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DLMF: §5.5 Functional Relations ‣ Properties ‣ Chapter 5 Gamma Function

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5 Gamma Function Properties 5.4 Special Values and Extrema 5.6 Inequalities

§5.5 Functional Relations

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Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/5.5
See also:: Annotations for Ch.5

Contents

§5.5(i) Recurrence

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Keywords:: gamma function, psi function, recurrence relation
Notes:: See Olver (1997b, pp. 32 and 39), or Temme (1996b, pp. 42 and 54).
Permalink:: http://dlmf.nist.gov/5.5.i
See also:: Annotations for §5.5 and Ch.5

5.5.1		$\Gamma\left(z+1\right)=z\Gamma\left(z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function and $z$ : complex variable A&S Ref: 6.1.15 Referenced by: §15.4(ii), §15.4(iii), (25.11.27), (25.5.6), §5.4(i) Permalink: http://dlmf.nist.gov/5.5.E1 Encodings: TeX, pMML, png See also: Annotations for §5.5(i), §5.5 and Ch.5

5.5.2		$\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}.$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $z$ : complex variable A&S Ref: 6.3.5 Permalink: http://dlmf.nist.gov/5.5.E2 Encodings: TeX, pMML, png See also: Annotations for §5.5(i), §5.5 and Ch.5

§5.5(ii) Reflection

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Keywords:: gamma function, psi function, reflection formula
Notes:: See Olver (1997b, pp. 35 and 39).
Permalink:: http://dlmf.nist.gov/5.5.ii
See also:: Annotations for §5.5 and Ch.5

5.5.3		$\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right),$
$z\neq 0,\pm 1,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 6.1.17 (without the condition on $z$ .) Referenced by: §10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15) Permalink: http://dlmf.nist.gov/5.5.E3 Encodings: TeX, pMML, png See also: Annotations for §5.5(ii), §5.5 and Ch.5

5.5.4		$\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right),$
$z\neq 0,\pm 1,\dots$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\tan\NVar{z}$ : tangent function and $z$ : complex variable A&S Ref: 6.3.7 (without the condition on $z$ .) Permalink: http://dlmf.nist.gov/5.5.E4 Encodings: TeX, pMML, png See also: Annotations for §5.5(ii), §5.5 and Ch.5

§5.5(iii) Multiplication

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Keywords:: gamma function, multiplication formulas
Notes:: For (5.5.5) see Olver (1997b, p. 35); for (5.5.6)–(5.5.8) see Temme (1996b, pp. 52–53 and 76). (5.5.9) follows from (5.5.6).
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/5.5.iii
Addition (effective with 1.0.5):: The reference to Sándor and Tóth (1989) was added at the end of this subsection.
See also:: Annotations for §5.5 and Ch.5

Duplication Formula

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Keywords:: duplication formula, gamma function
See also:: Annotations for §5.5(iii), §5.5 and Ch.5

For $2z\neq 0,-1,-2,\dots$ ,

5.5.5		$\Gamma\left(2z\right)=\pi^{-1/2}2^{2z-1}\Gamma\left(z\right)\Gamma\left(z+% \tfrac{1}{2}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable A&S Ref: 6.1.18 Referenced by: §14.19(ii), §15.4(i), §15.4(iii), (25.9.2), §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E5 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

Gauss’s Multiplication Formula

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Keywords:: Gauss’s multiplication formula, gamma function, multiplication formula, psi function
See also:: Annotations for §5.5(iii), §5.5 and Ch.5

For $nz\neq 0,-1,-2,\dots$ ,

5.5.6		$\Gamma\left(nz\right)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left% (z+\frac{k}{n}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable A&S Ref: 6.1.20 Referenced by: §15.4(iii), §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E6 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

5.5.7		$\prod_{k=1}^{n-1}\Gamma\left(\frac{k}{n}\right)=(2\pi)^{(n-1)/2}n^{-1/2}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/5.5.E7 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

5.5.8		$\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}% {2}\right)\right)+\ln 2,$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 6.3.8 Referenced by: §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E8 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

5.5.9		$\psi\left(nz\right)=\frac{1}{n}\sum_{k=0}^{n-1}\psi\left(z+\frac{k}{n}\right)+% \ln n.$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable Referenced by: §5.5(iii) Permalink: http://dlmf.nist.gov/5.5.E9 Encodings: TeX, pMML, png See also: Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

See also Sándor and Tóth (1989).

§5.5(iv) Bohr–Mollerup Theorem

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Keywords:: Bohr-Mollerup theorem, convexity, gamma function, logarithm
Notes:: See Andrews et al. (1999, pp. 34–36).
Referenced by:: §5.18(ii), Figure 5.3.2, Figure 5.3.2
Permalink:: http://dlmf.nist.gov/5.5.iv
See also:: Annotations for §5.5 and Ch.5

If a positive function $f(x)$ on $(0,\infty)$ satisfies $f(x+1)=xf(x)$ , $f(1)=1$ , and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\Gamma\left(x\right)$ .

5.4 Special Values and Extrema 5.6 Inequalities

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