Abstract
The paper deals with general expansions which give as special cases new results involving the Bessel functions, Jacobi, ultraspherical, and Laguerre polynomials, where the degree of the function is incorporated in the argument. In fact, the theorems unify and extend the Neumann-Gegenbauer expansion and its generalization by Fields and Wimp, Cohen, and others, the Kapteyn expansion theory, and the Kapteyn expansion of the second kind. New expressions are given for the Neumann-type degenerate form of a Gegenbauer addition theorem, the Feldheim expansions for the Jacobi and ultraspherical polynomials, and other expressions. Also of interest is the new method of proof, involving differential and integral operators.
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