Supercongruences involving Domb numbers and binary quadratic forms

In this paper, we prove two recently conjectured supercongruences (modulo \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^3$$\end{document}p3, where p is any prime greater than 3) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.


Introduction
The Domb numbers {D n }, defined by for non-negative integers n, first appeared in an extensive study by C. Domb [4] on interacting particles on crystal lattices.In particular, Domb showed that D n counts the number of 2n-step polygons on the diamond lattice.The Domb numbes also appear in a variety of other settings, such as in the coefficients in several known series for 1/π.For example, from [1,Equation (1.3)] we know that 2010 Mathematics Subject Classification.Primary 11A07; Secondary 11B65, 11B83, 11E16.
The first author was supported by the Natural Science Foundation of China (grant 12001288) and the China Scholarship Council (202008320187).
The second author was partially supported by FWF Austrian Science Fund grant P 32305.
In [10,Theorem 3.1], M.D. Rogers showed the following generating function for the Domb numbers by applying a rather intricate method: where |u| is sufficiently small.Y.-P.Mu and Z.-W.Sun [9,Equation (1.11)] proved a congruence involving the Domb numbers by applying the telescoping method: For any prime p > 3, we have the supercongruence where q p (a) denotes the Fermat quotient (a p−1 − 1)/p.In [5], J.-C. Liu proved a couple of conjectures of Z.-W.Sun and Z.-H.Sun.In particular he confirmed [5,Theorem 1.3] that for any positive integer n the two sums are also positive integers.
Z.-H.Sun [17,Conjecture 4.1] conjectured the following congruence for the Domb numbers: Let p > 3 be a prime.Then where {B n } are the Bernoulli numbers given by This conjecture was confirmed by the first author and J. Wang [7].
The main result of this paper is Theorem 1.1 which contains two supercongruences that were originally conjectured by Z.-H.Sun in [19,Conjecture 3.5,Conjecture 3.6].What makes them interesting is that their formulations involve the binary quadratic form x 2 +3y 2 for primes p that are congruent to 1 modulo 3. (It is well-known that any prime p ≡ 1 (mod 3) can be expressed as p = x 2 + 3y 2 for some integers x and y, an assertion first made by Fermat and subsequently proved by Euler, see [3].In his paper [19], Sun stated further conjectures of similar type, involving different moduli, and other binary quadratic forms.)First, Sun defined The two supercongruences which we will confirm are as follows.
Theorem 1.1.Let p > 3 be a prime.Then Our preparations for the proof of this theorem consist of seven lemmas that we give in Section 2. These are used in Section 3, devoted to the actual proof of Theorem 1.1.As tools for establishing the results in Sections 2 and 3 we utilize some congruences from [6,8] and several combinatorial identities that can be found and proved by the package Sigma [12] via the software Mathematica.
Lemma 2.3.Let p > 3 be a prime.For any p-adic integer t, we have which completes the proof of Lemma 2.3.
Again, by [6, pp.14-15], we have It is easy to check that the right-side of the above congruence is congruent to 4 25 4x 2 − 2p − p 2 4x 2 modulo p 3 .Therefore we immediately get the desired result stated in Lemma 2.4.