harmonic number

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1: 25.16 Mathematical Applications

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25.16.5

H\left(s\right)=\sum_{n=1}^{\infty}\frac{H_{n}}{n^{s}},

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Symbols:: $H\left(\NVar{s}\right)$ : Euler sums, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer, $s$ : complex variable and $h(n)$ : harmonic number
Keywords:: Euler sum, infinite series, series representation
Source:: Apostol and Vu (1984, p. 85)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E5
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonicnumber $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

►where

H_{n}

is given by (25.11.33). … ►

25.16.13

\sum_{n=1}^{\infty}\left(\frac{H_{n}}{n}\right)^{2}=\frac{17}{4}\zeta\left(4% \right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $H_{\NVar{n}}$ : harmonic number, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonicnumber, infinite series, special value
Source:: Flajolet and Salvy (1998, (4-1), p. 24)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.16.E13
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonicnumber $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.16(ii), §25.16 and Ch.25

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2: 25.11 Hurwitz Zeta Function

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25.11.32

\int_{0}^{a}x^{n}\psi\left(x\right)\,\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n% \right)+(-1)^{n}H_{n}\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{% 0.0pt}{}{n}{k}H_{k}\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}% \genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},

n=1,2,\dots

\Re a>0

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $H_{\NVar{n}}$ : harmonic number, $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $h(n)$ : harmonic number
Keywords:: definite integral, integral representation
Source:: Adamchik (1998, (17), p. 197)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E32
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonicnumber $h(n)$ (resp. $h(k)$ ) has been replaced to be $H_{n}$ (resp. $H_{k}$ ).
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

►where

H_{n}

are the harmonic numbers: ►

25.11.33

H_{n}=\sum_{k=1}^{n}k^{-1}.

ⓘ

Defines:: $H_{\NVar{n}}$ : harmonic number
Symbols:: $k$ : nonnegative integer, $n$ : nonnegative integer and $h(n)$ : harmonic number
Keywords:: harmonicnumber
Source:: Knuth (1968, Section 1.2.7, p. 73)
Referenced by:: §25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E33
Encodings:: pMML, png, TeX
Notation (effective with 1.1.4):: The notation previously used for the harmonicnumber $h(n)$ has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

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3: Errata

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Chapter 18 Additions

The following additions were made in Chapter 18:

Section 18.2

In Subsection 18.2(i), Equation (18.2.1_5); the paragraph title “Orthogonality on Finite Point Sets” has been changed to “Orthogonality on Countable Sets”, and there are minor changes in the presentation of the final paragraph, including a new equation (18.2.4_5). The presentation of Subsection 18.2(iii) has changed, Equation (18.2.5_5) was added and an extra paragraph on standardizations has been included. The presentation of Subsection 18.2(iv) has changed and it has been expanded with two extra paragraphs and several new equations, (18.2.9_5), (18.2.11_1)–(18.2.11_9). Subsections 18.2(v) (with (18.2.12_5), (18.2.14)–(18.2.17)) and 18.2(vi) (with (18.2.17)–(18.2.20)) have been expanded. New subsections, 18.2(vii)–18.2(xii), with Equations (18.2.21)–(18.2.46),
Section 18.3

A new introduction, minor changes in the presentation, and three new paragraphs.
Section 18.5

Extra details for Chebyshev polynomials, and Equations (18.5.4_5), (18.5.11_1)–(18.5.11_4), (18.5.17_5).
Section 18.8

Line numbers and two extra rows were added to Table 18.8.1.
Section 18.9

Subsection 18.9(i) has been expanded, and 18.9(iii) has some additional explanation. Equations (18.9.2_1), (18.9.2_2), (18.9.18_5) and Table 18.9.2 were added.
Section 18.12

Three extra generating functions, (18.12.2_5), (18.12.3_5), (18.12.17).
Section 18.14

Equation (18.14.3_5). New subsection, 18.14(iv), with Equations (18.14.25)–(18.14.27).
Section 18.15

Equation (18.15.4_5).
Section 18.16

The title of Subsection 18.16(iii) was changed from “Ultraspherical and Legendre” to “Ultraspherical, Legendre and Chebyshev”. New subsection, 18.16(vii) Discriminants, with Equations (18.16.19)–(18.16.21).
Section 18.17

Extra explanatory text at many places and seven extra integrals (18.17.16_5), (18.17.21_1)–(18.17.21_3), (18.17.28_5), (18.17.34_5), (18.17.41_5).
Section 18.18

Extra explanatory text at several places and the title of Subsection 18.18(iv) was changed from “Connection Formulas” to “Connection and Inversion Formulas”.
Section 18.19

