Congruences concerning generalized central trinomial coefficients

Chen, Jia-Yu; Wang, Chen

doi:10.1090/proc/15985

Congruences concerning generalized central trinomial coefficients
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by Jia-Yu Chen and Chen Wang PDF

Proc. Amer. Math. Soc. 150 (2022), 3725-3738 Request permission

Abstract:

For any $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and $b,c\in \mathbb {Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summations $\sum _{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), pp. 1375–1400].

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