Bilinear generating functions for Laguerre and Lauricella polynomials in several variables

(1)

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

L. C ^ARLITZ

Bilinear generating functions for Laguerre and Lauricella polynomials in several variables

Rendiconti del Seminario Matematico della Università di Padova, tome 43 (1970), p. 269-276

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AND LAURICELLA POLYNOMIALS IN SEVERAL VARIABLES

L. CARLITZ

*)

1.

Erdélyi [ 1 ]

^hasdefined the

Laguerre polynomial

^{in k}^variables

by means

^of ^.

This is

equivalent

^to

He obtained the

following

^bilinear

generating

^function.

*) Supported ⁱⁿpart by ^NSFgrant ^GP-7855.

Ind. dell’A.: Depart. ^ofMathematics, ^DukeUniversity, Durham, North Caro-

lina, ^U.S.A.

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270

where

and

(There

^is^a

slight

^error^{in the}definition of

(u;

x,

y)

^as

given

ⁱⁿ

[ 1 ] ).

Fork =1, (1.3)

^reduces^to^the

Hardy-Hille

^formula

[2,

p.

101]:

The

object

^{of the}

present

paper is ^to

give

^an

elementary proof

^of

(1.3)

^andindeed of the

following

^more

general

^bilinear^formula.

where

(4)

For

k=1, (1.7)

reduces ^to

a formula due to Weisner

[ 3 ] .

2. It will be convenient to define

where the summation is ower all

nonnegative

n; , r; , s; such that

Thus O

(n;

r,

s)

^is

homogeneous

^of

weight n

ⁱⁿ^{ui ,}^...,^{uk ,}^of

weight

r in _{xi ,}..., xk and of

weight

^sⁱⁿ^yi,^...,yk .

Alternatively

^wemay define

4)(n;

r,

s) by

^means^of

where

U, X, Y,

^{W are}^defined

by (1.4).

We rewrite

(1.3)

ⁱⁿ^the^form

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272

Making

^use^of

(1.2)

^and

(2.2),

^{it is}

easily

^seen^{that the}left hand side of

(2.4)

^is

equal

^to

Hence that

part

^ofthe left hand side of

(2.4)

^that^is^of

weight

^rⁱⁿ

XI ..., xk and of

weight

^sⁱⁿyi , ^...,Yk is

equal

^to

We set the

right

hand side of

(2.4) equal

^to

where

~(n;

^r,

^s)

^is

homogeneous

^of

degree n

ⁱⁿ^{ui ,}^...,uk , of

weight

r in xi , ..., xk and of

weight

^sⁱⁿyi, ..., yk . Then

clearly ^(2.4)

^is

equiv-

alent to

The

right

hand side of

(2.4)

^is

equal

^to

Since

( 1- U) W -I- XY

is of

weight

ôneⁱⁿ^{xl , ...,}^xkândôf

weight

^oneⁱⁿ

yi , ..., yk , it follows from

(2.9)

^that

(6)

In the next

place, ^(2.8)

^is

equivalent

^to

But, by (2.1 )

^and

(2.2),

On the other

hand, by (2.10),

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274

so that

In view of

(2.12)

^and

(2.13), (2.11)

^issatisfied. This

completes

the

proof

^of

^(2.4)

and therefore of

(1.3).

3. We shall now prove

(1.7). By (1.8)

the left hand side of

(1.7)

is

equal

^to

where

~(n;

r,

s)

^is^defined

by (2.1).

Now

by ^(2.8)

^we^have

(8)

where

~’(n;

^r,

^s)

^isthe result of

replacing by a

ⁱⁿ

^T(n;

^r,

^s).

Consequently

the left member of

(1.7)

^is

equal

^to

Then, by ^(2.10),

^this^sum^is

equal

^to

This

completes

^the

proof

^of

^(1.7).

4.

Exactly

^asⁱⁿ

proving (1.7),

^we^can^establish^the

following

more

general

^result.

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276

REFERENCES

[1] ERDÉLYI, A.: Beitrag ^zurTheorie der konfluenten hypergeometrischen ^Funk-

tionen von mehreren Veränderlichen, Akademie der Wissenschaften in Wien, Sitzungsberichte, ^Abt.IIa, Math.-Nat. Klasse, ^vol.¹⁴⁶(1937), pp. 431-467.

[2] SZEGÖ, ^G.:Orthogonale Polynomials, ^NewYork, ^1939.

[3] WEISNER, ^L.: Group-theoretic origin of ^certaingenerating functions, ^Pacific Journal ^ofMathematics, ^vol.⁵(1955), pp. 1033-1039.

Manoscritto pervenuto ⁱⁿredazione il 3 novembre 1969.

Bilinear generating functions for Laguerre and Lauricella polynomials in several variables

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

L. C ARLITZ

Bilinear generating functions for Laguerre and Lauricella polynomials in several variables

Rendiconti del Seminario Matematico della Università di Padova, tome 43 (1970), p. 269-276

*)

Erdélyi [ 1 ]

Laguerre polynomial

by means

equivalent

following

generating

(There

slight

(u;

y)

given

[ 1 ] ).

Fork =1, (1.3)

Hardy-Hille

[2,

101]:

object

present

give

elementary proof

(1.3)

following

general

k=1, (1.7)

[ 3 ] .

nonnegative

(n;

s)

homogeneous

weight n

weight

weight

Alternatively

4)(n;

s) by

U, X, Y,

by (1.4).

(1.3)

Making

(1.2)

(2.2),

easily

(2.4)

equal

part

(2.4)

weight

weight

equal

right

(2.4) equal

~(n;

s)

homogeneous

degree n

weight

weight

clearly (2.4)

equiv-

right

(2.4)

equal

( 1- U) W -I- XY

weight

weight

(2.9)

place, (2.8)

equivalent

But, by (2.1 )

(2.2),

R ^ENDICONTI

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L. C ^ARLITZ

^s)

clearly ^(2.4)

place, ^(2.8)

^(2.4)

by ^(2.8)

^s)

^T(n;

^s).

Then, by ^(2.10),

^(1.7).