen polynomials in k - 1 variables, say (01,(xi , xa ,..., S-J}, because a sequence { pn(zl , xa ,..., zJ} in A, is uniquely determined by (2.6) if we specify P&~l

9 x2 ,..*, %-I)

=

%(%

, x2 3*.*, %-A

(n = 0, 1, 2 )...).

THEOREM 3. Let f (t) be any formalpower series in t (convergent OY divergent) and let f en)(t) denote the formal power series obtained by dzyerentiating f term by term n times with respect to t. If { pn} E A, and ;f h and TVare any complex numbers, then

(The equality sign is not intended to imply convergence but only the equivalence of formal series.) Conversely, if (2.10) holds for p = 0, some value of z1 , all values of .z2 , z3 ,..., zk , X, and some f such that f (“J(0) # 0, (n = 0, 1,2 ,... ), then 1 PJ E A, . Proof.

If (p,J E A, , then

POLYNOMIALS SATISFYING A BINOMIAL THEOREM

549

The symmetry of the last expression in h and /.Limplies (2.10). (It implies no more, as one sees by assuming (2.10) to be true and replacing z by z - p and h by X + CL.) To prove the converse, we put p = 0 in (2.10), differentiate r times with respect to t, and put t = 0. Applying Leibnitz’ rule to g(t) = j’(“)(U) ~“/Kz! with h fixed, we have

and

Provided that f(r)(O) # 0, (r = 0, 1,2,...), it follows that

pr(.z+ A) = f: (i) p&g A’-“,

(r = 0, 1,2 ,**.),

?I=0

for some value of zr and all values of zu , za ,..., zk , h, whence { p,J E A, by Theorem 2. COROLLARY 3.

If f (t) is any formal power series and if ( p,J E Ak , then

Conversely, if (2.11) h oId s ( as an equivalence between formal series) for all values ,-**, zk and for some f such that f (la)(O) # 0, (n = 0, 1,2,...), then

of 21 9 z2 {PJ ~4.

Proof. In Theorem 3 put p = zi = 0. Then replace zj by zj - zi , (j = 2,3 ,..., k), and X by .zi . Since the case K = 1 has special interest, we restate Corollary 3 for this case, recalling from (2.7) that an arbitrary sequence of constants {p,(O)} determines uniquely a sequence { pn(z)) in A, . COROLLARY 4.

Iff(t)

is any formal power series in t and if { pn} E A, , then

jJo Pn(O)f (“W) 5 = go Pn(4f YO) $ * Conversely, if {pn(0)} is an arbitrary power series such that f cn)(0) f 0, generated by (2.12) is in A, .

(2.12)

sequence of constants and if f is a formal

(n = 0, 1, 2,...), then the sequence (p,(z)}

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CARLSON

By choosing f to be the formal series development of et, 1 - log(1 - t), and (1 - t)-c in turn (c being any complex number), we find the following special cases of (2.12): (2.13)

-p, log(l - tz) + f p&o;

(j&J”

= alp&)

f

?l=l

(1- tx>-cf p,(O)9 (j&g), = ZoP.(49 tn,

5

(2.14)

(2.15)

?l=O

where (c)~ = 1 and (c)~ = c(c + l)(c + 2) ... (c + n - 1) if n 3 1. Given any sequence of constants (p,(O)}, we have proved that the unique sequence (p,(z)} obtained from all three generating relations is in A, , and conversely that any sequence in A, satisfies all three relations. In particular, (2.13) is the same as (1.4) if p, # 0. The equivalent of (2.15) has been given previously by Rainville (see Section 3, Example 16).

