SOME BASICS OF BESSEL GENERATING AND RELATED FUNCTIONS THE STUDY OF SINGLE AND DOUBLE INTEGRATION FORMULA

Abstract

The Bessel functions fall into two categories, those with even symmetry in x for even orders n and those with odd symmetry in x for odd orders n. Solving Laplace's equation and the Helmholtz equation separately in cylindrical or spherical dimensions leads to Bessel's equation. Thus, Bessel functions play a crucial role in many issues involving wave propagation and static potentials. Bessel functions may be understood in a unique way via the lens of algebraic geometry. Many families of generating functions including products of Bessel and related functions are derived using a unique methodology developed by the authors, which relies on the combination of operational techniques with certain particular multivariable and multi-index polynomials. We determine the generating function and use it to establish a number of important classical conclusions and recurrence connections. To further understand the Bessel functions at certain values, we employ these recurrence relations. We also demonstrate a different way to derive the first Bessel function from the generating function.