Congruences for sums of Delannoy numbers and Legendre polynomials

In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Legendre polynomial [Formula: see text] and central Delannoy numbers [Formula: see text]. Let [Formula: see text] and p be an odd prime. It is proved that, for any [Formula: see text], there exist [Formula: see text] and [Formula: see text], both free of p, such that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the Legendre symbol. When n is a power of 2, there exist odd integers [Formula: see text] and even integers [Formula: see text], both independent of n, such that [Formula: see text] and [Formula: see text] The case [Formula: see text] in the last congruence confirms a conjecture of Guo and Zeng [New congruences for sums involving Apéry numbers or central Delannoy numbers, Int. J. Number Theory 8 (2012) 2003–2016].