3. EXAMPLES Many of the classical polynomials form Appell sets. The simpler examples are well known and can usually be verified by pointing to a generating relation of the form (2.13) or (2.15); some of the more complicated examples do not seem to be well known as Appell sets. We keep to the notation of the chapter on generating functions in the Bateman series ([5] Chap. 19). The sequences in Examples 1-17 are elements of A, and in most cases are also Appell sets in the strict sense that p, is of exact degree n. EXAMPLE 1. The

monomials ((r) .P-~} for each s = 0, 1, 2,... (see

WN. EXAMPLE 2. The polynomials {(f) x+~ CA=, &P/(1 + n - s),~}, for each s = 0, I,2 ,... and for any complex constants PO, /3r ,..., /3, . The quantity (y)/(l + n - s)~ is to be interpreted as n!/s!(n - s + m)! if n < s, but it is left unsimplified here for easier comparison with Ex. 14. By taking s - m as a new summation index, the polynomials can be put in the form (1.2). EXAMPLE 3. The Bernoulli polynomials (B,(z)} izations thereof ([5] Vol. 3, p. 253).

and certain general-

POLYNOMIALS SATISFYING A BINOMIAL THEOREM

551

EXAMPLE 4. The Euler polynomials {E,(z)} and certain generalizations thereof ([5] Vol. 3, p. 253). EXAMPLE 5. The Hermite polynomials {(2or)-” H,(olz)} for any complex 01# 0. We make use here of Corollary 1, for two choices of Hermite polynomials in common use are H,(z) and He,(z) = 2-n~zH,(2-1Fz). Both are Appell sets. Only under special conditions is -C-“K(4) and VGG4 (2.1) compatible with the three-term recurrence relation which is necessary and sufficient ([6] p. 153) for orthogonality. The conditions are easily found to be such that Hermite polynomials can be characterized as follows: if p,(z) has exact degree n in the single variable a, and if (p,(z)} is both an Appell set and an orthogonal set, then p,(z) = y(2a)-” H, (az+ /3)for some constants 01,p, y. A different proof of this result was given by Shohat [7]. EXAMPLE 6. Three sequences of Laguerre polynomials: (n! ~~L,~(l/z)/ (1 + a)%},where 01# -1, -2, -3,...; ((-l)n n!L”,-%(z)}; and {x+~L~-“(z)} for each s = 0, 1,2 ,... . The third sequence is essentially the special case 01= s of the second sequence, for ( -z)-s s! L:-“(z) = ,F,( -s, --n; - 1/x) is symmetric in s and n. The three sequences are special cases of Exs. 12-14. EXAMPLE 7.

The Chebyshev polynomials

((22- c?)nPT&/(z2- cY2)‘/2]} = {gz + Lx)”+ $(z- a)“}, for any complex 0~. EXAMPLE 8. The Legendre polynomials {(.~a- G>n12Pn[z/(z2 - a2)li2]}, for any complex 01.This sequence is a special case of Ex. 9. Its binomial theorem is equivalent to a formula ([6] p. 168) discovered by Rainville in 1945. Putting 01~= - 1 and z = cot j3, and applying the binomial theorem (2.1) with h = cot y - cot /3, we find

P,(cos y) = (e&)”

f

(Z) p&

~qn-mPm(cosp).

(3.1)

WL=O

It does not appear to have been observed previously that Rainville’s formula is a binomial theorem in disguise, although Carlitz [9] has formulated it as a multiplication theorem. Similar formulas are known for the Appell sets of Exs. 7, 9, 10, and 17. EXAMPLE 9. The associated Legendre functions ((x2 - a2)(n-~)/2P&, [z/(z2 - ~~)l/~]} and {(-l)n(z2 - ~~)o-~)/~ P~z/Lx)/(-2v),},where 01is any nonzero complex number, 2~ # 1,2, 3,..., and 2v # 0, 1, 2,... . We use

552

CARLSON

Hobson’s definition of Pvu. The two sequences are equivalent to each other and to the sequences of Ex. 10 except for changes of notation and multiplicative constants independent of n. EXAMPLE 10. The Gegenbauer polynomials

{(z” - ay

Cn”[z/(z2 - a2yq n!/(24}

= (( --(u/2y C;‘“-“-“(z/~)

n!/(v + 8),>,

where 2v # 0, -1, -2,... and 01is any complex number. This sequence is discussed following Ex. 19. The extension of Rainville’s formula (see Ex. 8) to the Gegenbauer polynomials is due to Car&. EXAMPLE 11. The Jacobi polynomials {Pf-“*B-“)(z) n! (-2)“/( ---a - &}, where 01+ /3 # 0, 1,2 ,... . See the discussion following Theorem 4 below. This is not an orthogonal sequence because the superscripts depend on n. EXAMPLE 12. The reversed hypergeometric polynomials {p,(x)}

