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

Abstract

We present a new formula for the highest power of $a+b$ that divides the sum $B(n,m,a,b)=\sum_{k=0}^{n}\binom{n}{k}^m a^{n-k}b^k$ 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 Schr\"{o}der 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 $m >2$.