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A New Kontorovich-Lebedev-Like Transformation

Commun. Math. Anal.
Volume 13, Number 1 (2012), 86 - 99

A New Kontorovich-Lebedev-Like Transformation

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Abstract

A different application of the familiar integral representation for the modified Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution operator and inversion formula are established. Solvability conditions and explicit solutions of the corresponding class of convolution integral equations are exhibited. Finally, as a valuable application it is shown, that the introduced transformation is a key ingredient for solving difference equations of the order $n \in \mathbb{N}$ with constant coefficients in a class of analytic functions in the right half-plane ${\rm Re} z > n.$