# General W-transform and its Particular Cases

• Semen B. Yakubovich
• Yurii F. Luchko
Part of the Mathematics and Its Applications book series (MAIA, volume 287)

## Abstract

In this section, following the idea of J. Wimp (1964), we will generalize by hypergeometric method the Kontorovich-Lebedev pair (6.1)–(6.2) with Macdonald function as the kernel to the case of the Meijer G-function (1.46) and moreover, using the Parseval equality (1.92) for the Mellin transform (1.81) we will introduce the so-called W-transform in the space (L) (see Definition 3.1). Historically, some known index transforms (the Mehler-Fock transform, the Olevskii transform, the Lebedev transform et c. (see J. Wimp (1964), O.I. Marichev (1983)), which we will discuss below, were investigated separately and independently by many authors. But the introduction in J. Wimp (1964) of his index transform, after simplification of the corresponding inversion formula in S.B. Yakubovich (1985) allowed to obtain the following pair of the Wimp-Yakubovich transform (as announced in Vu Kim Tuan et al. (1986), Vu Kim Tuan (1987), (1988a), S.B. Yakubovich et al. (1987), S.B. Yakubovich (1986), (1987a))
$$g\left( \tau \right) = \int_0^\infty {G_{p + 2,q}^{m,n + 2} \left( {u\begin{array}{*{20}c} {1 - v + i\tau ,1 - v - i\tau ,\left( {\alpha _p } \right)} \\ {\beta _q } \\ \end{array} } \right)} f\left( u \right)du,\tau > 0,$$
(1)
$$f\left( x \right) = \frac{1} {{\pi ^2 }}\int_0^\infty {G\begin{array}{*{20}c} {q - m,p - n + 2} \\ {p + 2,q} \\ \end{array} } \left( {x\begin{array}{*{20}c} {v + i\tau ,v - i\tau , - \left( {\alpha _p } \right)} \\ { - \left( {\beta _q^{m + 1} } \right), - \left( {\beta m} \right)} \\ \end{array} } \right)$$
(2)
where
$$\begin{gathered} m,n,p,q \in N,0 \leqslant n \leqslant p,0 \leqslant m \leqslant q,\left( {\alpha _p } \right) = \left( {\alpha _{1,...,} \alpha _p } \right),\left( {\beta _q } \right) = \hfill \\ \left( {\beta _{1,...,} \beta _q } \right), - \left( {\alpha \begin{array}{*{20}c} {n + 1} \\ p \\ \end{array} } \right) = \left( { - \alpha _{n + 1,...,} - \alpha _p } \right), - \left( {\beta \begin{array}{*{20}c} {m + 1} \\ q \\ \end{array} } \right) = \left( { - \beta _{m + 1,...,} - \beta _q } \right) \hfill \\ \end{gathered}$$
(3)
are the parameters of G-functions. We observe that the transform (7.1) is essentially a non-convolution one and the function g(τ) depends on the variable which is contained in the list of parameters of Meijer’s G-function. In its inversion formula (7.2) similar to formula (6.2) for the Kontorovich-Lebedev transform, the operation of integration is realized by the parameters (or indices) of respective G-function. Hence, in accordance with the table of particular cases for G-functions from A.P. Prudnikov et al. (1989a) it is not difficult to find all mentioned index transform and to investigate their properties from unified point of view of hypergeometric approach.

## Keywords

Hypergeometric Function Inversion Formula Composition Structure Whittaker Function Convolution Type
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