American Institute of Mathematical Sciences

The pentagonal numbers are the integers given by\begin{document}$ p_5(n) = n(3n-1)/2\ (n = 0,1,2,\ldots) $\end{document}.Let \begin{document}$ (b,c,d) $\end{document} be one of the triples \begin{document}$ (1,1,2),(1,2,3),(1,2,6) $\end{document} and \begin{document}$ (2,3,4) $\end{document}.We show that each \begin{document}$ n = 0,1,2,\ldots $\end{document} can be written as \begin{document}$ w+bx+cy+dz $\end{document} with \begin{document}$ w,x,y,z $\end{document} pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat.

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

Using Watson's terminating \begin{document}$ _8\phi_7 $\end{document} transformation formula, we prove a family of \begin{document}$ q $\end{document}-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo \begin{document}$ p^4 $\end{document} (\begin{document}$ p $\end{document} is an odd prime) and their \begin{document}$ q $\end{document}-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

Via symbolic computation we deduce 97 new type series for powers of \begin{document}$ \pi $\end{document} related to Ramanujan-type series. Here are three typical examples:

\begin{document}$ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $\end{document}

with

\begin{document}$ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $\end{document}

\begin{document}$ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $\end{document}

and

\begin{document}$ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $\end{document}

where the generalized central trinomial coefficient \begin{document}$ T_k(b,c) $\end{document} denotes the coefficient of \begin{document}$ x^k $\end{document} in the expansion of \begin{document}$ (x^2+bx+c)^k $\end{document}. We also formulate a general characterization of rational Ramanujan-type series for \begin{document}$ 1/\pi $\end{document} via congruences, and pose 117 new conjectural series for powers of \begin{document}$ \pi $\end{document} via looking for corresponding congruences. For example, we conjecture that

\begin{document}$ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $\end{document}

Eighteen of the new series in this paper involve some imaginary quadratic fields with class number \begin{document}$ 8 $\end{document}.