A note on Legendre polynomials

(1)

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

S. K. C HATTERJEA

A note on Legendre polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 30 (1960), p. 232-236

<http://www.numdam.org/item?id=RSMUP_1960__30__232_0>

L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions).

Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

Nota

(*)

S. K. CHATTERJEA ( a Calcutta)

ln a recent note

[1],

^B.^S.

Popov

^has

given

^aiialternative

proof

^ofmy result

[2’] :

^:

where

and

The

objeet

^of^this

present

note is to add- some more results to- those

already

^obtainedⁱⁿ^the

[1]

^and

[2].

First we prove

(* ^I^PervenutoⁱⁿRedazione iI 21 giugno 1960.

Indirizzo dell’A.: Bangabasi College. University ^of ^Calcutta (India).

(3)

233 where

and

denote

respectively

^the

Legendre

^{and the}^Jacobi

polynomial

of

degree

^n.

In

proving (2)

^we^shall

require

^the

following

^formulae:

and

The formula

(3)

is well-known and the formulae

(4)

^and

(5),

^which^are

generalization

^of Grosswald’s formula

[3]:

are obtained

by

^Carlitz

[4]

^and^are

presented

in correct forms

by

B. R. Bhonsle

[5].

(4)

Now.

using (3)

^we

get

the

integral

^on ^the

right

hand member of

(7) vanishes,

since m - r > n - s.

Thus we

easily

^obtain

(2)

^{f rom}

(7)

^with^the

help

^of

(4), (5)

^and

(6).

Now we like to make some

applications

of the formula

(2).

^Weremark that

for a Q. (2)

reduces to

where

and

. Next

putting

^a

i~ 1

ⁱⁿ

₍₈₎

^and

_observing

that

(5)

235

where

Pi (x)

denotes the

ultraspherical polynomial

^of

degree

n and a = A - 1, ²

^we^obtain

where

Finally puttin A =1 2

^and

^observing

^that

when

1

y we obtain from

(9)

2’

where

which is

(1).

In

precisely

^the^sameway, ^we^also^obtain

(6)

where

and

We remark that for

a = ~ = 0, (11)

> reduces to a

general integral formula,

^obtained

by

^{B. S.}

Popov,

^for^the

product

of the derivatives of

Legendre polynomials.

In

conclusion,

^I

acknowledge

my

grateful

^thanks^{to Dr.}

H. )1.

Sengupta

for his kind

help

^{ill tlte}

preparation

^of^this

note.

REFERENCES

[1] B. S. POPOV, A ^noteconcerning ^theintegrals involving the derivatives of Legendre polynomials. Rend. Sem. Mat. Univ. Padova.

vol. XXIX, 1959, pp. 316-317.

[2] S. K. CHATTERJEA. On certain definite integrals involving Legen-

dre’s polynomials, Rend. Sem. Mat. Univ. Padova, vol. XXVII, 1957, pp. 144-148.

[3] ^E.GROSSWALD. On a simple property of ^thederivatives of Legen- dre’s polynomials, Proc. Amer. Math. Soc. vol. I. 1950, p. 553.

[4] ^L.CARLITZ, ^Note^onLegendre polynomials. ^Bull.Cal. Math. Soc.

vol. 46. 1954, p. 93.

[5] B. R. BHONSLE. On ^someresults involving Jacobi polynomials,

Bull. Cal. Math. Soc. vol. 50. 1958. p. 162.

A note on Legendre polynomials

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

S. K. C HATTERJEA

A note on Legendre polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 30 (1960), p. 232-236

(*)

[1],

Popov

given

proof

[2’] :

objeet

present

already

previous

[1]

[2].

respectively

Legendre

polynomial

degree

proving (2)

require

following

(3)

(4)

(5),

generalization

[3]:

by

[4]

presented

by

[5].

using (3)

get

integral

right

(7) vanishes,

easily

(2)

(7)

help

(4), (5)

(6).

applications

(2).

for a Q. (2)

putting

i~ 1

(8)

observing

Pi (x)

ultraspherical polynomial

degree

n and a = A - 1, 2

Finally puttin A =1 2

observing

1

(9)

(1).

precisely

a = ~ = 0, (11)

general integral formula,

by

Popov,

product

Legendre polynomials.

conclusion,

acknowledge

grateful

Sengupta

help

preparation

R ^ENDICONTI

S ^EMINARIO M ^ATEMATICO

U NIVERSITÀ DI P ^ADOVA

₍₈₎

_observing

n and a = A - 1, ²

^observing