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 [Formula: see text] 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 [Formula: see text], 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 [Formula: see text] is a power of [Formula: see text], there exist odd integers [Formula: see text] and even integers [Formula: see text], both independent of [Formula: see text], 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 in 2012.