Generalized Bessel functions: A group theoretic view (Q1340751)

The paper is devoted to the study of the generalized Bessel functions (GBF), defined by \[ J_ n^{(m)} (x,y;t)= \sum_{\ell=-\infty}^ \infty t^ \ell J_ \ell (x) J_{n+m\ell} (y), \] where the parameter \(t\) is complex. The above expression is the generalization of the series \[ J_ n(x,y)= \sum_{\ell=-\infty}^ \infty J_ \ell(x) J_{n-2\ell} (y). \] Various properties such as recurrence relations, differential recurrence relations, generating functions, integral representation and Graf type addition theorem are established. The generating function is given by \[ \exp \textstyle {\bigl\{ {y\over 2} (r- {1\over r})+ {x\over 2} ({1\over {r^ m}}- {{r^ m} \over t} )\bigr\}= \sum_{n=-\infty}^ \infty r^ n J_ n^{(m)} (x,y; t).} \] The connection of the GBF's with the theory of group representation is also investigated. In their concluding remark the authors indicate the possibility of further generalizations of BF's to three or more variables, to three or more indices. According to them such type of generalizations can also be introduced and studied for other special functions of relevance in mathematical physics.