Polygamma function

The first polygamma functions in real terms m = 0 m = 1 m = 2 m = 3 m = 4
            

In mathematics, the polygamma functions are a series of special functions that are defined as the derivatives of the function . The gamma function and the natural logarithm denote . ${\displaystyle \psi _{n}(z)}$ ${\displaystyle \ln \Gamma (z)}$ ${\displaystyle \Gamma (z)}$ ${\displaystyle \ln }$

The first two polygamma functions are called the digamma function and the trigamma function .

Representation of the first five polygamma functions in the complex plane

${\displaystyle \ln \Gamma (z)}$	${\displaystyle \psi ^{(0)}(z)}$	${\displaystyle \psi ^{(1)}(z)}$	${\displaystyle \psi ^{(2)}(z)}$	${\displaystyle \psi ^{(3)}(z)}$	${\displaystyle \psi ^{(4)}(z)}$

notation

The polygamma functions are identified with the small Greek letter Psi . In the case of the first polygamma function, the index is usually left out or defined as 0; it is called the digamma function . The second polygamma function, i.e. the trigamma function , has the symbol (or less often ) and is the second derivative of . In general, the is -th polygamma function or polygamma function of the order with or referred to and than the th derivative of defined. ${\displaystyle \psi }$ ${\displaystyle \psi (z)}$ ${\displaystyle \psi _{1}}$ ${\displaystyle \psi ^{(1)}}$ ${\displaystyle \ln \Gamma (z)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \psi _{n}}$ ${\displaystyle \psi ^{(n)}}$ ${\displaystyle (n+1)}$ ${\displaystyle \ln \Gamma (x)}$

Definition and further representations

It is

${\displaystyle \psi _{m}(z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z)={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\,\psi (z)}$

with the digamma function . Such derivatives are also referred to as logarithmic derivatives of . ${\displaystyle \psi (z)}$ ${\displaystyle \Gamma (\cdot )}$

An integral representation is

${\displaystyle \psi _{m}(z)=(-1)^{m+1}\int \limits _{0}^{\infty }{\frac {t^{m}\mathrm {e} ^{-zt}}{1-\mathrm {e} ^{-t}}}\,\mathrm {d} t}$

for and ${\displaystyle {\rm {{Re}\,z>0}}}$ ${\displaystyle m>0.}$

properties

Difference equations

The polygamma functions have the difference equations

${\displaystyle \psi _{m}(z+1)=\psi _{m}(z)+(-1)^{m}\;m!\;z^{-m-1}.}$

Reflection formula

Another important relationship is

${\displaystyle (-1)^{m}\psi _{m}(1-z)-\psi _{m}(z)=\pi {\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\cot {(\pi z)}.}$

Multiplication formula

The multiplication formula is given for by ${\displaystyle m>0}$

${\displaystyle \sum _{k=0}^{n-1}\psi _{m}\left({\frac {z+k}{n}}\right)=n^{m+1}\psi _{m}(z).}$

For the case of the digamma function , see there. ${\displaystyle m=0,}$

Series representations

A series representation of the polygamma function is

${\displaystyle \psi _{m}(z)=(-1)^{m+1}\;m!\;\sum _{k=0}^{\infty }{\frac {1}{(z+k)^{m+1}}}}$

where and is any complex number other than negative integers. The formula is easier to write using Hurwitz's zeta function than ${\displaystyle m>0}$ ${\displaystyle z\not =-1,-2,-3,\ldots }$ ${\displaystyle \zeta (x,y)}$

${\displaystyle \psi _{m}(z)=(-1)^{m+1}\;m!\;\zeta (m+1,z).}$

The generalization of the polygamma functions to arbitrary, non-whole orders is given below . ${\displaystyle m}$

Another series representation is

${\displaystyle \psi _{m}(z)=-\gamma \delta _{m,0}\;-\;{\frac {(-1)^{m}m!}{z^{m+1}}}\;+\;\sum _{k=1}^{\infty }\left({\frac {1}{k}}\delta _{m,0}\;-\;{\frac {(-1)^{m}m!}{(z+k)^{m+1}}}\right),}$

where the Kronecker delta denotes, which follows from the decomposition of the gamma function according to Weierstrass' product theorem. ${\displaystyle \delta _{n,0}}$

