The Gegenbauer function

(1)

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

W. R. S CHUCANY The Gegenbauer function

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3

^e

série, tome 21, n

^o

3 (1967), p. 349-351

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(2)

THE

GEGENBAUER FUNCTION by

^W.^R.^SCHUCANY

(*)

~

In the notes of the American Mathematical

Monthly, [1]

G. G. Bilodeau

gives

^{a new}

expression

^for^the

Gegenbauer polynomials by employing

^the

concept

^of^afractional derivative. It is

interesting

^{to note}^that

by employing

the results obtained in

[3],

^{one can} ^obtain^a

^single

formula for both the classical

polynomials

^and^the

Gegenbauer

^functions

(C)

^as^defined

^by

^Ba-

teman

[2].

The

only

definition necessary for

clarity

^{is the}

following:

z

where n is a

positive integer,

^then^the

Holmgren-Riesz

^transform

of f

^is

given by

Now one solution of the

equation

may be written in its H - R transform form as

Pervenuto alla Redazione il 18 Agosto ^1966.

(if) ^LTVELECTROSYSTEMS, INC. GREENVILLE DIVISION.

(3)

350

If the

arbitrary

constant k

equals

then

when fl

⁼it,

equation (2)

^becomes^the^standard

Rodrigues

formula for the

Gengenbauer polynomials.

^Also^for

positive non-integer

^values

of f,

p > -

1/2 and IA # 0, equation (2)

^can^{be shown}^to^be

equivalent

^to

for x E

(1, 3),

^where

F [a, b ;

c ;

z]

^{is the}

Hypergeometric

^function.

(See [2]).

Consider

(2)

m, ^an

integer,

and note that when n

= 11- fl] I

^~

Letting z == 201320132013.

^the

^right

hand side of the above

equation becomes,

x-1

Then

setting

^we

have,

(4)

351

Now

applying

^alinear transformation formula we

get

Therefore

Hence the

single expression (2)

^with^the

given

^value^{of lc}^is^avalid formula for the

Gegenbauer polynomials,

^{and with}^the

generality

^afforded

by

the H - .R transform the

expression

may be

interpreted

^as

including

^the

associated function as well.

REFERENCES

[1] BILODEAU, ^G.G., ^«Gegenbauer Polynomials and Fractional Derivatives », American Mathe- matical Monthly, ^Vol.71, No. 9, ^1964.

[2] ÊRDELYI,Â.,êtâl.^HigherTranscendental Functions, ^Vols.¹ând2, McGraw-Hill, ^1953.

[3] GRAY, ^H.L., «Application of ^theHolmgren-Riesz Transform », Annali della Scuola Nor- male Superiore ^{di Pisa}^Scienze^Fisiche^eMatematiche, ^SerieIII, ^Vol.^XVIII,

Fasc. I, ^1964.

The Gegenbauer function

A NNALI DELLA

S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze

W. R. S CHUCANY The Gegenbauer function

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3

série, tome 21, n

3 (1967), p. 349-351

GEGENBAUER FUNCTION by

(*)

Monthly, [1]

gives

expression

Gegenbauer polynomials by employing

concept

interesting

by employing

[3],

single

polynomials

Gegenbauer

(C)

by

[2].

only

clarity

following:

positive integer,

Holmgren-Riesz

of f

given by

equation

arbitrary

equals

when fl

equation (2)

Rodrigues

Gengenbauer polynomials.

positive non-integer

of f,

1/2 and IA # 0, equation (2)

equivalent

(1, 3),

F [a, b ;

z]

Hypergeometric

(See [2]).

(2)

integer,

= 11- fl] I

Letting z == 201320132013.

right

equation becomes,

setting

have,

applying

get

single expression (2)

given

Gegenbauer polynomials,

generality

by

expression

interpreted

including

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

^single

^by

^right