Zhi-Hong Sun

Generalizations of the Christoffel-Darboux formula and congruences involving Ap\'ery-like numbers

In this paper, we first extend the Christoffel-Darboux formula for orthogonal polynomials to general three-term recurrence sequences, and then investigate the identities and congruences for $g_n(x)$ and $v_n(x)$ given by \begin{align*}&g_0(x)=1,\ g_1(x)=\frac{x+1}2,\ (n+1)^2g_{n+1}(x)=\Big(2n(n+1)+\frac{x+1}2\Big)g_n(x...