bibasic sums and series

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11: 27.5 Inversion Formulas

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27.5.6 $$G(x)=\sum _{n\le x}F\left(\frac{x}{n}\right)⟺F(x)=\sum _{n\le x}\mu \left(n\right)G\left(\frac{x}{n}\right),$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E6

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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27.5.7 $$G(x)=\sum _{m=1}^{\mathrm{\infty }}\frac{F(mx)}{m^s}⟺F(x)=\sum _{m=1}^{\mathrm{\infty }}\mu \left(m\right)\frac{G(mx)}{m^s},$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $m$: positive integer, $x$: real number, $F(s)$: Dirichlet series and $G(s)$: Dirichlet series

Referenced by:

§27.17, §27.5

Permalink:

http://dlmf.nist.gov/27.5.E7

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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12: 5.7 Series Expansions

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5.7.3 $$\ln \mathrm{\Gamma }\left(1+z\right)=-\ln \left(1+z\right)+z(1-\gamma )+\sum _{k=2}^{\mathrm{\infty }}{(-1)}^k(\zeta \left(k\right)-1)\frac{z^k}{k},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.33

Referenced by:

§5.21, §5.7(i)

Permalink:

http://dlmf.nist.gov/5.7.E3

Encodings:

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See also:

Annotations for §5.7(i), §5.7 and Ch.5

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13: 8.7 Series Expansions

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

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Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

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14: 1.8 Fourier Series

… ►Then the series (1.8.1) converges to the sum … ►

1.8.16 $$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n+x)}^2\omega }=\sqrt{\frac{\pi }{\omega }}\left(1+2\sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2{\pi }^2/\omega }\cos \left(2n\pi x\right)\right),$$ $\mathrm{\Re }\omega >0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $n$: nonnegative integer

Referenced by:

§1.8(iv)

Permalink:

http://dlmf.nist.gov/1.8.E16

Encodings:

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See also:

Annotations for §1.8(iv), §1.8 and Ch.1

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15: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series ►

28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.30(i), §28.30 and Ch.28

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16: 25.8 Sums

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25.8.1 $$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (1.4), p. 132)

Permalink:

http://dlmf.nist.gov/25.8.E1

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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25.8.2 $$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$ $s\ne 1,0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Landau (1953, (2), p. 273); with (5.2.4), (5.2.5)

Permalink:

http://dlmf.nist.gov/25.8.E2

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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25.8.10 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $k$: nonnegative integer

Keywords:

infinite series

Source:

Ewell (1990, (1), p. 219)

Permalink:

http://dlmf.nist.gov/25.8.E10

Encodings:

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See also:

Annotations for §25.8 and Ch.25

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17: 25.16 Mathematical Applications

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25.16.1 $$\psi \left(x\right)=\sum _{m=1}^{\mathrm{\infty }}\sum _{p^m\le x}\ln p,$$

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Defines:

$\psi \left(x\right)$: Chebyshev $\psi$-function

Symbols:

$\ln z$: principal branch of logarithm function, $m$: nonnegative integer, $p$: prime number and $x$: real variable

Keywords:

definition, double series, infinite series, series representation

Source:

Apostol (1976, p. 75)

Permalink:

http://dlmf.nist.gov/25.16.E1

Encodings:

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See also:

Annotations for §25.16(i), §25.16 and Ch.25

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25.16.5 $$H\left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{H_n}{n^s},$$

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Symbols:

$H\left(s\right)$: Euler sums, $H_n$: 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:

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Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ 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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25.16.11 $$H(s,z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\sum _{m=1}^n\frac{1}{m^z},$$ $\mathrm{\Re }\left(s+z\right)>1$,

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Symbols:

$H(s,z)$: generalized Euler sums, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

Euler sum, infinite series, series representation

Source:

Apostol and Vu (1984, (1), p. 86)

Permalink:

http://dlmf.nist.gov/25.16.E11

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.14 $$\sum _{r=1}^{\mathrm{\infty }}\sum _{k=1}^r\frac{1}{rk(r+k)}=\frac{5}{4}\zeta \left(3\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series, special value

Source:

Apostol and Vu (1984, (14), p. 94)

Permalink:

http://dlmf.nist.gov/25.16.E14

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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25.16.15 $$\sum _{r=1}^{\mathrm{\infty }}\sum _{k=1}^r\frac{1}{r^2(r+k)}=\frac{3}{4}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series, special value

Source:

Apostol and Vu (1984, (15), p. 94)

Permalink:

http://dlmf.nist.gov/25.16.E15

Encodings:

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See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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18: 9.19 Approximations

… ►The constants $a$ and $b$ are chosen numerically, with a view to equalizing the effort required for summing the series. …

19: 24.8 Series Expansions

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer

Keywords:

Euler polynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

20: 28.11 Expansions in Series of Mathieu Functions

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28.11.7 $$\sin (2m+2)z=\sum _{n=0}^{\mathrm{\infty }}{B}_{2m+2}^{2n+2}(q){\mathrm{se}}_{2n+2}(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\sin z$: sine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11, §28.11 and Ch.28