On certain definite integrals involving Legendre's polynomials

(1)

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

S. K. C ^HATTERJEE

On certain definite integrals involving Legendre’s polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 27 (1957), p. 144-148

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INVOLVING LEGENDRE’S POLYNOMIALS

Nota

(*)

^diS. K. CHATTERJEE

(a Calcutta)

1. - The

object

^{of the}

present

^noteis to evaluate certain definite

integrals involving Legendre’s Polynomials. Using

^the

symbol

^for ^we^write

2. - Let _{m, n,}r and s be four

positive integers

^such^that

1 m n and Then

(*) ^Pervenutaⁱⁿ^Redazione^il24 ottobre 1956.

Indirizzo dell’A.: Department ^{of Pure}Mathematics, University, Calcutta (India).

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145

3. - CASE I. - If 2 Ç r

-~-

^s ¹ⁿ^we^have

The last

integral

^vanishes^on^account^{of the}

orthogonal property

^of

Legendre’s Poly.nomials.

If however r -.+-

+

1,

the process ^comesto an end earlier.

CASE II. - Let

+

^{1 ~}^r.-~- s

Ç

^2~rt.

Writing r +

^s^-^m+ ^b

we have 1 C b ln. In this case

Here,

^we^have

Now,

it may be

easily

^shewn^that

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provided

^r

^{-~- t ~}

^rr~ ^and

~2013~20131~0,

^which ^is

certainly

true in the

present

^cases.

4. - It

follows, therefore,

ⁱⁿ^case¹

where

In case II

And in case III

If ,. = . = 1 and

m > 2,

^case¹

gives

the well known

result [1] (Clare, 1M)-

From the above it is also easy to

verify

^that

[2] (Math.

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147

Trip, 1897)

5. - ^We^nextevaluate the

integral

^where

n1, 1t2’ 3 ..., 3 n, ^arek

positive integers

^{8uch that}

where

Again

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Each of the k - 1

integrals

vanishes if

where

So,

under the conditions stated above

If

k = 2,

^we

again get

^the^result

[1].

where and ii ⁼⁼r~2 .

The results of this paper ^areembodied in formulae

(4,1), (4.2), (4.3)

^and

5.3).

1 am indebted to Dr. H. M.

Sengupta

for the kind inte- rest he has taken in the work.

REFERENCES

[1] ^WHITTAKER^E.T. and WATSON G. N. - Modern Analysis. 1952, p. 309.

[2] WHITTAKER E. T. and WATSON G. N. - Modern Analysis. 1952, p. 309.

On certain definite integrals involving Legendre's polynomials

R ENDICONTI

del

S EMINARIO M ATEMATICO

della

U NIVERSITÀ DI P ADOVA

S. K. C HATTERJEE

On certain definite integrals involving Legendre’s polynomials

Rendiconti del Seminario Matematico della Università di Padova, tome 27 (1957), p. 144-148

INVOLVING LEGENDRE’S POLYNOMIALS

(*)

(a Calcutta)

object

present

integrals involving Legendre’s Polynomials. Using

symbol

positive integers

-~-

integral

orthogonal property

Legendre’s Poly.nomials.

+

+

Ç

Writing r +

Here,

Now,

easily

provided

-~- t ~

~2013~20131~0,

certainly

present

follows, therefore,

m &#x3E; 2,

gives

result [1] (Clare, 1M)-

verify

[2] (Math.

Trip, 1897)

integral

positive integers

Again

integrals

So,

k = 2,

again get

[1].

(4,1), (4.2), (4.3)

5.3).

Sengupta

R ^ENDICONTI

S ^EMINARIO M ^ATEMATICO

U NIVERSITÀ DI P ^ADOVA

S. K. C ^HATTERJEE

^{-~- t ~}

m > 2,