[Paper Review] Abel's Lemma and Identities on Harmonic Numbers

Hai-Tao Jin, Daniel K. Du|arXiv (Cornell University)|Nov 25, 2014

Mathematical and Theoretical Analysis12 references4 citations

TL;DR

This paper introduces the Abel-Gosper and Abel-WZ methods to derive and verify identities involving harmonic and generalized harmonic numbers. By combining Abel's lemma on summation by parts with Gosper's algorithm and the WZ method, the authors establish a new inversion formula for generalized harmonic numbers and prove several novel identities, including a generalized form of the classical harmonic sum identity involving binomial coefficients and harmonic numbers.

ABSTRACT

Recently, Chen, Hou and Jin used both Abel's lemma on summation by parts and Zeilberger's algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums involving harmonic numbers. In this paper, we use the Abel-Gosper method to prove an identity involving the generalized harmonic numbers. Special cases of this result reduce to many famous identities. In addition, we use both Abel's lemma and the WZ method to verify and to discover identities involving harmonic numbers. Many interesting examples are also presented.

Motivation & Objective

• To develop and apply the Abel-Gosper method for evaluating indefinite sums involving generalized harmonic numbers.
• To introduce the Abel-WZ method as a systematic framework for discovering and verifying identities on harmonic numbers using hypergeometric summation techniques.
• To generalize the classical inversion formula ∑(-1)^{k-1} C(n,k) H_k = 1/n to include generalized harmonic numbers and binomial coefficients.
• To provide closed-form evaluations of definite sums involving harmonic numbers through recurrence relations and algorithmic verification.

Proposed method

• The Abel-Gosper method combines Gosper's algorithm for indefinite hypergeometric summation with Abel's lemma on summation by parts to transform sums of the form ∑ f_k H_k into hypergeometric sums.
• The Abel-WZ method couples the WZ method for hypergeometric identities with Abel's lemma to generate recurrence relations and closed forms for sums involving harmonic numbers.
• The authors use Zeilberger's algorithm to derive and verify recurrence relations for sums involving non-hypergeometric terms such as harmonic numbers.
• Boundary value analysis and telescoping techniques are applied to simplify sums after transformation via Abel’s lemma.
• The method is applied to known hypergeometric identities to generate new identities involving H_k, H_k^{(2)}, and H_{mk+s}(x).
• Special cases are derived by substituting specific values for parameters m, s, p, n, and x in the generalized identity.

Experimental results

Research questions

• RQ1 Can the Abel-Gosper method be used to derive a generalized inversion formula for generalized harmonic numbers?
• RQ2 How can the Abel-WZ method be systematically applied to discover new identities involving harmonic numbers?
• RQ3 What are the closed-form evaluations of sums involving binomial coefficients, generalized harmonic numbers, and alternating signs?
• RQ4 Can the method recover known identities such as Prodinger’s identity on H_k^{(2)} and Calkin’s identity?
• RQ5 What are the special cases of the generalized identity that reduce to classical harmonic number identities?

Key findings

• The paper proves a generalized inversion formula: ∑_{k=p}^n (-1)^{k-1} C(n,k) C(k,p) H_{mk+s}(x) = (-1)^p m^{n-p-1} n! / ((n-p)p!) × ∑_{i=1}^m 1 / ∏_{u=p}^{n-1} (mu + s + x + i) for n > p.
• For m=1, s=0, x=0, the identity reduces to the classical result ∑_{k=p}^n (-1)^{k-1} C(n,k) C(k,p) H_k = (-1)^p / (n-p) × C(p,0) / C(n,0), which includes the known identity ∑(-1)^{k-1} C(n,k) H_k = 1/n as a special case.
• The Abel-WZ method successfully re-derives Prodinger’s identity: ∑_{k=0}^n (-1)^{n-k} C(n,k) C(n+k,k) H_k^{(2)} = 2 ∑_{k=1}^n (-1)^{k-1}/k^2.
• The method generates new identities such as ∑_{k=0}^n (-1)^{n-k} C(n,k) C(n+k,k) H_{2k} = 3H_n - H_{⌊n/2⌋} for n ≥ 1.
• The paper derives identities like ∑_{k=p}^n C(n,k)^2 C(k,p) H_k = C(2n-p,n) C(n,p) (2H_n - H_{2n-p}) from known binomial identities.
• For alternating sums, the method yields ∑_{k=0}^{2n} (-1)^k C(2n,k)^2 H_k = (-1)^n C(2n,n) (H_n + H_{2n})/2 and similar results for H_k^{(2)}.