eISSN:
 2688-1594

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)
Abstract:

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)
Abstract:

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)
Abstract:

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)
Abstract:

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:

with

and

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}.

Some multivariate polynomials for doubled permutationsSpecial Issues
Bin Han
2020,  doi: 10.3934/era.2020098 +[Abstract](15)+[HTML](11) +[PDF](374.51KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

More bijections for Entringer and Arnold familiesSpecial Issues
Heesung Shin and Jiang Zeng
2020,  doi: 10.3934/era.2020111 +[Abstract](15)+[HTML](11) +[PDF](418.67KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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