Ramanujan series upside-down
Based on the known example NEWLINE\[NEWLINE\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{6n}}\begin{pmatrix} 2n,\\ 2\end{pmatrix}{}^3\left(\frac{1}{2}+2n\right)NEWLINE\]NEWLINE and the so-called \textit{companion series identity} NEWLINE\[NEWLINE8\,L_{-4}(2)=\sum_{n=1}^{\infty}\frac{(-1)^n2^{6n}}{n^3\begin{pmatrix} 2n,\\ n\end{pmatrix}{}^3}\left(\frac{1}{2}-2n\right),NEWLINE\]NEWLINE where \(L_{-4}(2)=1-\frac{1}{3^2}+\frac{1}{5^2}+\ldots\) is Catalan's constant and \(L_k(s):=\sum_{n=1}^{\infty}\chi_k(n)/n^s\) denotes the general Dirichlet \(L\)-series with \(\chi_k(x)=\left(\frac{k}{n}\right)\) being the Jacobi symbol, the authors study seventeen formulas identified by Ramanujan of the form NEWLINE\[NEWLINE\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(s)_n(\frac{1}{2})_n(1-s)_n}{(1)_n^3}(a+bn)z^n,NEWLINE\]NEWLINE where \((x)_n=\Gamma(x+n)/\Gamma(x)\) is the Pochhammer symbol. In each case one has \(s\in\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\,\frac{1}{6}\}\) and \((a,b,z)\) parametrized by modular functions. Preserving these modular parametrizations for \((a,b,z)\), the general \textit{companion series} is given by NEWLINE\[NEWLINE\sum_{n=1}^{\infty}\frac{(1)_n^3}{(s)_n(\frac{1}{2})_n(1-s)_n}\frac{(a-bn)}{n^3}z^{-n}.NEWLINE\]NEWLINE For \(s\in\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\}\) many values of this companion series reduce to linear combinations of two Epstein zeta functions and elementary constants. If this linear combination of Epstein zeta functions reduces to Dirichlet \(L\)-values, then the companion series also reduces to Dirichlet \(L\)-values. Proofs are based upon a new idea called \textit{completing the hypergeometric function}. This approach fails completely when \(s=\frac{1}{6}\), which is explained. The methods used by the authors lead to several explicit formulas involving rapidly converging hypergeometric functions.
scientific article; zbMATH DE number 6334050
scientific article