defined

by

wherep, Q = 0, 1,2 ,... and ,& # 0, --I, -2 ,... (1 ,< i ,< 4). The polynomial p, has the form (1.2). EXAMPLE 13. The sequence {qn(z)} defined by

fh(4 = C-1)nhl+a+1) .9+1

FP

[

(1 -&>?a

..* (1 -P*,>n

(1 -

*** (1 -

42

%)?a

-72, 01~- n,..., LYE- n p1 - n,..., & - n ’ ’ I ’

where p, q = 0, I,2 ,... and c+ # 1,2, 3,... (I < i < p). Ex. 13 is obtained from Ex. 12 by applying the easily verified reversal formula Pi-1F * [

-n, aI ,..., a9 p &3 1 9”‘) '9+1

I =(-4” F9

[

(4i *** b,), ca>n ... RJ?a

-72, 1 - & - n,..., 1 -& - n (-l)p+* ;~ z 1 - a1 - n,..., 1 - 01~- n

1. (3.2)

POLYNOMIALS

SATISFYING

A BINOMIAL

THEOREM

553

EXAMPLE 14. The sequence (Y&Z)> defined by

wherep, q, s = 0, 1,2 ,... and fii + 0, - 1, -2 ,.,. (1 < i < q). This sequence is a special case of Ex. 2. EXAMPLE

15. The sequence {s&)} defined by

wherep, q = 0, I,2 ,... and fii # 0, -1, -2 ,... (1 < i < q). The polynomial S, can be reduced to the form (1.2) by using the identity (+r/2)k(1/2

- n/2),, = 2F(--n),,

= (-2)-a”

n!/(n - 2k)!

and taking m = 2k as a new summation index. The Hermite polynomials of Ex. 5 can be obtained as the special casep = q = 0 by using Corollary 1. EXAMPLE 16. Rainville [6, pp. 132, 1451 has discussed polynomials a,(z) defined by the generating relation et+(&) = CL, u,(z) tn, where 4(t) is analytic at t = 0. As Rainville observed, a,(x) is a reversed Appell polynomial; replacing z by 1/z and t by tz, defining p,(z) = n! ~“a,( l/z), and comparing with (2.13), we find {p,(z)} E A, . The equivalent of (2.15) is given in Rainville’s Theorem 46, and the binomial theorem for p, becomes

U&Z)

=

f

Xm(l - h)“+ a,(z)/(n - m)! .

m=O EXAMPLE 17. The polynomials F,(x) defined by the generating relation ezt#[t2(x2 - l)] = Czz,,F,(x) t”/n!, where $(t) is analytic at t = 0, were

used recently by Chaudhuri [4] to obtain a generalization of Rainville’s formula (see Ex. 8). Putting x = x/(z2 - a2)lj2 and replacing t by t(z2 - 012)1/a, where (Yis any complex number, we have e”$G(a”t”) = Czdpa(z) P/n!, where p,(z) = (z” - o~~)~/~F,[z/(z~- a”)‘/“]. It follows from (2.13) that {p,(z)} E A,, and the binomial theorem (2.1) can be rewritten in terms of F, as a Rainville formula ([4] Eq. (3.1)) just as in Ex. 8. The equivalent of (2.15) for F, is given by Rainville ([6] p. 186). EXAMPLE 18. The elementary symmetric functions &(x1 , x2 ,..., zk) defined by nfz, (t - zi) = C”,;=, (-1)n J?&(Z)tkmn are (except for normal-

554

CARLSON

ization) the first k + 1 members of a sequence {(t)-’ E,(z)} E A, . This sequence is a special case of Ex. 19, and those elements of the sequence with n > R are defined by putting bi = -1 - E, (1 < i < K), in Ex. 19 and letting E tend to zero. For example the normalized elementary symmetric functions of two variables are the first three members of the sequence

i

T

(xn + y”) + ; (Ply

+ xy”-‘)

1 E A, .

EXAMPLE 19. The hypergeometric R-polynomials generated by

fi (1 - tzi)-bi = 2 (P/n!)(b, + *.* + b,& R,(b, )...) b, ; zi )...) Zk) TL=O (3.3)

i=l

are easily shown to satisfy Cr=, D&(b, z) = nR,-,(b, x). Theorem 1, {R,(b, z)} E A, . The p oly nomial R,(b, z) is the a = --n of the hypergeometric R-function [3]; that is, R,(b, x) = It follows from (3.3) and the generating relation for the polynomials (7%~that

Hence, by special case R(-KZ, b, z). Gegenbauer

(3.4) Appropriate choice of x and y gives the polynomials of Ex. 10, (9 - c2y

Cn+f/(z” - cy]

It is obvious from Definition

n!/(2&

= R,(v, u; x + 01,z - cd).

