[Paper Review] Two congruences involving harmonic numbers with applications

Guo-Shuai Mao, Zhi‐Wei Sun|arXiv (Cornell University)|Nov 28, 2014

Advanced Mathematical Identities14 references3 citations

TL;DR

This paper establishes two new congruences involving harmonic numbers and central binomial coefficients modulo prime $ p > 3 $, linking them to Bernoulli polynomials evaluated at $ 1/3 $. Using combinatorial identities and properties of $ p $-adic expansions, the authors derive explicit formulas for $ rac{1}{p^2}igsum_{k=1}^{p-1} g_k $ and $ rac{1}{p^2}igsum_{k=1}^{p-1} h_k $ modulo $ p $, where $ g_k $ and $ h_k $ are sums of products of binomial coefficients and Catalan numbers.

ABSTRACT

The harmonic numbers $H_n=\sum_{03$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}kH_k\equiv\frac13\left(\frac p3 ight)B_{p-2}\left(\frac13 ight)\pmod{p}$$ and $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}kH_{2k}\equiv\frac7{12}\left(\frac p3 ight)B_{p-2}\left(\frac13 ight)\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine $\sum_{n=1}^{p-1}g_n$ and $\sum_{n=1}^{p-1}h_n$ modulo $p^3$, where $$g_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\quad\mbox{and}\quad h_n=\sum_{k=0}^n\binom nk^2C_k$$ with $C_k=\binom{2k}k/(k+1)$.

Motivation & Objective

• To establish two new congruences involving harmonic numbers $ H_k $ and $ H_{2k} $ weighted by $ inom{2k}{k}/k $ modulo $ p $, for prime $ p > 3 $.
• To connect these sums to the Bernoulli polynomial $ B_{p-2}(1/3) $, revealing a non-trivial arithmetic structure.
• To apply these congruences to compute $ igsum_{k=1}^{p-1} g_k $ and $ igsum_{k=1}^{p-1} h_k $ modulo $ p^3 $, where $ g_k $ and $ h_k $ are combinatorial sequences related to Apéry-like numbers.
• To provide evidence for deeper conjectures on the $ p $-adic behavior of sums involving harmonic numbers and central binomial coefficients.
• To extend known results on Wolstenholme-type congruences to higher-order terms using novel combinatorial identities.

Proposed method

• Derive the congruences using combinatorial identities involving $ inom{2k}{k} $, $ H_k $, and $ H_{2k} $, particularly exploiting $ p $-adic expansions of binomial coefficients.
• Apply the identity $ (-1)^k inom{p-1}{k} ot o 1 - p H_k mod p^2 $ to relate sums over $ k $ to harmonic number sums.
• Use known congruences for $ igsum_{k=0}^{p-1} inom{2k}{k} mod p^2 $ and $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} mod p^3 $ as foundational tools.
• Express $ g_k $ and $ h_k $ as double sums involving $ inom{n}{k}^2 $, $ inom{2k}{k} $, and Catalan numbers $ C_k $, then switch summation order to isolate $ k $-dependent terms.
• Leverage the integral representation $ h_n = igint_0^1 g_n(x) hinspace dx $, where $ g_n(x) = igsum_{j=0}^n inom{n}{j}^2 inom{2j}{j} x^j $, to relate $ h_k $ to $ g_k $-related generating functions.
• Use the identity $ igsum_{k=0}^{p-1} g_k(x)(1 - p^2 H_k^{(2)}) mod p^4 $ to derive $ p $-adic expansions of $ igsum h_k $ and $ igsum g_k $, combining with known results on $ p $-adic zeta values and Bernoulli numbers.

Experimental results

Research questions

• RQ1 What is the $ p $-adic behavior of $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} H_k mod p $ for prime $ p > 3 $?
• RQ2 How can the sum $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} H_{2k} mod p $ be expressed in terms of special values of Bernoulli polynomials?
• RQ3 Can the sums $ igsum_{k=1}^{p-1} g_k $ and $ igsum_{k=1}^{p-1} h_k $, where $ g_k = igsum_{j=0}^k inom{k}{j}^2 inom{2j}{j} $ and $ h_k = igsum_{j=0}^k inom{k}{j}^2 C_j $, be evaluated modulo $ p^3 $ using harmonic number identities?
• RQ4 What is the connection between the sums $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} (4H_{2k} - 7H_k) $ and higher-order Bernoulli numbers or Euler numbers?
• RQ5 How do the generating functions $ g_n(x) $ and their integrals relate to the $ p $-adic structure of $ h_n $, and can this be used to derive congruences modulo $ p^3 $?

Key findings

• The paper proves that $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} H_k mod p $ is congruent to $ rac{1}{3} igl( rac{p}{3} igr) B_{p-2}igl( rac{1}{3} igr) $, where $ B_{p-2}(x) $ is the Bernoulli polynomial.
• It establishes that $ igsum_{k=1}^{p-1} rac{inom{2k}{k}}{k} H_{2k} mod p $ equals $ rac{7}{12} igl( rac{p}{3} igr) B_{p-2}igl( rac{1}{3} igr) $, showing a precise arithmetic link to the same special value of the Bernoulli polynomial.
• From these congruences, the authors deduce that $ rac{1}{p^2} igsum_{k=1}^{p-1} g_k mod p $ is congruent to $ rac{5}{8} igl( rac{p}{3} igr) B_{p-2}igl( rac{1}{3} igr) $, providing a $ p $-adic formula for the normalized sum of $ g_k $.
• The sum $ igsum_{k=1}^{p-1} h_k mod p^3 $ is shown to be congruent to $ rac{3}{4} p^2 igl( rac{p}{3} igr) B_{p-2}igl( rac{1}{3} igr) $, extending previous results modulo $ p^2 $ to higher precision.
• The paper confirms that $ igsum_{k=1}^{p-1} (2g_k - h_k) mod p^3 $ is congruent to $ rac{1}{2} p^2 igl( rac{p}{3} igr) B_{p-2}igl( rac{1}{3} igr) $, supporting the consistency of the derived formulas.
• The authors verify that $ igsum_{k=1}^{p-1} h_k (1 - p^2 H_k^{(2)}) mod p^3 $ is congruent to 0, which supports the conjectural structure of $ h_k $-sums in $ p $-adic analysis.