Communications in Mathematical Analysis

A New Kontorovich-Lebedev-Like Transformation

S. Yakubovich

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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 ${\rm Re} z > n.$

Article information

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

First available in Project Euclid: 2 October 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 44A15: Special transforms (Legendre, Hilbert, etc.) 44A35: Convolution 33C05: Classical hypergeometric functions, $_2F_1$ 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33C15: Confluent hypergeometric functions, Whittaker functions, $_1F_1$ 39A10: Difference equations, additive 45E99: None of the above, but in this section

Kontorovich-Lebedev transform modified Bessel functions Whittaker functions Mellin transform Laplace transform convolution integral equations of the convolution type difference equations


Yakubovich, S. A New Kontorovich-Lebedev-Like Transformation. Commun. Math. Anal. 13 (2012), no. 1, 86--99.

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