On Bessel, Jacobi and Laguerre polynomials

(1)

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

H. M. S RIVASTAVA

On Bessel, Jacobi and Laguerre polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 35, n

^o

2 (1965), p. 424-432

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(2)

Making

^use^of^twoformulae of

SRIV ASTA VA,

a number of Neumann

type expansions involving Bessel,

^Jacobi

and

Laguerre polynomials

^are^obtained.

1. Srivastava

[5]

^has

recently proved

the formulae:

and

*) Pervenuta in redazione il 15 marzo 1965.

Indirizzo dell’A.:

Department

^ofMathematics, ^The

University,

Jodhpur (India).

(3)

425

where the notation for the double

hypergeometric

functions is due to Burchnall and

Chaundy [3,

p.

112]

ⁱⁿ

preference,

^for

the sake of

brevity,

^to^thatintroduced

by Kampé

^De^Fériet.

We note that when

p==.P==:~20131==~20131==0~

^which

we shall call a confluent case ’ of the functions

involved,

^the

double series in

(1.1)

^reduces^to

Appell’s F4

^which^can^{be further}

expressed

^{as a}

product

^{of two}

2F¡’s [2, § 9.6],

^and

by

^a^suitable

choice of )1. ^wehave Bateman’s well-known

expansion [6,

p.

370].

It is not difficult to show that

(1.2)

^reduces^toCarlitz’s formula

[4,

p.

134]

^undersimilar restrictions ^onthe

integers p, q,

^P^and

Q.

Now the

hypergeometric

^series^on^the

right

^of

(1.1)

^is

equal

^to:

and this reduces to a when b ⁼0.

Therefore,

^{if in}

(1.1)

we set b ⁼0 and

change

the notation

slightly

^we

get:

Similarly, (1.2) yields:

(4)

and on

summing

^the

well-poised terminating aF2 by

^Dixon’s

theorem

[2, § 3.1]

^{we see}^that:

From

(1.3)

^we^therefore^have:

and

(1.4) gives:

Al-Salam and Carlitz

[1,

p.

157] proved

the last formula in a different way.

(5)

427

Next we consider the Jacobi

polynomials:

and we find that:

A

special

^case^of

(1.3)

^is:

and from

(1.4)

^we

similarly

^have:

(6)

where t ⁼

~/ (1 - z2 ) .

^It^is^not^difficult^to

give

^a^direct

proof

of

(2.5).

Expansions involving product

^of^two

terminating ~F1

^’s

can also be derived from

(1.3)

^and

(1.4).

^Thus^we^have:

and since the

Laguerre polynomial:

from

(2.6)

^and

(2.7)

^we^obtain:

and:

(7)

429

respectively.

We remark in

passing

that formulae

involving product

^of

two

Gegenbauer polynomials

^or^two

Legendre polynomials

^are

particular

^cases^of

(2.3), (2.4)

^and

(2.5),

^andthat from

(2.1)

and

(2.2)

^{we can}^deduce

expansions involving product

^{of two}

polynomials

^of^the

type:

3. Another confluent case of

(1.3)

^{i~ :}

and

replacing

^the

2F¡ by

^Jacobi

polynomial

^we^have:

In a similar _way

(1.4) gives [1,

p.

153]:

(8)

and ^anecessary consequence of

(3.3)

is the formula

[4,

p.

136]:

which is

obviously

^a

special

^case^of^the

Gegenbauer

^addition

theorem

[6,

p.

362].

a formula that has been

proved

ⁱⁿ

[1]

ⁱⁿ^a^differentway.

glso,

for p = q ^-¹⁼

2, (1.3)

^leads^to:

(9)

431

and

similarly,

^y ^from

(1.4)

^we

get:

From

(3.7)

^we^have:

and

(3.8) readily gives:

a formula

proved by

^Carlitz

[4].

REFERENCES

[1] AL-SALAM W. A., ^CARLITZ^L.:^Some

expansions

ⁱⁿ

products of

Bessel

functions. Portugaliae

Mathematica, ^vol. ²²(1963), pp.

153-160.

(10)

terly (Oxford)

[5] SRIVASTAVA H. M.: Some

expansions

ⁱⁿ

products of hypergeometric functions, to

appear.

[6] WATSON G. N.:

Theory of

^Bessel

functions,

^secondedition, ^Cam-

bridge

^{and New}York, ^1944.

On Bessel, Jacobi and Laguerre polynomials

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

H. M. S RIVASTAVA

On Bessel, Jacobi and Laguerre polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 35, n

2 (1965), p. 424-432

Making

SRIV ASTA VA,

type expansions involving Bessel,

Laguerre polynomials

[5]

recently proved

Department

University,

Jodhpur (India).

hypergeometric

Chaundy [3,

112]

preference,

brevity,

by Kampé

p==.P==:~20131==~20131==0~

involved,

(1.1)

Appell’s F4

expressed

product

2F¡’s [2, § 9.6],

by

expansion [6,

370].

(1.2)

[4,

134]

integers p, q,

Q.

hypergeometric

right

(1.1)

equal

Therefore,

(1.1)

change

slightly

get:

Similarly, (1.2) yields:

summing

well-poised terminating aF2 by

[2, § 3.1]

(1.3)

(1.4) gives:

[1,

157] proved

polynomials:

special

(1.3)

(1.4)

similarly

~/ (1 - z2 ) .

give

proof

(2.5).

Expansions involving product

terminating ~F1

(1.3)

(1.4).

Laguerre polynomial:

(2.6)

(2.7)

respectively.

passing

involving product

Gegenbauer polynomials

Legendre polynomials

particular

(2.3), (2.4)

R ^ENDICONTI

S ^EMINARIO M ^ATEMATICO

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