Congruences for sums of MacMahon's q-Catalan polynomials

2024/06/18 by Tewodros Amdeberhan, Amdeberhan, Tewodros, Roberto Tauraso +1
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2406.12332

Abstract

One variant of the q-Catalan polynomials is defined in terms of Gaussian polynomials by C_k(q)=\genfrac[]0pt2kk_q-q\genfrac[]0pt2kk+1_q. Liu studied congruences of the form ∑_k=0^n-1 q^kC_k modulo the cyclotomic polynomial Φ_n(q)², provided that n≡± 1\pmod3. Apparently the case n≡ 0\pmod3 has been missing from the literature. It is our primary purpose to fill this gap by the current work. In addition, we discuss certain fascinating link to Dirichlet character sum identities.

Related

• Some q-congruences arising from certain identities
• Congruences modulo cyclotomic polynomials and algebraic independence for q-series
• Congruences for the coefficients of MacMahon-like q-series
• A Generalization of MacMahon Series via Cyclotomic Polynomials
• A family of q-congruences modulo the square of a cyclotomic polynomial