Recently, I found a (conjectural) new series for $\sqrt3\pi$: $$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$ The series converges fast with converging rate $1/9$, so one can easily check the identity \eqref{1} numerically.

Motivated by \eqref{1}, I also conjecture that $$\sum_{k=1}^\infty\frac{\binom{4k}{2k}\left((8k-3)(5H_{2k-1}-4H_{k-1})-6\right)}{k(4k-1)9^k\binom{2k}k^2}=\frac32\sum_{n=0}^\infty\left(\frac1{(3n+1)^2}-\frac1{(3n+2)^2}\right) \label{2}\tag{2}$$ and $$\sum_{k=1}^\infty\frac{\binom{4k}{2k}(8k-3)\left(2H_{2k-1}^{(2)}-5H_{k-1}^{(2)}\right)}{k(4k-1)9^k\binom{2k}k^2}=\frac{\pi^3}{36\sqrt3}, \label{3}\tag{3}$$ where $$H_n:=\sum_{0<k\le n}\frac1k\ \ \ \text{and}\ \ \ \ H_n^{(2)}:=\sum_{0<k\le n}\frac1{k^2}$$ for each $n=0,1,2,\ldots$.

**QUESTION.** Can one prove the new identities \eqref{1},\eqref{2} and \eqref{3} by using known tools (such as the WZ method and various hypergeometric series identities) ?

For the identities \eqref{1},\eqref{2} and \eqref{3}, we also have corresponding conjectural $p$-adic congruences. For example, I conjecture that for any prime $p>3$ we have $$\sum_{k=1}^{(p-1)/2}\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}\equiv-\frac5{36}pB_{p-2}\left(\frac13\right)\pmod{p^2}\tag{4}$$ and $$\sum_{k=1}^{(p-1)/2}\frac{\binom{4k}{2k}\left((8k-3)(5H_{2k-1}-4H_{k-1})-6\right)}{k(4k-1)9^k\binom{2k}k^2}\equiv\frac16B_{p-2}\left(\frac13\right)\pmod p,\tag{5}$$ where $B_{p-2}(x)$ denotes the Bernoulli polynomial of degree $p-2$.

Your comments are welcome!

parametric excessof a hypergeometric is the sum of its bottom parameters, minus the sum of its top parameters, and is an important quantity in the study of hypergeometric series, and it happens to be $-1/2$ in this case… $\endgroup$inverse symbolic calculatorto guess a nice closed form in terms of common irrational numbers such as square roots of natural numbers, $e$, and $\pi$. I think in the literature on experimental mathematics there should be discussions of exhaustive searches of certain input values for interesting hypergeometric identities. $\endgroup$4more comments