1 that (R,(v, V; zi , z2>E A, implies

VW, v; x + 01,.z - 4) E A, , and thus the sequence of Ex. 10 (and hence also Exs. 8 and 9) is shown to be in A, by relating it to a sequence which is known to be in A, . This method is so useful that we state formally THEOREM 4. P&l

If{p~(z1,x2,...,zm))

3 3 Y-*.9zk>

=

%dzl,

x2

E A, and ,-**, xk-l

, zk

+

% zk

+

b*-,

xk

+

r>v

where k < m and a, /3,..., y are constants, then {p,} E A, .

The proof is obvious from Definition 1. Another application of the theorem, again with m = 2 and k = 1, consists in showing the Appell character of the Jacobi polynomials (Ex. ll), which are related to the Rpolynomials by ppGJ-“‘(z)

= ‘-“?-~

- 8111R,(-

a, -p; z + 1, x - 1).

(3.5)

POLYNOMIALS SATISFYING A BINOMIAL THEOREM

A third application, not connected with R-polynomials, Section 4.

555

will be found in

EXAMPLE 20. Let u = (ui , ua ,..., ule) satisfy Et=, ui = 1, and let P(u) be any function such that SEP(u) du, du, ..’ dukwl exists, where E is some bounded domain in (k - 1)-dimensional Euclidean space. Then the polynomials p,(z) = SE(Cf=, uizJn P(u) du, du, ... du,-, satisfy {p,} f A, . The polynomials R,(b, a) of Ex. 19 are a special case according to ([3] Eq. (7.10)), provided that Re bi > 0, (; = I,2 ,..., K). EXAMPLE 21. The polynomials defined in terms of Appell’s function F, ([S] Vol. 1, p. 230) by the equation p,(x, y, a) = xnFz(-n, /3,p’, y, y’; 1 - x/z, 1 - y/z) satisfy (p,(x, y, z)} E A, . A similar result in any number of variables holds for Lauricella’s function FA . EXAMPLE 22. Let za ,..., zk be fixed and let R denote the hypergeometric R-function [3]. --n, P + n, b3 ,..., 4 ; X/Y, Then P&G Y) = Y"%; z,J satisfies {p,(x, y)} E A, . This result follows from a single1, 23 ,.**, integral representation of R ([3] Eq. (7.2)). Like Ex. 19, it can be stated alternatively in terms of Lauricella’s function F, , which includes Appell’s Fl as a special case. EXAMPLE 23. Unlike Fl and F2 , Appell’s functions F3 and F4 ([SJ Vol. 1, p. 230) do not appear to have binomial theorems such as those of Exs. 19 and 21, in which no variables are held fixed. For fixed y, however, p,(x, z) = znF3(-n, a’, /z$/3’, y; 1 - x/x, y) satisfies (p,(x, a)} E A, ; and for fixed X, satisfies 44 Y, 4 = ,99-3@, --n, 8,8', Y; x, 1 - r/4 MY, 41 E A, . Sequences in A, can be constructed similarly from Lauricella’s function FB .

4. A COMPOSITION THEOREM Before stating the theorem, we illustrate it by two special cases. The polynomials of a sequence {p,} E A, necessarily have the form (2.7). If we replace P--~ in (2.7) by CJ~+(Z), where (qn} E A, , the resulting polynomials (4.1) satisfy {Q%} E A, . We consider next {2-n(x + y>“} E A, . On the right side of 2-“(x +yp

= 2-n 5 (1) W&=0

40913213-7

xmyn--m

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CARLSON

we replace a? by p,(x) and yn-m by pnWm( y), where {p,} E A, and {qn} E A, , thus forming

Q&Y) Then

l&J

( ;) P&> qn-m(y).

