5 Gamma FunctionProperties5.4 Special Values and Extrema5.6 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

1. §5.5(i) Recurrence
2. §5.5(ii) Reflection
3. §5.5(iii) Multiplication
4. §5.5(iv) Bohr–Mollerup Theorem

§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		$$\mathrm{\Gamma }\left(z+1\right)=z\mathrm{\Gamma }\left(z\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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: $\psi \left(z\right)$: 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		$$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$
$z\ne 0,\pm 1,\mathrm{\dots }$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: 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		$$\psi \left(z\right)-\psi \left(1-z\right)=-\pi /\tan \left(\pi z\right),$$
$z\ne 0,\pm 1,\mathrm{\dots }$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tan z$: 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 $2z\ne 0,-1,-2,\mathrm{\dots }$,

5.5.5		$$\mathrm{\Gamma }\left(2z\right)={\pi }^{-1/2}{2}^{2z-1}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(z+\frac{1}{2}\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 $nz\ne 0,-1,-2,\mathrm{\dots }$,

5.5.6		$$\mathrm{\Gamma }\left(nz\right)={(2\pi )}^{(1-n)/2}{n}^{nz-(1/2)}\prod _{k=0}^{n-1}\mathrm{\Gamma }\left(z+\frac{k}{n}\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$\prod _{k=1}^{n-1}\mathrm{\Gamma }\left(\frac{k}{n}\right)={(2\pi )}^{(n-1)/2}{n}^{-1/2}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$\psi \left(2z\right)=\frac{1}{2}\left(\psi \left(z\right)+\psi \left(z+\frac{1}{2}\right)\right)+\ln 2,$$
ⓘ Symbols: $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: 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: $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: 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 $(0,\mathrm{\infty })$ satisfies $f(x+1)=xf(x)$, $f(1)=1$, and $\ln f(x)$ is convex (see §1.4(viii)), then $f(x)=\mathrm{\Gamma }\left(x\right)$.

5.4 Special Values and Extrema5.6 Inequalities

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