Functions orthogonal to polynomials and their application in axially symmetric problems in physics

We investigate properties of sets of functions comprising countably many elements $A_n$ such that every function $A_n$ is orthogonal to all polynomials of degrees less than $n$. We propose an effective method for solving Fredholm integral equations of the first kind whose kernels are generating functions for these sets of functions. We study integral equations used to solve some axially symmetric problems in physics. We prove that their kernels are generating functions that produce functions in the studied families and find these functions explicitly. This allows determining the elements of the matrices of systems of linear equations related to the integral equations for considering the physical problems.