# Ramanujan-like series for $1/\pi^m$ and Dirichlet L-values

A rational Ramanujan-like series for $$\pi^{-m}$$ and character $$\chi$$ is a series with rational parameters which is of the following form:

$$\sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n}}{(1)_{n}} \right) \sum_{k=0}^{m} a_k n^k z_0^{n} = \frac{\sqrt{(-1)^m \chi}}{\pi^m}.$$

Consider the following extension with $$x$$:

$$\sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n+x}}{(1)_{n+x}} \right) \sum_{k=0}^{m} a_k (n+x)^k z_0^{n+x},$$

and replace $$z_0^{n+x}$$ with $$(-1)^n (-z_0)^x$$ if $$z_0<0$$. I can justify by means of a $$p$$-adic argument that for any particular series the coefficient of $$x^{2m+1}$$ in the development of the series is of the form $$r L(\chi, m+1)$$, where $$r$$ is a rational number, but I do not know how to prove that this is the case for all these kind of series.

We show two examples:

1-) If we develop in powers of $$x$$ the series

$$\sum_{n=0}^{\infty} \frac{\left(\frac12\right)_{n+x}\left(\frac14\right)_{n+x}\left(\frac34\right)_{n+x}}{(1)_{n+x}^3}(65(n+x)+8) (-1)^n \left( \frac{16^2}{63^2} \right)^{n+x},$$

that has $$\chi=-7$$, then the coefficient of $$x^3$$ is a rational multiple of $$L(-7, 2)$$.

2-) If we develop in powers of $$x$$ the series:

$$\begin{multline} \sum_{n=0}^{\infty} \frac{\left( \frac12 \right)_{n+x} \left( \frac13 \right)_{n+x} \left( \frac23 \right)_{n+x} \left( \frac16 \right)_{n+x} \left(\frac56 \right)_{n+x}}{(1)_{n+x}^5} (-1)^n \left( \frac{1}{80^3} \right)^{n+x} \\ \left(5418(n+x)^2+693(n+x)+29\right), \end{multline}$$

that has $$\chi=5$$, then the coefficient of $$x^5$$ is a rational multiple of $$L(5, 3)$$.

See arXiv:2312.16827

Any ideas?