A class of polynomials and discrete transformations associated with the Kontorovich–Lebedev operators

Abstract

We consider a class of polynomials related to the kernel K iτ(x) of the Kontorovich–Lebedev transformation. Algebraic and differential properties are investigated and integral representations are derived. We draw a parallel and establish a relationship with the Bernoulli's and Euler's numbers and polynomials. Finally, as an application we invert a discrete transformation with the introduced polynomials as the kernel, basing it on a decomposition of Taylor's series in terms of the Kontorovich–Lebedev operator.