## Communications in Mathematical Analysis

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

### A New Kontorovich-Lebedev-Like Transformation

#### 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 ${\rm Re} z > n.$

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 2 October 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.cma/1349204800

**Mathematical Reviews number (MathSciNet)**

MR2998349

**Zentralblatt MATH identifier**

1262.44003

**Subjects**

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

**Keywords**

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

#### Citation

Yakubovich, S. A New Kontorovich-Lebedev-Like Transformation. Commun. Math. Anal. 13 (2012), no. 1, 86--99. https://projecteuclid.org/euclid.cma/1349204800