Some expansions of generalized Whittaker functions

1965 ◽  
Vol 61 (4) ◽  
pp. 895-896 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Whittaker Functions ◽  
Image Position ◽  
Usual Notation

In the usual notation letwhere (a)n = a(a + 1)(a + 2)…(a + n − 1), (a)0 = 1.Dr L. J. Slater ((3), page 628) gave an expansion of the formfor all values of n and t, real or complex.

scholarly journals An Inversion Formula for the Generalised Laplace Transform

1949 ◽  
Vol 8 (3) ◽  
pp. 126-127 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Whittaker Functions ◽  
Laplace Integral ◽  
Image Position

Recently I have given a generalisation of the Laplace integralin the formwhere Wk, m (x) stands for Whittaker Functions.

The Inverse Laplace Transform of the Product of Two Whittaker Functions

1962 ◽  
Vol 58 (4) ◽  
pp. 580-582 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Laplace Transform ◽  
Whittaker Function ◽  
Inverse Laplace Transform ◽  
Whittaker Functions ◽  
Original Function ◽  
The Laplace Transform ◽  
Image Position

The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters

Certain integral representations for Whittaker functions

1948 ◽  
Vol 44 (3) ◽  
pp. 453-455
Author(s):  
Hari Shanker
Keyword(s):  
Integral Representations ◽  
Whittaker Functions ◽  
Image Position

1. Shastri (1) has shown that if and thenBut (Goldstein (2))hence substituting from (1·2) in (1·1) and changing the order of integration, which we suppose to be permissible, we find that ifthenassuming of course that the integrals converge.

Some integrals involving Gauss's hypergeometric function and Meijer's G-function

1967 ◽  
Vol 63 (4) ◽  
pp. 1049-1053 ◽  
Author(s):  
S. D. Bajpai
Keyword(s):  
Positive Integer ◽  
Hypergeometric Function ◽  
Whittaker Functions ◽  
Type Integral ◽  
Image Position

The object of this paper is to evaluate some integrals involving the product of Gauss's hypergeometric function and Meijer's G-function by expressing the G-function as a Mellin–Barnes type integral and interchanging the order of integrations. The integrals are important because on specializing the parameters they lead to many results for MacRobert's E-function, Bessel, Legendre, Whittaker functions and other related functions. In what follows δ is a positive integer and Δ(δ, α) represents the set of parameters

Definite integrals involving Whittaker functions

1968 ◽  
Vol 64 (4) ◽  
pp. 1033-1039 ◽  
Author(s):  
F. M. Ragab ◽  
M. A. Simary
Keyword(s):  
Whittaker Functions ◽  
Definite Integrals ◽  
Image Position

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are establishedwhere the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.

Two singular integral equations involving confluent hypergeometric functions

1969 ◽  
Vol 66 (1) ◽  
pp. 71-89 ◽  
Author(s):  
Tilak Raj Prabhakar
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Singular Integral Equations ◽  
Whittaker Functions ◽  
Confluent Hypergeometric Functions ◽  
The Laplace Transform ◽  
Similar Transform ◽  
Image Position

Widder(1) obtained an inversion of the convolution transformby the method of the Laplace transform, Ln(x) being the Laguerre polynomial. Buschman (2) inverted a similar transform with a generalized Laguerre polynomial as kernel and also solved (3) the singular integral equationusing Mikusinski operators. Srivastava(4, 4a) solved singular integral equations with kernels involving and Whittaker functions Mk,μ(x).

Double Meijer transformations of certain hypergeometric functions

1968 ◽  
Vol 64 (2) ◽  
pp. 425-430 ◽  
Author(s):  
H. M. Srivastava ◽  
J. P. Singhal
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Image Position ◽  
Usual Notation

Following the usual notation for generalized hypergeometric functions we let(a) denotes the sequence of A parametersthat is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretationwith a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parametersand for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.

On a group of 1440 birational transformations of four variables that arises in considering the projective equivalence of double sixes

1926 ◽  
Vol 23 (2) ◽  
pp. 103-108
Author(s):  
W. Burnside
Keyword(s):  
The Other ◽  
Birational Transformations ◽  
Actual Position ◽  
Projective Equivalence ◽  
Image Position ◽  
Usual Notation

The lines of a double-six will here be represented by the usual notationwhere two lines whose symbols are in the same line or same column of this scheme are non-intersectors and all other pairs of lines intersect. Any six of the lines, no two of whose symbols are in the same column, and just three are in the same row, are generators of a quadric, and the actual position in space of each of the other six is determined by the two points in which it intersects this quadric.

scholarly journals Integral operators involving Whittaker functions

1983 ◽  
Vol 24 (2) ◽  
pp. 139-148 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Integral Equations ◽  
Laplace Operator ◽  
Fractional Integral ◽  
Integral Operators ◽  
Whittaker Functions ◽  
Fractional Integral Operators ◽  
Good Account ◽  
Image Position

We define the integral operators and asandwhereand Wk, u and Mk, u are the Whittaker's confluent hyper-geometric functions. These operators, in their slightly less general form, have been dealt with in [2] and [4]. There the authors have used the fact that these integral operators can be expressed as compositions of the Kober's fractional integral operators and the modified Laplace operator. Then these operators are inverted accordingly. Generally, this type of technique has been very useful for inverting many kinds of integral equations; and a good account of the procedures involved is given [5].

scholarly journals A Type of Alternant

1953 ◽  
Vol 9 (1) ◽  
pp. 20-27
Author(s):  
F. W. Ponting
Keyword(s):  
Image Position ◽  
Usual Notation

We definewhere ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we setAll partitions will be in descending order and the usual notation for repeated parts will be used.

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