= 2-” i.

(4.2)

whence it follows by Theorem 4 that the polynomials

E A,,

P&4 = Q& 4 =

2”

i ( ;) P&) qn-m(4

(4.3)

m=o

satisfy {P,) E A, . THEOREM 5. the form

Assume that {p,} E A,

P&l

3z2 ,a.*, 4

= c Y&,

and write

each polynomial

p,

in

s,..., t) zlr~2* *** %ct,

where ylz(r, s,..., t) is a numerical coej&nt and C denotes a finite summation over the nonnegative integers r, s,..., t. For each i = 1,2 ,..., k, let (q$} be a sequence of polynomials in the variables w1 , w2 ,..., w, such that {qg)} E A,,, . (The polynomials of a particular sequence may possibly depend on only a proper subset of the w’s, and the subset in question may d#er from one sequence to another.) If Qn(wl , w2,...,w,) is dejned by Q&>

= c y,(r, s,.-, t) 41”(w) q;‘(w)

0-eq1”‘(w),

(n = 0, 1, 2 )...),

then {QJ E A, Proof.

By Theorem 1, {p,} E A, implies

Comparing coefficients of zIrzz* a**zkt, we find w&r,

s,..., t) = (r + 1) m(r + 1, s,..., t) + (s + 1) y&, s + L..., t> + *.* + (t + 1) b(r, s,..., t + 1).

(4.4)

Again by Theorem 1, {q$} E A, implies %:I c $

= C YJr, s,..., t)[rqi!Iqi2)

e-e4;“) + . . . + tq;lJq;)

. . . qit;~.

POLYNOMIALS

SATISFYING

A BINOMIAL

THEOREM

551

In each term we make a unit shift in one summation index to obtain

“aQ, c ami = c [m(r + 1, s,..., t)(y + 1) + *** i=l + Y&Y SY.9 t + l)(t + l)] q$l’qF’ .** qik’. Because of (4.4), we now have

whence {Qn} E A, by Theorem 1. As an application of Theorem 5, let Y be any fixed positive integer and define

P,(x, y) = (x - y)’ xn = to (j x”+y-yp,

(n = 0, 1,2 ,... ).

It is obvious that {P,} E A,. If {qn(x)} E A, it follows from Theorem 5 that

Q&

Y) = to (1) qn+M-YY-’

satisfies {Qn} E A, and

satisfies {p,} E A, . The last equation exhibits the relation, verified by another method in [lo], between the Appell sets {p,(x)} and {qJx)} satisfying p,(O) = qn+r(0) for every n = 0, 1,2 ,... . The present proof illustrates how properties of Appell sets in one variable can sometimes be established or understood more easily by using Appell sets in several variables.

REFRRBNCES

1. P. APPELL, Sur une classe de polynomes, Ann. Sci. &ok Norm. Sup. (2) 9 (1880), 119-144. 2. R. P. Bogs AND R. C. BUCK, “Polynomial Expansions of Analytic Functions,” Springer, Berlin, 1958. 3. B. C. CARLSON, Lauricella’s hypergeometric function FD , 1. Math. Anal. Appl. 7 (1963), 452-470. 4. J. CHAUDHURI, On the generalization of a formula of Rainville, Proc. Amer. Math. Sot. 17 (1966), 552-556. 5. A. ERDBLYI, W. MAGNUS, F. OBERHETTINGER, AND F. G. TRICOMI, “Higher Transcendental Functions,” McGraw-Hill, New York, 1953.

558

CARLSON

6. E. D. RAINVILLE, “Special Functions,” Macmillan, New York, 1960. 7. J. SHOHAT, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. /. Math. 58 (1936), 453-464. 8. C. A. TRUESDELL, “An Essay Toward a Unified Theory of Special Functions,” Princeton University Press, Princeton, 1948. formulas, Rend. Sem. Mat. Univ. Padova 32 9. L. CARLITZ, Some multiplication (1962), 239-242. 10. C. SCARAVELLI, Su i polinomi di Appell, Riv. Mat. Univ. Parma (2) 6 (1965), 103-l 16.