Some multiplication formulas

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

L EONARD C ARLITZ Some multiplication formulas

Rendiconti del Seminario Matematico della Università di Padova, tome 32 (1962), p. 239-242

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Some years ago Rainville

[5] proved

the formula

where

P.(x)

^{is the}

Legendre polynomial.

The writer

[11

^showed

that

where

Cf(z)

^{is the}

ultraspherical polynomial

^defined

by

For A ⁼

1/2, (2)

^{reduces to}

(1).

Chatterjea [2]

^has

recently proved

^the

following

^formulas:

where

~"(x)

is defined

by

(*)

Pervenuta in redazione il 3 dicembre 1961.

Indirizzo dell’A.:

Department

^of Mathematics, Duke

University,

Durham, North Carolina

(U.S.A.).

(**) Supported

ⁱⁿ

part by

National Science Foundation

grant

^{G 16485.}

240

and

where

L,~(~)

^{is the}

Laguerre polynomial.

These results and ⁱⁿ

particular ^{(3) suggest}

^the consideration of the

functional equation

If we

take fl =

^{it is}

^easily

^{verified that}

⁽⁵⁾

^{reduces to}

For

^{= tan a} this becomes

For x ⁼

1, (6)

^becomes

where

Conversely

^if

f n(~1)

^is

^{defined by (7),}

^{where the c, are}

^arbitrary

..constants,

^{it is}

easily ^verified

^that

⁽⁶⁾

holds. Indeed ^{we have}

We see therefore that

(6)

^is

equivalent

^to

(7)

and that the

general

^{solution of}

(6)

^is

given by (7)

^with

arbitrary

^{c, . Moreover}

it follows from

(6)

^{with x}

replaced by

pz ^and A

replaced by

^that

If mow take A ⁼

tan a, It

^{= tan}

~B

^we

get (5).

^Thus

(5)

^is

equi-

valent to

(7).

We recall that Feldheim

[3]

^has

proved

^{that the}

Laguerre polynomials

^{are the}

only orthogonal polynomials

^with

a

multiplacation

theorem of the form

It may be of interest ^{to note} that

(7) implies

or, if ^we

prefer,

with

242

where the last is

symbolic.

^Rainville

[4,

p.

244]

has noted that the

polynomials f n (~ )

^defined

by (9) satisfy (6).

It follows from

(9)

^{that the}

polynomials

constitute an

Appell set,

^{that is}

B~2013/ ^/

Also

(6)

^becomes

Conversely

^{if the}

Appell

^set

{g"(x)}

^is

assigned

^{then the}

polynomials

^defined

by (11) satisfy (5).

REFERENCES

[1]

CARLITZ L.: Note on a

formula of

Rainville. Bulletin of the Calcutta Mathematical

Society,

^{vol. 51}

(1959),

pp. 132-133.

[2] CHATTERJEA S. K. : Notes on a

formula of

Carlitz. Rendiconti del Seminario Matematica della Università di Padova, vol. 31 (1961), pp. 243-248.

[3] FELDHEIM E.: Une

propriété caracteristique

^des

polynomes

^{de La-}

guerre. Commentarii Mathematici Helvetici, vol. 13 (1940), pp. ^6-10.

[4] RAINVILLE E. D.: Certain

generating functions

and associated

poly-

nomials. American Mathematical

Monthly,

^{vol. 52} (1945), pp. ^239- 250.

[5] ^RAINVILLE E. D.: Notes on

Legendre polynomials.

Bulletin of the American Mathematical

Society,

^{vol. 51} (1945), pp. 268-271.