Discrete Mehler-Fock transforms

Discrete analogs of the classical Mehler-Fock transforms are introduced and investigated. It involves series with the associated Legendre function $P_{in-1\mathrm{/}2}^{\mu }(x),x>1,\mathrm{R}\mathrm{e}\mu <1\mathrm{/}2,n\in \mathbb{N},i$ is the imaginary unit, and the so-called incomplete Legendre integrals. Several expansions of arbitrary functions and sequences in terms of these series and integrals are established.