§5.5 Functional Relations

Referenced by:: §10.59
Permalink:: http://dlmf.nist.gov/5.5

§5.5(i) Recurrence
§5.5(ii) Reflection
§5.5(iii) Multiplication
§5.5(iv) Bohr–Mollerup Theorem

§5.5(i) Recurrence

Notes:: See Olver (1997b, pp. 32 and 39), or Temme (1996b, pp. 42 and 54).
Keywords:: gamma function, psi function
Permalink:: http://dlmf.nist.gov/5.5.i

5.5.1 $\mathop{\Gamma\/}\nolimits\!\left(z+1\right)=z\mathop{\Gamma\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $z$ : complex variable
A&S Ref:: 6.1.15
Referenced by:: §15.4(ii), §15.4(iii), §25.5(i), §5.4(i)
Permalink:: http://dlmf.nist.gov/5.5.E1
Encodings:: TeX, pMML, png

5.5.2 $\mathop{\psi\/}\nolimits\!\left(z+1\right)=\mathop{\psi\/}\nolimits\!\left(z\right)+\frac{1}{z}.$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : 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

§5.5(ii) Reflection

Notes:: See Olver (1997b, pp. 35 and 39).
Keywords:: gamma function, psi function
Permalink:: http://dlmf.nist.gov/5.5.ii

5.5.3 $\mathop{\Gamma\/}\nolimits\!\left(z\right)\mathop{\Gamma\/}\nolimits\!\left(1-z\right)=\pi/\mathop{\sin\/}\nolimits\!\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\sin\/}\nolimits z$ : 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), §25.11(vii), §5.21, §9.12(vi)
Permalink:: http://dlmf.nist.gov/5.5.E3
Encodings:: TeX, pMML, png

5.5.4 $\mathop{\psi\/}\nolimits\!\left(z\right)-\mathop{\psi\/}\nolimits\!\left(1-z\right)=-\pi/\mathop{\tan\/}\nolimits\!\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$ .

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\tan\/}\nolimits z$ : 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

§5.5(iii) Multiplication

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).
Keywords:: gamma function
Referenced by:: Version 1.0.5 (October 1, 2012)
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.
Reported 2012-04-02

¶ Duplication Formula

Keywords:: gamma function

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

5.5.5 $\mathop{\Gamma\/}\nolimits\!\left(2z\right)=\pi^{{-1/2}}2^{{2z-1}}\mathop{\Gamma\/}\nolimits\!\left(z\right)\mathop{\Gamma\/}\nolimits\!\left(z+\tfrac{1}{2}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $z$ : complex variable
A&S Ref:: 6.1.18
Referenced by:: §14.19(ii), §15.4(i), §15.4(iii), §5.5(iii)
Permalink:: http://dlmf.nist.gov/5.5.E5
Encodings:: TeX, pMML, png

¶ Gauss’s Multiplication Formula

Keywords:: gamma function, psi function

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

5.5.6 $\mathop{\Gamma\/}\nolimits\!\left(nz\right)=(2\pi)^{{(1-n)/2}}n^{{nz-(1/2)}}\prod _{{k=0}}^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(z+\frac{k}{n}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $n$ : 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

5.5.7 $\prod _{{k=1}}^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(\frac{k}{n}\right)=(2\pi)^{{(n-1)/2}}n^{{-1/2}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $n$ : nonnegative integer and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.5.E7
Encodings:: TeX, pMML, png

5.5.8 $\mathop{\psi\/}\nolimits\!\left(2z\right)=\tfrac{1}{2}\left(\mathop{\psi\/}\nolimits\!\left(z\right)+\mathop{\psi\/}\nolimits\!\left(z+\tfrac{1}{2}\right)\right)+\mathop{\ln\/}\nolimits 2,$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : 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

5.5.9 $\mathop{\psi\/}\nolimits\!\left(nz\right)=\frac{1}{n}\sum _{{k=0}}^{{n-1}}\mathop{\psi\/}\nolimits\!\left(z+\frac{k}{n}\right)+\mathop{\ln\/}\nolimits n.$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\mathop{\ln\/}\nolimits z$ : 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

§5.5(iv) Bohr–Mollerup Theorem

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

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