Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials

Abstract

The Apery polynomials are defined by $A_n(x)={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)}^2{x}^k$ for all nonnegative integers $n$. We confirm several conjectures of Z.-W. Sun on the congruences for the sum ${\sum }_{k=0}^{n-1}(-1{)}^k(2k+1)A_k(x)$ with $x\in Z$.

11 pages, to appear in Journal of Number Theory

References in corpus (6)

• Binomial coefficients, Catalan numbers and Lucas quotients
• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$
• On sums of Apéry polynomials and related congruences
• On Apery numbers and generalized central trinomial coefficients
• Congruences involving generalized central trinomial coefficients
• New congruences for sums involving Apery numbers or central Delannoy numbers

Cited by in corpus (5)

• Proof of Sun's conjectures on integer-valued polynomials
• On two congruences involving Apéry and Franel numbers
• Two congruences concerning Apéry numbers
• On divisibility of sums of Apery polynomials
• A Simple Proof of Schmidt's Conjecture