[Paper Review] Some Generalized Harmonic Number Identities

M. J. Kronenburg|arXiv (Cornell University)|Mar 28, 2011

Scientific Research and Discoveries2 references13 citations

TL;DR

This paper derives closed-form identities for finite sums involving generalized harmonic numbers using summation by parts and the Euler-Maclaurin formula. It provides explicit polynomial expressions for sums of the form $\sum k^p H_k^{(m)}$ and $\sum k^p H_{n-k}^{(m)}$, with results valid for nonnegative integers $p$, complex $m$, and arbitrary $n$, enabling efficient evaluation of harmonic series with polynomial weights.

ABSTRACT

Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite series of generalized harmonic numbers involving nonnegative integer powers, which can be used to evaluate the sums of the finite series of generalized harmonic numbers involving polynomials. Many examples and a computer program are provided.

Motivation & Objective

• To derive general closed-form expressions for finite sums of generalized harmonic numbers weighted by polynomials.
• To extend classical harmonic number identities to generalized forms with complex order $m$ and offset $c$.
• To provide computationally efficient formulas for evaluating sums involving $k^p H_k^{(m)}$ and $k^p H_{n-k}^{(m)}$.
• To support symbolic computation of such sums via a Mathematica program provided in the paper.
• To unify and generalize known results on harmonic number sums using advanced summation techniques.

Proposed method

• Applies summation by parts with $x_k = (k+1)^w - k^w$ and $y_k = H_k^{(m)}$ to derive identity (3.1) for generalized harmonic sums.
• Uses the Euler-Maclaurin formula to express power sums $\sum k^p$ as polynomials in $n$, leveraging Bernoulli numbers $B_{p-k+1}^+$.
• Reverses summation order in double sums $\sum_k k^p \sum_i 1/i^m$ to express $\sum k^p H_k^{(m)}$ in terms of generalized harmonic numbers and Bernoulli-weighted sums.
• Applies binomial expansion and coefficient reversal techniques to derive $\sum k^p H_{n-k}^{(m)}$ in terms of $H_n^{(-p)}$, $H_n^{(m)}$, and Bernoulli-weighted harmonic terms.
• Introduces offset $c = s$ via $H_{s,n}^{(m)} = H_{s+n}^{(m)} - H_s^{(m)}$ to generalize results to non-zero starting indices.
• Implements a Mathematica program using symbolic manipulation to compute and verify the derived identities efficiently.

Experimental results

Research questions

• RQ1 How can finite sums of the form $\sum_{k=0}^n k^p H_k^{(m)}$ be expressed in closed form for nonnegative integer $p$ and complex $m$?
• RQ2 What is the general structure of sums $\sum_{k=0}^n k^p H_{n-k}^{(m)}$ involving generalized harmonic numbers with polynomial weights?
• RQ3 Can the Euler-Maclaurin formula be effectively used to derive polynomial expressions for generalized harmonic number sums?
• RQ4 How can generalized harmonic numbers with non-zero offsets $c$ be handled in finite sum identities?
• RQ5 What symbolic computation framework enables automated evaluation of such harmonic sums?

Key findings

• The identity $\sum_{k=0}^n [(k+1)^w - k^w] H_k^{(m)} = (n+1)^w H_n^{(m)} - H_n^{(m-w)}$ provides a general tool for telescoping harmonic series with rational or polynomial weights.
• For $\sum k^p H_k^{(m)}$, the closed-form expression is $H_n^{(-p)}H_n^{(m)} + H_n^{(m-p)} - \frac{1}{p+1} \sum_{k=1}^{p+1} \binom{p+1}{k} B_{p-k+1}^+ H_n^{(m-k)}$, valid for nonnegative integer $p$.
• The sum $\sum k^p H_{n-k}^{(m)}$ is given by a more complex formula involving $H_n^{(-p)}$, $H_n^{(m)}$, and Bernoulli-weighted harmonic terms with alternating signs.
• Explicit evaluations for $p = 0$ to $5$ and $m = 2, 3, 4$ are tabulated, showing polynomial dependence on $n$ and harmonic numbers.
• The paper derives identities like $\sum_{k=0}^n k^4 H_{n+k}^{(2)} = H_n^{(-4)}(2H_{2n+1}^{(2)} - H_n^{(2)}) - \frac{1}{30}(30n^4 + 60n^3 + 30n^2 - 1)(H_{2n+1} - H_n) + \frac{1}{60}(n+1)(35n^3 + 47n^2 + 5n - 2)$.
• A Mathematica program is provided that computes these sums symbolically, using $0^0 = 1$ and modified Bernoulli numbers $B_n^+$, enabling automated verification and application.