June 2023 CONGRUENCES INVOLVING SQUARES OF GENERALIZED CENTRAL TRINOMIAL COEFFICIENTS
Chen Wang, Jia-Yu Chen
Rocky Mountain J. Math. 53(3): 959-968 (June 2023). DOI: 10.1216/rmj.2023.53.959

Abstract

The generalized central trinomial coefficient Tn(b,c) introduced by Noe is defined as the coefficient of xn in the expansion of (x2+bx+c)n, where n={0,1,2,} and b,c. Let p be an odd prime. In this paper, we determine k=0p1kTk(b,c)2mk modulo p2 for any integer m satisfying the equation (md)2=16mc, where d=b24c. As applications, we prove that for any prime p>3, we have

n=0p1n(k=0n12knk2kk)234(3p)p(1p)(modp2)

and

n=0p1n(k=0n1(6)knk2kk)214(3p)p(modp2),

as conjectured by Sun (Finite Fields Appl. 46 (2017), 179–216), where (p) stands for the Legendre symbol.

Citation

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Chen Wang. Jia-Yu Chen. "CONGRUENCES INVOLVING SQUARES OF GENERALIZED CENTRAL TRINOMIAL COEFFICIENTS." Rocky Mountain J. Math. 53 (3) 959 - 968, June 2023. https://doi.org/10.1216/rmj.2023.53.959

Information

Received: 29 November 2021; Revised: 10 May 2022; Accepted: 24 July 2022; Published: June 2023
First available in Project Euclid: 21 July 2023

MathSciNet: MR4617924
zbMATH: 07731158
Digital Object Identifier: 10.1216/rmj.2023.53.959

Subjects:
Primary: 05A10
Secondary: 11A07 , 11B65 , 11B75

Keywords: binomial coefficients , congruences , generalized central trinomial coefficients

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 3 • June 2023
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