On the Big Hartley transform

Abstract

In this paper, we introduce a new type of singular first-order differential-difference operator of Dunkl type on the real line. This operator is obtained as a limiting case from both the first-order Dunkl-type operators corresponding to Bannai-Ito and Big 1-Jacobi orthogonal polynomials. We provide an explicit expression for the eigenfunction of this operator in terms of Bessel functions. The obtained kernel is called the Big Hartley function, which is a generalization of the usual Hartley kernel and the little Hartley function studied in Bouzeffour [The generalized Hartley transform. Integral Transforms Spec Funct. 2014;25(3):230–239]. Additionally, we present a new product formula for the little Hartley function, which is related to the Kingman-Bessel hypergroup and the Rosler-Dunkl signed hypergroup. Finally, we investigate inversion formulae for the transforms of both the little Hartley function and the big Hartley function.