A new introduction.
Section 18.21

Equation (18.21.13).
Section 18.25

Extra explanatory text in Subsection 18.25(i) and the title of Subsection 18.25(ii) was changed from “Weights and Normalizations: Continuous Cases” to “Weights and Standardizations: Continuous Cases”.
Section 18.26

In Subsection 18.26(i) an extra paragraph on dualities has been included, with Equations (18.26.4_1), (18.26.4_2).
Section 18.27

Extra text at the start of this section and twenty seven extra formulas, (18.27.4_1), (18.27.4_2), (18.27.6_5), (18.27.9_5), (18.27.12_5), (18.27.14_1)–(18.27.14_6), (18.27.17_1)–(18.27.17_3), (18.27.20_5), (18.27.25), (18.27.26), (18.28.1_5).
Section 18.28

A big expansion. Six extra formulas in Subsection 18.28(ii) ((18.28.6_1)–(18.28.6_5)) and three extra formulas in Subsection 18.28(viii) ((18.28.21)–(18.28.23)). New subsections, 18.28(ix)–18.28(xi), with Equations (18.28.23)–(18.28.34).
Section 18.30

Originally this section did not have subsections. The original seven formulas have now more explanatory text and are split over two subsections. New subsections 18.30(iii)–18.30(viii), with Equations (18.30.8)–(18.30.31).
Section 18.32

This short section has been expanded, with Equation (18.32.2).
Section 18.33

Additional references and a new large subsection, 18.33(vi), including Equations (18.33.17)–(18.33.32).
Section 18.34

This section has been expanded, including an extra orthogonality relations (18.34.5_5), (18.34.7_1)–(18.34.7_3).
Section 18.35

This section on Pollaczek polynomials has been significantly updated with much more explanations and as well to include the Pollaczek polynomials of type 3 which are the most general with three free parameters. The Pollaczek polynomials which were previously treated, namely those of type 1 and type 2 are special cases of the type 3 Pollaczek polynomials. In the first paragraph of this section an extensive description of the relations between the three types of Pollaczek polynomials is given which was lacking previously. Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), (18.35.4_5), (18.35.6_1)–(18.35.6_6), (18.35.10).
Section 18.36

This section on miscellaneous polynomials has been expanded with new subsections, 18.36(v) on non-classical Laguerre polynomials and 18.36(vi) with examples of exceptional orthogonal polynomials, with Equations (18.36.1)–(18.36.10). In the titles of Subsections 18.36(ii) and 18.36(iii) we replaced “OP’s” by “Orthogonal Polynomials”.
Section 18.38

The paragraphs of Subsection 18.38(i) have been re-ordered and one paragraph was added. The title of Subsection 18.38(ii) was changed from “Classical OP’s: Other Applications” to “Classical OP’s: Mathematical Developments and Applications”. Subsection 18.38(iii) has been expanded with seven new paragraphs, and Equations (18.38.4)–(18.38.11).
Section 18.39

This section was completely rewritten. The previous 18.39(i) Quantum Mechanics has been replaced by Subsections 18.39(i) Quantum Mechanics and 18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom, containing the same essential information; the original content of the subsection is reproduced below for reference. Subsection 18.39(ii) was moved to 18.39(v) Other Applications. New subsections, 18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences, 18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods; Equations (18.39.7)–(18.39.48); and Figures 18.39.1, 18.39.2.

The original text of 18.39(i) Quantum Mechanics was:

“Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$ :

errata.1 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)% \psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$

where $\hbar$ is the reduced Planck’s constant. On substituting $\psi(x,t)=\eta(x)\zeta(t)$ , we obtain two ordinary differential equations, each of which involve the same constant $E$ . The equation for $\eta(x)$ is

errata.2 $\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x% )\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

errata.3 $V(x)=\tfrac{1}{2}m\omega^{2}x^{2},$

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $-\infty<x<\infty$ , the constant $E$ (the total energy of the particle) must satisfy

errata.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$ .

The corresponding eigenfunctions are

errata.5 $\eta_{n}(x)=\pi^{-\frac{1}{4}}2^{-\frac{1}{2}n}(n!\,b)^{-\frac{1}{2}}H_{n}% \left(x/b\right){\mathrm{e}}^{-x^{2}/2b^{2}},$

where $b=(\hbar/m\omega)^{1/2}$ , and $H_{n}$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

A second example is provided by the three-dimensional time-independent Schrödinger equation

errata.6 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).”
Section 18.40

The old section is now Subsection 18.40(i) and a large new subsection, 18.40(ii), on the classical moment problem has been added, with formulae (18.40.1)–(18.40.10) and Figures 18.40.1, 18.40.2.

