Proof of Sun's conjectures on integer-valued polynomials

TL;DR

This paper proves two of Z.-W. Sun's conjectures on integer-valued polynomials related to Delannoy numbers by establishing new summation formulas and demonstrating integrality of certain rational sums at integers.

Contribution

It introduces new summation formulas for polynomials related to Delannoy numbers and confirms Sun's conjectures on their integer-valued properties.

Findings

Proved Sun's conjectures on integer-valued polynomials

Derived new summation formulas for polynomial squares

Showed certain rational sums are integers at integer points

Abstract

Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to prove that certain rational sums involving even powers of those polynomials are integers whenever they are evaluated at integers. This confirms two conjectures of Z.-W. Sun. We also conjecture that many of these results have neat $q$-analogues.