On an expansion of the confluent hypergeometric function in series of Bessel functions
In this paper, the author gives a proof of \[ \phi^*(a,c;wx)= \sum^{\infty}_{n=0} a_ nR_ n^*(a,c,\lambda;w)h_ n(x) \] where \[ \phi^*(a,c,x)= \sum^{\infty}_{n=0} \frac{(a)_ nx^ n}{\Gamma(c+n)n!}, a_ n=\frac{\sqrt{\pi}(\lambda +n)\Gamma(2\lambda +n)}{2^{2\lambda -1}n!},(-2\lambda \not\in N), \] \[ R_ n^*(a,c,\lambda;w)=_ 3F_ 2^*(-n,n+2\lambda;\lambda +\frac{1}{2},c;w)=\sum^{n}_{k=0}\frac{(-n)_ k(2\lambda +n)_ k(a)_ kw^ k}{\Gamma(\lambda +frac{1}{2}+k)\Gamma(c+k)k!} \] and \[ h_ n(x)=2^{\lambda}e^{x/2}(x/2)^{-\lambda}I_{\lambda +n}(x/2) \] using the integral representation \[ \phi^*(a,c;x)=\frac{1}{\Gamma(a)\Gamma(c-a)(e^{2\pi ia}-1)(e^{2\pi i(c-a)}-1)}\times \] \[ \times \int_{(1^+,0^+,1^-,0^- )}e^{xt}t^{a-1}(1-t)^{c-a-1}dt \] where the contour is a closed curve on the Riemann surface of the integrand, going round 1 and 0, firstly in the positive sense and then in the negative sense. Further, the author obtains a four term recursion formula, a generating function, a set of difference and differential recursion formulae for \(R_ n^*(a,c,\lambda;w)\). Some special cases are also obtained.
scientific article; zbMATH DE number 3842260