Electronic Research Archive

Special issue on combinatorial number Th. & enumerative combinatorics

Zhi-Wei Sun, Department of Mathematics, Nanjing University, Nanjing 210093, China zwsun@nju.edu.cn

Jiang Zeng, Institut Camille JordanUniversité Claude Bernard Lyon 143, boulevard du 11 novembre 1918F-69622 Villeurbanne Cedex, France zeng@math.univ-lyon1.fr

On sums of four pentagonal numbers with coefficientsSpecial Issues
Dmitry Krachun and Zhi-Wei Sun
2020, 28(1): 559-566 doi: 10.3934/era.2020029 +[Abstract](11)+[HTML](10) +[PDF](252.97KB)

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.

Proof of Sun's conjectural supercongruence involving Catalan numbersSpecial Issues
Ji-Cai Liu
2020, 28(2): 1023-1030 doi: 10.3934/era.2020054 +[Abstract](23)+[HTML](12) +[PDF](324.81KB)

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.

A family of $ q $-congruences modulo the square of a cyclotomic polynomialSpecial Issues
Victor J. W. Guo
2020, 28(2): 1031-1036 doi: 10.3934/era.2020055 +[Abstract](22)+[HTML](11) +[PDF](258.92KB)

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.

New series for powers of $ \pi $ and related congruencesSpecial Issues
Zhi-Wei Sun
2020, 28(3): 1273-1342 doi: 10.3934/era.2020070 +[Abstract](12)+[HTML](10) +[PDF](720.4KB)

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:



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

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

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