On a new class of special functions generated by integral-difference operators

We introduce and investigate a new class of special functions ΞN[k](x), x∈[0,1). Originally these functions naturally appeared upon spectral analysis of integral-difference operators. We discuss issues of generating functions; decompositions in series; links to Legendre polynomials, complete elliptic integrals and Gauss hypergeometric functions; functional relations for functions ΞN[k], mostly in case N=1.