On new divisibility properties of generalized central trinomial coefficients and Legendre polynomials

2023/11/24 by Jovan Mikić, Mikić, Jovan
Mathematics · Computer Science · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2311.14623

Abstract

We present a new formula for the highest power of a+b that divides the sum B(n,m,a,b)=∑_k=0ⁿ\binomnk^m a^n-kb^k for the case m=2. By using this formula, we give complete 3-adic valuation for central Dellanoy numbers. Also, we find the highest power of an odd integer x that divides Legendre's polynomial P_n(x). By using the same idea, generalized trinomial coefficients and generalized Motzkin numbers are treated. As a result, we give complete 3-adic valuation for little Schröder numbers and restricted hexagonal numbers. By using new class of binomial sums, we examine divisibility of B(n,m, a,b) by powers of a+b for m >2.

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