Abstract
We introduce and investigate a new class of special functions Ξ N [ 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 Ξ N [ k ] , mostly in case N = 1 .
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