Generalized Harmonic Numbers Overview

Updated 28 December 2025

Generalized harmonic numbers are finite sums that extend classical harmonic numbers using adjustable orders and parameter shifts.
They are derived via hypergeometric transformations, generating functions, and telescoping identities, yielding closed-form evaluations.
Their applications span combinatorics, algorithm analysis, and analytic number theory, underpinning series convergence and zeta function relations.

A generalized harmonic number is a finite sum that extends the classical harmonic numbers to incorporate higher orders, parameter shifts, and algebraic, combinatorial, or analytic generalizations. The concept is ubiquitous in combinatorics, analytic number theory, special function theory, and algorithm analysis, appearing in the context of summation identities, generating functions, modular congruences, and as structural building blocks for zeta and polylogarithm transformations.

1. Definitions, Notation, and Classical Framework

Let $n\ge1$ and $r\in\mathbb{C}$ . The classical generalized harmonic numbers of order $r$ are

$H_n^{(r)} = \sum_{k=1}^n \frac{1}{k^r},$

with $H_0^{(r)}=0$ by convention. For $r=1$ this recovers the ordinary harmonic numbers $H_n$ .

Shifted (or parametrically deformed) generalized harmonic numbers introduce a complex parameter $x$ :

$H_n^{(r)}(x) = \sum_{k=1}^n \frac{1}{(x + k)^r},\qquad H_n^{(r)} = H_n^{(r)}(0).$

Variants relevant in combinatorics include:

Recursive harmonic numbers: $H_n^{(m)} = \sum_{k=1}^n H_k^{(m-1)}$ , with $H_n^{(0)} = 1/n$ (Maw et al., 2017).
Exponentialized/generalized harmonic numbers: For $\alpha\in\mathbb{C}$ ,

$H_n(\alpha) = \sum_{k=1}^n \frac{\alpha^k}{k}.$

The normalized binomial inversion and telescoping identities allow analytic continuation in order $r$ and parameter shifts $x$ (Kronenburg, 2011). Recurrence relations, such as $H_n^{(r)} = H_{n-1}^{(r)} + 1/n^r$ , are foundational.

2. Summation Identities and Algebraic Transformations

Generalized harmonic numbers are central ingredients in a wide family of summation formulas. The most developed framework involves hypergeometric transformations and the use of the derivative operator $D_x$ combined with binomial sums or telescoping products.

Watson's $_3F_2$ identities and their derivatives yield explicit closed forms for binomial sums involving harmonic factors. For example, for $n=2m$ or $2m+1$, $x\in\mathbb{C}$ ,

$\sum_{k=0}^n (-1)^k \binom{n}{k} \frac{x+k}{2x+n+k} H_k(x) = \begin{cases} H_m(x) + 2H_{2m} - H_m, & n=2m, \ H_m(x) - 2H_{2m+1} - H_m(x), & n=2m+1, \end{cases}$

with parameter conditions as above (Wei, 2016).

When combined with the derivative operator or higher-order differentiation, these identities extend to sums with higher degree polynomial weights, higher-order harmonic numbers, and shifted parameters. The general approach is to start from a hypergeometric $_3F_2$ identity, differentiate with respect to an auxiliary variable or parameter, and reinterpret resulting sums as generalized harmonic numbers (Wei, 2016, Wei et al., 2012).

Combinatorial/Ablian framework: The product-difference equation and summation by parts are the foundation of many combinatorial identities involving generalized harmonic numbers, allowing for complex parameter shifts and negative orders. Binomial and gamma-function representations provide generality over both integer and complex parameters (Kronenburg, 2011, Kronenburg, 2011).

3. Generating Functions and Analytic Properties

Generalized harmonic numbers admit generating functions, both ordinary and exponential, which encode their algebraic structure and analytic properties. For $|x|<1$ , the classical formula is:

$\sum_{n\ge1} H_n^{(r)} x^n = \frac{\operatorname{Li}_r(x)}{1-x},$

where $\operatorname{Li}_r(x)$ is the polylogarithm of order $r$ (Adegoke, 2015, Kargın et al., 2020).

For exponentialized harmonic numbers $H_n(\alpha)$ ,

$\frac{\ln(1-\alpha t)}{1-t} = -\sum_{n=1}^{\infty} H_n(\alpha) t^n, \quad |\alpha t|<1,$

so specializations to $\alpha = 1$ and $\alpha = -1$ recover the generating functions for classical and skew-harmonic numbers respectively (Peregrino, 20 Dec 2025).

Recursion and differential properties include

$\frac{d}{dx} H_n^{(r)}(x) = -r\, H_n^{(r+1)}(x),$

and higher derivatives yield polynomials in $H_n^{(s)}(x)$ , central in the derivation of closed-form combinatorial identities (Kim et al., 2022).

4. Connections to Zeta Values, Euler Sums, and Series Acceleration

Generalized harmonic numbers provide rapidly converging series representations for special values of the Riemann and Hurwitz zeta functions. For $k\in\mathbb{N}$ :

$\zeta(k,1) = \frac{2^{k-1}}{2^{k-1}-1} \sum_{n=1}^\infty \frac{H_n^{(k-1)}}{n\, 2^n},$

where $H_n^{(k-1)}$ denotes the (iterated) generalized harmonic numbers (Szabłowski, 2015).

