Congruences involving Franel and Catalan-Larcombe-French numbers

2015/02/09 by Zhi-Hong Sun, Sun, Zhi-Hong
Mathematics · #05A10 #05A19 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11A07 #Secondary 11E25

paper · pdf · doi:10.48550/arxiv.1502.02499

openalex publication_date 2015/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \f_n\ be the Franel numbers given by f_n=∑_k=0ⁿ\binom nk³, and let p>5 be a prime. In this paper we mainly determine ∑_k=0^p-1 \binom2kk(f_k)/(m^k)\pmod p for m=5,-16,16,32,-49,50,96. Let S_n=∑_k=0ⁿ\binom nk\binom2kk\binom2n-2kn-k. We also determine ∑_k=0^p-1 \binom2kk(S_k)/(m^k)\pmod p for m=7,16,25,32,64,160,800,1600, 156832.

Citations

• Some conjectures on congruences
• Super congruences involving Bernoulli and Euler polynomials
• Congruences for Domb and Almkvist-Zudilin numbers

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