Infinite-variable Bessel functions of the Anger type and the Fourier expansion
This paper is a continuation of an earlier one by the authors [Rend. Mat. Appl., VII Ser. 15, No. 3, 405-420 (1995; Zbl 0848.33002)] in which some basic properties of an infinite-dimensional generalization denoted \(J_n(\{\alpha_m\})\) \((n\) integer) for the Bessel function of the first kind and several variables \(J_n(\alpha_1, \alpha_2, \dots, \alpha_m)\), were obtained. Here they present the following infinite-variable analogue version for the ordinary Anger function [see \textit{G. N. Watson}, A Treatise on the theory of Bessel functions (1995; Zbl 0849.33001), Chapter X, p. 308]: \[ A_\nu\bigl( \{\alpha_m\}\bigr) ={1\over\pi} \int^\pi_0 \cos\left( \nu\theta- \sum^\infty_{m=1} \alpha_m\sin m\theta \right)d \theta\;(\nu\in\mathbb{R}) \tag{*} \] with \(\{\alpha_m\}\) real and the series \(\sum^\infty_{m=1} m|\alpha_m |\) being assumed convergent. For \(\nu=n\) (integer) one has \(A_n(\{\alpha_m\}) \equiv J_n(\{\alpha_m\})\) while for \(\nu\neq n\), (*) can be put in the form \(A_\nu (\{\alpha_m\})= {\sin\nu\pi \over\pi} \sum^\infty_{r=-\infty} (-1)^r {J_r (\{\alpha_m\}) \over(\nu-r)}\) which, as function of \(\nu\), is determined by its values \(J_r(\{\alpha_m\})\). The authors study some of its basic properties (recurrence relations, addition formula,\dots) as well as the connection with Fourier expansions of certain class of smooth functions having proper symmetry properties. The corresponding modified version of (*) and the infinite-variable function of the Weber type \(W_\nu (\{\alpha_m\})\) are also discussed. Finally, some numerical aspects are given.
scientific article; zbMATH DE number 1091042