Symbolic methods for the evaluation of sum rules of Bessel functions

Abstract

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.

Operational Versus Umbral Methods and the Borel Transform

“…Other practical outcome concerns the handling of the associated integrals and of many other technicalities related to the Ramanujan Master Theorem (RMT) 1 [1,2]. Furthermore, the same formalism provides the possibility of recovering a large body of the properties of Bessel functions and of other special functions as well, using genuine algebraic tools [5,6].…”

“…Such a restyling allows noticeable simplifications in the theory of Bessel functions themselves [5,6]. Other practical outcome concerns the handling of the associated integrals and of many other technicalities related to the Ramanujan Master Theorem (RMT) 1 [1,2].…”

“…In a number of previous papers [5,6] it has been established that the umbral image of the Bessel function of the first kind is the Gaussian. Indeed, if we define the shift operator c z satisfing c α z c β z = c α+β…”

“…The extension of the criterion of permanence, under umbral correspondence, to the properties of the gamma function yields the identities [5,6] [26], and…”

Quasi Exact Solution of the Fisher Equation

“…From the dynamical point of view the equation describes a spatial diffusion in a homogeneous medium (the coefficient D is assumed to be independent of the spatial coordinate) embedded with a growth of logistic nature [4].The coefficients r and K are the growth rate and the carrying capacity of the environment respectively [4]. If the process is purely diffusive (r = 0), the solution of our evolution problem is just provided by the following Gauss Weierstrass transform [5] …”