Linear Euler sums, defined as $E(m, n) = \sum_{v=1}^{\infty} \frac{H_v^{(m)}}{v^n}$ , decompose into combinations of Tornheim double series $T(r, s, t)$ and reduce in specific combinations to polynomials in zeta constants (Adegoke, 2015). The interplay between harmonic numbers and zeta values explains the centrality of these sums in evaluation of transcendental constants and in analytic number theory.

Explicit closed-forms for series such as

$\sum_{n=1}^{\infty} \frac{H_n}{n^3} = \frac{\pi^4}{72}$

are derived via polylogarithm integral representations, reflecting the deep connection with poly-Bernoulli numbers, Bernoulli numbers, and special values of the polylogarithm (Li, 2021, Kargın et al., 2020).

Nonlinear analogues arise, e.g.,

$\sum_{n=1}^\infty \frac{(H_n^{(k)})^2}{n^{k}} = \zeta(k)^3 - 5\zeta(3k),$

and alternating or even-odd parity analogues are also characterized (Adegoke et al., 2015).

5. Generalizations: q-Analogues, Hyperharmonic, and Parameteric Extensions

q-analogues: Three distinct q-generalizations of harmonic numbers have been developed (Mező, 2011). The first is

$H_{n,q}^{(1)} = \sum_{k=1}^n \frac{1}{[k]_q},$

where $[k]_q = (1 - q^k)/(1-q)$ . Other formulations use q-Stirling numbers or basic-hypergeometric expansions. As $q\to 1$ , these reduce to their classical counterparts.

Hyperharmonic numbers, defined recursively as $H_n^{(p,q)}$ , with multiple indices, relate to poly-Bernoulli polynomials. Their exponential generating function is

$\sum_{n=0}^\infty H_n^{(p,q)} t^n = \frac{\operatorname{Li}_p(t)}{(1-t)^q},$

and their structure encodes combinatorial information about set partitions, congruences, and structural symmetries (Kargın et al., 2020).

Further generalizations include binomial transformations, shifted harmonic numbers, and mixed polynomial weights, and direct connections to classical orthogonal polynomials (Laguerre, Fibonacci, etc.) (Peregrino, 20 Dec 2025).

6. Combinatorial, Algorithmic, and Number-Theoretic Applications

Generalized harmonic numbers feature in:

Enumerative combinatorics (e.g., counting problems with weighted binomial sums) (Wei, 2016).
Analysis of algorithms (e.g., expected running times; average path length in trees), where harmonic sums often appear in the analysis of average-case performance (Kim et al., 2022).
Symbolic summation algorithms (Zeilberger's algorithm, Abel–Gosper methods) for systematically reducing summations with harmonic terms to closed forms (Wei, 2016, Jin et al., 2014).

Tables of identities and recursive formulae enable both direct evaluations and algorithmic reduction of sums. For number theory, $p$ -adic congruence results (such as generalized Eisenstein and Wolstenholme congruences) for $H_{p-1}^{(j)}$ extend classical properties to higher orders and connect Bernoulli numbers and harmonic sums (Gy, 2019).

7. Binomial Sums, Recursive Structures, and Algorithmic Summation

Finite sum identities involving generalized harmonic numbers and binomial coefficients underlie inversion theorems and structure the transformation of indefinite to definite sums. Canonical results include

$H_n^{(m)} = \sum_{k=1}^n (-1)^{k+1} \binom{n}{k} \frac{1}{k^m},$

mirroring the recursive, Pascal-triangle-like structure, and admitting generating function representations

$\sum_{n\ge 1} H_n^{(m)} x^n = \frac{-\ln(1-x)}{(1-x)^m},\quad |x|<1,$

(Maw et al., 2017). Abelian and Gosper-type approaches are used to systematically translate binomial-harmonic double sums into products or determinants, enabling automated proof and discovery of new identities (Jin et al., 2014).

Algorithmic frameworks for symbolic computation of such identities have been implemented and facilitate further application in computational mathematics and experimental number theory (Kronenburg, 2011).

References

(Wei, 2016) Watson-type $_3F_2$ -series and summation formulae involving generalized harmonic numbers
(Wei, 2016) Saalschütz's theorem and summation formulae involving generalized harmonic numbers
(Kronenburg, 2011) Some Combinatorial Identities some of which involving Harmonic Numbers
(Kronenburg, 2011) Some Generalized Harmonic Number Identities
(Adegoke et al., 2015) New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers
(Maw et al., 2017) Recursive Harmonic Numbers and Binomial Coefficients
(Peregrino, 20 Dec 2025) Generalized Harmonic Numbers: Identities and Properties
(Adegoke, 2015) On generalized harmonic numbers, Tornheim double series and linear Euler sums
(Kargın et al., 2020) Generalized harmonic numbers via poly-Bernoulli polynomials
(Mező, 2011) Some possible $q$ -generalizations of harmonic numbers
(Sousa, 2018) Generalized Harmonic Numbers
(Kim et al., 2022) Some identities on generalized harmonic numbers and generalized harmonic functions
(Li, 2021) Integrals of polylogarithms and infinite series involving generalized harmonic numbers
(Szabłowski, 2015) On connection between values of Riemann zeta function at integers and generalized harmonic numbers
(Jin et al., 2014) Abel's Lemma and Identities on Harmonic Numbers
(Sousa, 2018) Generalized Harmonic Progression
(Gy, 2019) Extended Congruences for Harmonic Numbers