Generalization of Bessel and hypergeometric functions (Q917749)

The generalized Bessel function in the form \[ (1)\quad J_{\ell mn}(z)=\sum^{\infty}_{k=0}\frac{(-1)^ k}{k!\Gamma ((n/\ell)+(mk/\ell)+1)}(\frac{z}{\ell +m})^{(n/\ell)+(mk/\ell)+k}, \] for \(z\in C\), \(\ell,m\in N\), \(n\in Z\) is presented. The author defines a generalised Bessel function \(J_{\lambda \mu \nu}(z)\) by the series (1) for z,\(\lambda\),\(\mu\),\(\nu\in {\mathbb{C}}.\) Analogously, replacing the symbol \[ (a)_{\ell}=a(a+1)...(a+\ell - 1)=\Gamma (a+1)/\Gamma (a)\text{ for } \ell \in N \] by the symbol \[ (a)_{\ell}=\Gamma (a+\lambda)/\Gamma (a)\text{ for } \lambda \in R \] the author defines the ``generalised hypergeometric function'' with \(p+q\) pairs of arguments in the form \[ (2)\quad_ pF_ q(\alpha_ 1:A_ 1,...,\alpha_ p:A_ p;\beta_ 1:B_ 1,...,\beta_ q:B_ q;z)=\sum^{\infty}_{k=0}\frac{(\alpha_ 1)_{A_ 1k}...(\alpha_ p)_{A_ pk}}{(\beta_ 1)_{B_ 1k}...(\beta_ q)_{B_ qk}}\frac{z^ k}{k!} \] (N.B.: in the (7.3) there are printing errors). The function \(J_{\lambda \mu \nu}(z)\) is a special case of the function (2). The preliminary analysis (e.g. differential equations, integral representations) is given.