On some conjectural supercongruences involving the sequence t_n(x)

2025/10/13 by Han, Hui-Li, Wang, Chen
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.11338

Abstract

In this paper, we study some supercongruences involving the sequence t_n(x)=∑_k=0ⁿ\binomnk\binomxk\binomx+kk2^k and solve some open problems. For any odd prime p and p-adic integer x, we determine ∑_n=0^p-1t_n(x)² and ∑_n=0^p-1(n+1)t_n(x)² modulo p²; for example, we establish that ∑_n=0^p-1t_n(x)²≡\begincases (\dfrac-1p)\pmodp²,amp;if 2x≡-1\pmodp,
(-1)^{⟨ x⟩_p}\dfracp+2(x-⟨ x⟩_p)2x+1\pmodp²,amp;otherwise, \endcases where ⟨ x⟩_p denotes the least nonnegative residue of x modulo p. This confirms a conjecture of Z.-W. Sun.

Citations

• On a conjectural supercongruence involving the dual sequence s_n(x)
• Congruences Related to Dual Sequences and Catalan Numbers

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