Power series with the Riemann zeta-function in the coefficients
The author first gives, by complex integration, an elegant proof of Ramanujan's formula \[ \zeta (\nu,1+x) = \sum^\infty_{n=0} {(\nu)_n \over n!} \zeta(\nu+n) (-x)^n\qquad \bigl(|x |<1,\;(s)_n= \Gamma (s+n)/ \Gamma(s) \bigr), \] where \(\zeta (s,\alpha)\) is the Hurwitz zeta-function, and \(\nu \neq-1,0,1,2, \dots\). Then he obtains an asymptotic expansion (in terms of the \(K\)-Bessel function) of \[ G_\nu (x) = \sum_{n> \text{Re}\, \nu+1} \zeta (n-\nu) {(-x)^n \over n!}\qquad (x>0), \] thus establishing a conjecture of Chowla and Hawkins. This is done by using the representation \[ G_\nu(x) = {1\over 2 \pi i} \int^{c+i \infty}_{c-i \infty} \Gamma (-s) \zeta(s-\nu) x^s\,ds \quad \bigl(\text{Re}\, \nu+ 1<c< [\text{Re}\,\nu+2] \bigr). \] Finally he defines the functions \[ {\mathcal F}_\nu (\alpha,\beta; \gamma;z) = \sum^\infty_{n=0} {(\alpha)_n (\beta)_n \over (\gamma)_n n!} \zeta (\nu+n) z^n \quad \bigl(|z |<1\bigr), \] \[ {\mathcal F}_\nu (\alpha; \gamma; z) = \sum^\infty_{n=0} {(\alpha)_n \over (\gamma)_n n!} \zeta (\nu +n) z^n, \] where \(\gamma \neq 0\), \(-1,-2, \dots\). Since \(\lim_{\text{Re}\, \nu \to \infty} \zeta (\nu+n) = 1\) uniformly for \(n=0,1,2, \dots\), these functions generalize the classical hypergeometric functions \(F(\alpha, \beta; \gamma; z)\) and \(F(\alpha; \gamma; z)\), respectively. Some of the properties of the new functions are discussed, and detailed proofs will appear elsewhere.
scientific article; zbMATH DE number 938081
scientific article