Combinations of cross-products of Bessel functions, associated addition formulas and biorthogonal systems

Abstract

This study deals with cross-products of Bessel functions of the first and second kinds χ ν ( p , w ) = Y ν ( w ) J ν ( p ) − Y ν ( p ) J ν ( w ) , ν , p , w ∈ C . Combinations χ ν ( p , w ) are shown to be generating functions for associated Legendre functions. This fact permits to establish new integral and series representations of χ ν ( p , w ) . Under special choices of parameters, the representations become integral transforms or coefficients of orthogonal series whose inverting leads to new addition formulas for Bessel functions and their continuous analogues. On other hand, integral representations in some cases are integral operators with known inversion formulas and can be treated as operator relations between two systems of functions. Hence theorems on expanding functions into χ ν ( p , w ) series are proved and biorthogonal systems of functions are found in special cases. As examples, it is applied to evaluate new integrals and series.