Congruences involving generalized central trinomial coefficients

By Zhi-Wei Sun. Science China Mathematics, 2010

84 citations recorded by the source corpus.

For integers b and c the generalized central trinomial coefficient Tn(b, c) denotes the coefficient of xn in the expansion of (x2 + bx + c)n. Those Tn = Tn(1, 1) (n = 0, 1, 2, …) are the usual central trinomial coefficients, and Tn(3, 2) coincides with the Delannoy number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_n = \sum\nolimits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} \left( {\begin{array}{*{20}c} {n + k} \\ k \\ \end{array} } \right)$$\end{document} in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each n = 1, 2, 3, …, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{k = 0}^{n - 1} {\left( {2k + 1} \right)T_k \left( {b,c} \right)^2 \left( {b^2 - 4c} \right)^{n - 1 - k} \equiv 0 \left( {\bmod n^2 } \right)}$$\end{document} and in particular \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left. {n^2 } \right|\sum\nolimits_{k = 0}^{n - 1} {\left( {2k + 1} \right)D_k^2 }$$\end{document} is an odd prime then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{k = 0}^{p - 1} {T_k^2 \equiv \left( {\frac{{ - 1}} {p}} \right)\left( {\bmod p} \right) and } \sum\limits_{k = 0}^{p - 1} {D_k^2 \equiv \left( {\frac{2} {p}} \right)\left( {\bmod p} \right), }$$\end{document} where (−) denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/f9baada47186002b1ed3752d95c28c37dd9be3f3