The Kontorovich-Lebedev Transform

Abstract

Discrete analogs of the classical Kontorovich-Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function K in (x), x > 0, n ∈ N, i is the imaginary unit, and incomplete Bessel integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established.

Discrete Mehler–Fock transforms

“…where the interchange of the order of integration is allowed by Fubini's theorem. Therefore under conditions of the theorem one can invert the latter discrete Kontorovich-Lebedev transform, appealing to Theorem 5 in [6]. As a result we find…”

“…Moreover, the sequence {a n } n≥1 satisfies the condition of Theorem 1 in [6] and the inversion formula for the discrete Kontorovich-Lebedev transform holds. This means that…”

“…Finally in this section we note that the discrete analogs of the Kontorovich-Lebedev transform were considered recently by the author in [6]. It involves the modified Bessel functions and incomplete Bessel integrals [1], Vol.…”

“…2. This means that the right-hand side of (2.4) is the discrete Kontorovich-Lebedev transform [6] and the equality reads…”

The Kontorovich–Lebedev Transform as a Map between d‐Orthogonal Polynomials

“…We notice that all these index transforms can be obtained through the composition of the KL with the Mellin type convolution transforms. For further reading on the subject we refer to [5][6][7].…”