On sums of Apéry polynomials and related congruences

Abstract

The Apéry polynomials are given by $$A_n(x)=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)}^2{x}^k(n=0,1,2,\dots ).$$ (Those $A_n=A_n(1)$ are Apéry numbers.) Let $p$ be an odd prime. We show that $$\sum _{k=0}^{p-1}(-1{)}^k{A}_k(x)\equiv \sum _{k=0}^{p-1}\frac{{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}{16^k}{x}^k(\mathrm{mod}{p}^2),$$ and that $$\sum _{k=0}^{p-1}{A}_k(x)\equiv \left(\frac{x}{p}\right)\sum _{k=0}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{4k}{k,k,k,k}\right)}{(256x{)}^k}(\mathrm{mod}p)$$ for any $p$-adic integer $x\not\equiv 0(\mathrm{mod}p)$. This enables us to determine explicitly ${\sum }_{k=0}^{p-1}(\pm 1{)}^k{A}_k$ mod $p$, and ${\sum }_{k=0}^{p-1}(-1{)}^k{A}_k$ mod $p^2$ in the case $p\equiv 2(\mathrm{mod}3)$. Another consequence states that $$\sum_{k=0}^{p-1}(-1)^kA_k(-2)\equiv\begin{cases}4x^2-2p\pmod{p^2}&\mbox{if}\ p=x^2+4y^2\ (x,y\in\mathbb Z),\\0\pmod{p^2}&\mbox{if}\ p\equiv3\pmod4.\end{cases}$$ We also prove that for any prime $p>3$ we have $$\sum _{k=0}^{p-1}(2k+1)A_k\equiv p+\frac{7}{6}{p}^4{B}_{p-3}(\mathrm{mod}{p}^5)$$ where $B_0,B_1,B_2,\dots$ are Bernoulli numbers.

29 pages, final published version

References in corpus (4)

• Open Conjectures on Congruences
• On congruences related to central binomial coefficients
• A generalization of Carlitz's congruence
• On sums involving products of three binomial coefficients

Cited by in corpus (7)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials
• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$
• Generalized Legendre polynomials and related congruences modulo $p^2$
• Congruences for Franel numbers
• New congruences for sums involving Apery numbers or central Delannoy numbers
• On divisibility of sums of Apery polynomials