The Journal of the Indian Mathematical Society

R. S. Varma ¹

Affiliations
1 Christ Church College, Cawnpore, IN

Abstract

In a recent paper*, Dr. S. C. Mitra has shown that the Parabolic cylinder function, D_n (z), satisfies the relation

D_m+n (x) + ( - 1)ⁿ (n + 1) R_{n, m-2}, (x) D_n (x)

+ ( - l )^m = R_n-1,m-1 (x) D_{n +1} (x) = 0, … (1)

where R_n,m (x) is a polynomial analogous to that of Lommelt in the theory of Bessel Functions.

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R. S. Varma ¹

Affiliations
1 Christ Church College, Cawnpore, IN

Abstract

The object of this paper is to evaluate some infinite integrals involving Weber's parabolic cylinder functions. Thus in 2, an integral involving a Whittaker function W_k,m(x) and a Kummer function ₁F₁ is investigated.

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R. S. Varma ¹

Affiliations
1 Cawnpore, IN

Abstract

On Laguerre's Polynomials which are Self-Reciprocal in the Hankel Transform.

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R. S. Varma ¹

Affiliations
1 Lucknow University, IN

Abstract

In general we can have two types of infinite series involving the polynomial:

Σ a_rT^r_m(x)                                         (A)

and

Σ b_rT_rⁿ(x)                                          (B).

Series of the type (A) have been discussed extensively from time to time by various authors, but very little is known about series of the type (B).

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R. S. Varma

Abstract

In a recent paper I have shown that the functions

x^v+1/2e^1/4x2D-2r-3(x)                                   R(v)>-1

and

x^v-1/2e^1/4x2D-2r(x)                                      R(v)>-1/2

are self-reciprocal in the Hankel-transform of order v.

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R. S. Varma ¹

Affiliations
1 Christ Church College, Cawnpore, IN

Abstract

In my previous paper bearing this title, I investigated the differential equation satisfied by the polynomial R_n,m(x), introduced by Dr. S. C. Mitra, and gave the following expressions for this polynomial in a series of Hermite's polynomial H_n(x).

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