A new index transform with the square of Whittaker's function
Semyon Yakubovich
TL;DR
This work introduces a novel index transform with the square of the Whittaker function as kernel and analyzes its mapping properties and inversion. The approach leverages the integral representation of $W^2_{\mu,i\tau}(x)$ via Gauss hypergeometric functions, and expresses the transform as a Laplace convolution composed with the Olevskii transform, enabling a rigorous inversion via Mellin transform techniques. The main contributions include a boundedness result (Theorem 1), an inversion framework (Theorems 2 and 3) for $0<\mu<1/4$, and an explicit inversion formula in the Lebedev-type special case $\mu=0$, involving modified Bessel functions. The results connect the new transform to Kontorovich-Lebedev and Lebedev-type transforms and provide concrete kernels and Parseval-type relations for practical use.
Abstract
An index transform, involving the square of Whittaker's function is introduced and investigated. The corresponding inversion formula is established. Particular cases cover index transforms of the Lebedev type with products of the modified Bessel functions.