Minkowski inequalities for sums and series

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21: 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

22: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►Such series diverge for Fuchs–Frobenius solutions. …Every Heun function can be represented by a series of Type II. ►

§31.11(v) Doubly-Infinite Series

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23: 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

24: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$ $K$ even,

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${\mathrm{\Psi }}_K\left(𝐱\right)={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$ $K$ odd,

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$a_{n+1}(𝐱)={\displaystyle \frac{\mathrm{i}}{n+1}}{\displaystyle \sum _{p=0}^{\min (n,K-1)}}(p+1)x_{p+1}{a}_{n-p}(𝐱),$ $n=0,1,2,\mathrm{\dots }$.

►For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982). …

25: 1.15 Summability Methods

… ►Here $u(x,y)=A(r,\theta )$ is the Abel (or Poisson) sum of $f(\theta )$, and $v(x,y)$ has the series representation …

26: 3.9 Acceleration of Convergence

… ►Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums. … ►If $S={\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k$ is a convergent series, then ►

3.9.2 $$S=\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{2}^{-k-1}{\mathrm{\Delta }}^k{a}_0,$$

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Symbols:

$\mathrm{\Delta }$: forward difference operator and $S$: a convergent series

Permalink:

http://dlmf.nist.gov/3.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

… ►If $s_n$ is the $n$th partial sum of a power series $f$, then $t_{n,2k}={\epsilon }_{2k}^{(n)}$ is the Padé approximant ${[(n+k)/k]}_f$ (§3.11(iv)). …

27: 7.6 Series Expansions

§7.6 Series Expansions

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§7.6(i) Power Series

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7.6.5 $$C\left(z\right)=\cos \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n}}{1\cdot 3\mathrm{\cdots }(4n+1)}{z}^{4n+1}+\sin \left(\frac{1}{2}\pi z^2\right)\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{\pi }^{2n+1}}{1\cdot 3\mathrm{\cdots }(4n+3)}{z}^{4n+3}.$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.12

Referenced by:

§11.10(vi), §7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

… ►The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$. ►

§7.6(ii) Expansions in Series of Spherical Bessel Functions

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28: 3.10 Continued Fractions

… ►can be converted into a continued fraction $C$ of type (3.10.1), and with the property that the $n$th convergent $C_n=A_n/B_n$ to $C$ is equal to the $n$th partial sum of the series in (3.10.3), that is, …

29: 25.2 Definition and Expansions

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25.2.1 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.$$

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Defines:

$\zeta \left(s\right)$: Riemann zeta function

Symbols:

$n$: nonnegative integer and $s$: complex variable

Keywords:

definition, infinite series

Sources:

Apostol (1976, Chapter 12); Riemann (1859, p. 136)

A&S Ref:

23.2.1

Referenced by:

(25.11.2), (25.2.2), (25.2.4), §25.2(ii), Erratum (V1.0.17) for Equation (25.2.4)

Permalink:

http://dlmf.nist.gov/25.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(i), §25.2 and Ch.25

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25.2.2 $$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.2.1).

A&S Ref:

23.2.20 (is the special case with integer values of $s$)

Referenced by:

(25.11.11)

Permalink:

http://dlmf.nist.gov/25.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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25.2.6 $${\zeta }^{\prime }\left(s\right)=-\sum _{n=2}^{\mathrm{\infty }}(\ln n)n^{-s},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Apostol (1976, (12), p. 236)

Permalink:

http://dlmf.nist.gov/25.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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30: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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