The Taylor series um is given by ${\displaystyle z=1}$

${\displaystyle \psi _{m}(z+1)=\sum _{k=0}^{\infty }(-1)^{m+k+1}(m+k)!\;\zeta (m+k+1)\;{\frac {z^{k}}{k!}},}$

which converges for. referred to the Riemann zeta function . ${\displaystyle |z|<1}$ ${\displaystyle \zeta }$

Special values

The values of the Polygammafunktionen for rational arguments can usually be expressed using constants and functions such as , square root , Clausen function , Riemann ζ function , Catalan's constant and Dirichlet β function ; z. B. ${\displaystyle \pi }$ ${\displaystyle \mathrm {Cl} (x)}$ ${\displaystyle G}$

${\displaystyle \psi _{m}({\tfrac {1}{2}})=(-1)^{m+1}m!\,(2^{m+1}-1)\zeta (m+1),\qquad m\in \mathbb {N} }$

In general, the following also applies:

${\displaystyle \psi _{m}(1)=(-1)^{m+1}m!\,\zeta (m+1),\qquad m\in \mathbb {N} }$.

The mth derivative of the tangent can also be expressed using the polygamma function:

${\displaystyle {\frac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}\tan x={\frac {\psi _{m}({\tfrac {1}{2}}+{\tfrac {x}{\pi }})-(-1)^{m}\,\psi _{m}({\tfrac {1}{2}}-{\tfrac {x}{\pi }})}{\pi ^{m+1}}}}$.

In addition, special values ​​of polygamma functions have repeatedly proven useful as universal constants in a closed limit value description of series or integrals, for example applies

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{4}}}={\frac {1}{768}}\left(\psi _{3}\left({\tfrac {1}{4}}\right)-8\pi ^{2}\right).}$

Generalized polygamma function

The generalized polygamma function satisfies for and the functional equation ${\displaystyle s\in \mathbb {C} }$ ${\displaystyle z\in \mathbb {C} \setminus -\mathbb {N} _{0}}$

${\displaystyle \psi _{s}(z+1)=\psi _{s}(z)+{\frac {\ln z-\psi (-s)-\gamma }{\Gamma (-s)}}\,z^{-(s+1)},}$

where denotes the Euler-Mascheroni constant . Because of ${\displaystyle \gamma }$

${\displaystyle {\frac {\psi (-m)}{\Gamma (-n)}}=(-1)^{n-1}n!}$

for integers , the difference equation given above for natural is included. ${\displaystyle m,n\geq 0}$ ${\displaystyle n}$

The relationship is then obtained with the aid of the Hurwitz function ${\displaystyle \zeta }$

${\displaystyle \psi _{s}(z)={\frac {1}{\Gamma (-s)}}\left({\frac {\partial }{\partial s}}+\psi (-s)+\gamma \right)\zeta (s+1,z)=\mathrm {e} ^{-\gamma \,s}{\frac {\partial }{\partial s}}\left(\mathrm {e} ^{\gamma \,s}\,{\frac {\zeta (s+1,z)}{\Gamma (-s)}}\right),}$

which satisfies the functional equation.

As a consequence, the doubling formula

${\displaystyle \psi _{s}\left({\frac {z}{2}}\right)+\psi _{s}\left({\frac {z+1}{2}}\right)=2^{s+1}\psi _{s}(z)+{\frac {2^{s+1}\ln 2}{\Gamma (-s)}}\zeta (s+1,z)}$

derive. A generalization of this is

${\displaystyle \sum \limits _{k=0}^{n-1}\psi _{s}\left({\frac {z+k}{n}}\right)=n^{s+1}\psi _{s}(z)+{\frac {n^{s+1}\ln n}{\Gamma (-s)}}\zeta (s+1,z),}$

which is an equivalent to the Gaussian multiplication formula of the gamma function and contains the multiplication formula as a special case for . ${\displaystyle s\in \mathbb {N} }$

q-polygamma function

The polygamma function is defined by ${\displaystyle q}$

${\displaystyle \psi _{n}^{q}(z)={\frac {\partial ^{n}\psi _{q}(z)}{\partial z^{n}}}}$.

literature

• Milton Abramowitz and Irene Stegun , Handbook of Mathematical Functions , (1964) Dover Publications, New York. ISBN 978-0-486-61272-0 . See §6.4
• Eric W. Weisstein : Polygamma Function on MathWorld , in functions.wolfram.com , in references.worlfram.com .

credentials

1. ↑ Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
2. ↑ Eric W. Weisstein : q-Polygamma Function . In: MathWorld (English).