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}.

Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.

The Euler number \begin{document}$ E_n $\end{document} (resp. Entringer number \begin{document}$ E_{n,k} $\end{document}) enumerates the alternating (down-up) permutations of \begin{document}$ \{1,\dots,n\} $\end{document} (resp. starting with \begin{document}$ k $\end{document}). The Springer number \begin{document}$ S_n $\end{document} (resp. Arnold number \begin{document}$ S_{n,k} $\end{document}) enumerates the type \begin{document}$ B $\end{document} alternating permutations (resp. starting with \begin{document}$ k $\end{document}). In this paper, using bijections we first derive the counterparts in André permutations and Simsun permutations for the Entringer numbers \begin{document}$ (E_{n,k}) $\end{document}, and then the counterparts in signed André permutations and type \begin{document}$ B $\end{document} increasing 1-2 trees for the Arnold numbers \begin{document}$ (S_{n,k}) $\end{document}.

Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.