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Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers · arXivDesk

2608.13192Aug 13, 202639 pages

Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers

Zhi-Hong Sun

Abstract

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 align* &g₀(x)=1,\ g₁(x)=x+12,\ (n+1)²g_n+1(x)=(2n(n+1)+x+12)g_n(x)-n²g_n-1(x)\ (n≥ 1), \&v₀(x)=1,\ v₁(x)=x,\ (n+1)³v_n+1(x)=(2n+1)(n(n+1)+x)v_n(x)-n³v_n-1(x)\ (n≥ 1).align*

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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 align* &g₀(x)=1,\ g₁(x)=x+12,\ (n+1)²g_n+1(x)=(2n(n+1)+x+12)g_n(x)-n²g_n-1(x)\ (n≥ 1), \&v₀(x)=1,\ v₁(x)=x,\ (n+1)³v_n+1(x)=(2n+1)(n(n+1)+x)v_n(x)-n³v_n-1(x)\ (n≥ 1).align*

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The paper addresses: Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-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 align* &g₀(x)=1,\ g₁(x)=x+12,\ (n+1)²g_n+1(x)=(2n(n+1)+x+12)g_n(x)-n²g_n-1(x)\ (n≥ 1), \&v₀(x)=1,\ v₁(x)=x,\ (n+1)³v_n+1(x)=(2n+1)(n(n+1)+x)v_n(x)-n³v_n-1(x)\ (n≥ 1).align*