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11: 14.5 Special Values

§14.5 Special Values

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14.5.14 $${𝖰}_{\nu }^{-1/2}\left(\cos \theta \right)={\left(\frac{\pi }{2\sin \theta }\right)}^{1/2}\frac{\cos \left(\left(\nu +\frac{1}{2}\right)\theta \right)}{\nu +\frac{1}{2}}.$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $\nu$: general degree and : variable

A&S Ref:

8.6.15 (corrected)

Referenced by:

Erratum (V1.0.16) for Equation (14.5.14)

Permalink:

http://dlmf.nist.gov/14.5.E14

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

Originally this equation was incorrect because of a minus sign in front of the right-hand side.

Reported 2017-04-10 by André Greiner-Petter

See also:

Annotations for §14.5(iii), §14.5 and Ch.14

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14.5.23 $${𝖰}_{-\frac{1}{2}}\left(\cos \theta \right)=K\left(\cos \left(\frac{1}{2}\theta \right)\right).$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind and : variable

A&S Ref:

8.13.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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14.5.27 $${𝑸}_{-\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}{\mathrm{e}}^{-\xi /2}K\left({\mathrm{e}}^{-\xi }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.4 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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12: 25.16 Mathematical Applications

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25.16.10 $$H\left(-2a\right)=\frac{1}{2}\zeta \left(1-2a\right)=-\frac{B_{2a}}{4a},$$ $a=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function and $a$: real or complex parameter

Keywords:

Euler sum, specialvalue

Source:

Apostol and Vu (1984, (7), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E10

Encodings:

pMML, png, TeX

See also:

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

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25.16.13 $$\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{H_n}{n}\right)}^2=\frac{17}{4}\zeta \left(4\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $H_n$: harmonic number, $n$: nonnegative integer and $h(n)$: harmonic number

Keywords:

harmonic number, infinite series, specialvalue

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

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Symbols:

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

Keywords:

infinite series, specialvalue

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, specialvalue

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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13: 24.4 Basic Properties

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§24.4(vi) Special Values

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14: 5.15 Polygamma Functions

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5.15.1 $${\psi }^{\prime }\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{(k+z)}^2},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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15: 27.9 Quadratic Characters

… ►Special values include: …

16: 25.11 Hurwitz Zeta Function

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§25.11(v) Special Values

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25.11.11 $$\zeta (s,\frac{1}{2})=(2^s-1)\zeta \left(s\right),$$ $s\ne 1$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function and $s$: complex variable

Keywords:

specialvalue

Proof sketch:

Derivable using (25.11.1), (25.2.2).

Referenced by:

§25.11(iv)

Permalink:

http://dlmf.nist.gov/25.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.12 $$\zeta (n+1,a)=\frac{{(-1)}^{n+1}{\psi }^{(n)}\left(a\right)}{n!},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

specialvalue

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29)

Permalink:

http://dlmf.nist.gov/25.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.13 $$\zeta (0,a)=\frac{1}{2}-a.$$

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function and $a$: real or complex parameter

Keywords:

specialvalue

Source:

Apostol (1976, p. 268)

Permalink:

http://dlmf.nist.gov/25.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.14 $$\zeta (-n,a)=-\frac{B_{n+1}\left(a\right)}{n+1},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

specialvalue

Source:

Apostol (1976, (17), p. 264)

Permalink:

http://dlmf.nist.gov/25.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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17: 8.19 Generalized Exponential Integral

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§8.19(iii) Special Values

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18: 8.21 Generalized Sine and Cosine Integrals

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§8.21(v) Special Values

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19: 19.6 Special Cases

§19.6 Special Cases

… ►Exact values of $K\left(k\right)$ and $E\left(k\right)$ for various special values of $k$ are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006). … ►

§19.6(v) $R_C(x,y)$

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20: 26.8 Set Partitions: Stirling Numbers

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§26.8(iii) Special Values

►For $n\ge 1$, …