A family of polynomials and related congruences and series

TL;DR

The paper introduces the polynomial family $S_n^{(m)}(x)$ as a natural generalization of Apéry-type sequences, deriving a general expansion and, for $m=1$, a closed form and a recurrence that reveal rich combinatorial structure and connections to Apéry numbers via $S_n^{(2)}(1)=A_n$. It establishes a prime-mod congruence for sums of $S_k^{(0)}(x)$ in terms of the Legendre symbol and provides $p$-adic and polynomial congruence results for averaged sums of $S_k^{(1)}(x)$, linking these congruences to Ramanujan-type series for $1/\pi$. The work further expresses $S_n^{(1)}(x)$ in terms of generalized central trinomial coefficients, obtaining integer-valued sum-averages and $p$-adic implications, and collects extensive conjectures on congruences for $S_n^{(2)}(x)$ across various quadratic forms, suggesting deep arithmetic connections and motivating new $1/\pi$-type series. Overall, the paper blends combinatorial identities, modular congruences, and conjectural $p$-adic phenomena to deepen the understanding of Apéry-like polynomials and their arithmetic.

Abstract

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac x{2x-1}\left(1+\left(\frac{1-4x^2}p\right)\right)\pmod p $$ for any odd prime $p$ and integer $x\not\equiv1/2\pmod p$, where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also formulate some open conjectures on related congruences and series for $1/π$. For example, we conjecture that $$\sum_{k=0}^\infty(7k+1)\frac{S_k^{(2)}(1/11)}{9^k}=\frac{5445}{104\sqrt{39}\,π}$$ and $$\sum_{k=0}^\infty(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt2\,π}.$$

Theorems & Definitions (54)

• Theorem 1.1
• Remark 1.1
• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.2
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3