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Notation

Previously the notation $h(n)$ was used for the harmonic number $H_{n}$ (defined in (25.11.33)). The more widely used notation $H_{n}$ will now be used throughout the DLMF. In particular, this change was made in (25.11.32), (25.11.33), (25.16.5) and (25.16.13) (suggested by Gergő Nemes on 2021-08-23).

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4: 1.2 Elementary Algebra

… ►The geometric mean

G

and harmonic mean

H

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by …

5: Bibliography T

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J. W. Tanner and S. S. Wagstaff (1987) New congruences for the Bernoulli numbers. Math. Comp. 48 (177), pp. 341–350.

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External Links:: ISSN 0025-5718, FullText, MathReview (Wells Johnson), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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N. M. Temme (1993) Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (3), pp. 233–243.

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External Links:: ISSN 0022-2526, FullText, MathReview (E. Rodney Canfield), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

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A. Terras (1988) Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt
Referenced by:: §35.1, §35.5(ii), Notations K.
Encodings:: BibTeX

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O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:: Document
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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T. Ton-That, K. I. Gross, D. St. P. Richards, and P. J. Sally (Eds.) (1995) Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:: Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994
External Links:: ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: Profile Donald St. P. Richards.
Encodings:: BibTeX

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6: 18.39 Applications in the Physical Sciences

… ►argument a) The Harmonic Oscillator … ►This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. … ►The eigenfunctions of

\mathrm{L}^{2}

are the spherical harmonics

Y_{{l},{m_{l}}}\left(\theta,\phi\right)

with eigenvalues

\hbar^{2}l(l+1)

, each with degeneracy

2l+1

m_{l}=-l,-l+1,\dots,l

. … … ►

b) The Bohr Quantum Number

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7: 1.9 Calculus of a Complex Variable

… ►

Harmonic Functions

… ►

Winding Number

… ►

Mean Value Property

►For

u(z)

harmonic, … ►

Poisson Integral

…

8: Bibliography H

… ►

E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.

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Notes:: Reprinted by Chelsea Publishing Company, New York, 1955 and 1965.
External Links:: ISBN 0-8284-0104-7, MathReview, ZentralBlatt
Referenced by:: §14.1, §14.11, §14.16(ii), §14.16(iii), §14.20(x), §14.27, §14.27, §14.3(i), §14.3(i), Ch.14, §29.18(iii), Ch.29.
Encodings:: BibTeX

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K. Horata (1989) An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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External Links:: ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

►

K. Horata (1991) On congruences involving Bernoulli numbers and irregular primes. II. Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.

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External Links:: ZentralBlatt
Referenced by:: §24.15(iii).
Encodings:: BibTeX

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L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:: Reprinted in 1979.
External Links:: ISBN 0-8218-1556-3, MathReview, ZentralBlatt
Referenced by:: §35.4(i).
Encodings:: BibTeX

►

I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.

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External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.14(ii).
Encodings:: BibTeX

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9: Bibliography M

… ►

T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

►

Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:: A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.
External Links:: Magma
Referenced by:: 4th item.
Encodings:: BibTeX

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I. Marquette and C. Quesne (2013) New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54 (10), pp. Paper 102102, 12 pp..

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External Links:: MathReview Entry
Referenced by:: §18.36(vi).
Encodings:: BibTeX

►

I. Marquette and C. Quesne (2016) Connection between quantum systems involving the fourth Painlevé transcendent and

k

-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:: ISSN 0022-2488, Document, Link, MathReview Entry
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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G. Matviyenko (1993) On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:: ISSN 1063-5203, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(iii).
Encodings:: BibTeX

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10: Bibliography

… ►

A. Adelberg (1996) Congruences of

p

-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.

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External Links:: ISSN 0022-314X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:: For corrigendum see same journal v. 90 (1998), no. 2, p. 265
External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §5.6(i).
Encodings:: BibTeX

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H. Alzer (2000) Sharp bounds for the Bernoulli numbers. Arch. Math. (Basel) 74 (3), pp. 207–211.

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External Links:: ISSN 0003-889X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.9.
Encodings:: BibTeX

… ►

T. M. Apostol and I. Niven (1994) Number Theory. In The New Encyclopaedia Britannica, Vol. 25, pp. 14–37.

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Referenced by:: §27.13(i), §27.15, §27.16, Ch.27.
Encodings:: BibTeX

… ►

T. M. Apostol (2000) A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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External Links:: MathReview (Giovanni Coppola), ZentralBlatt
Referenced by:: (25.16.2), (25.16.4), §25.16(i), §25.16.
Encodings:: BibTeX

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