Balanced polygamma function: Difference between revisions

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In mathematics, the '''generalized polygamma function''' or '''balanced negapolygamma function''' is a function introduced by Olivier Espinosa Aldunate and [[Victor Moll|Victor H.Hugo Moll]].<ref>[{{cite journal|url=http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf |first1=Olivier |last1=Espinosa|first2=Victor Hugo|last2=Moll |author-link2=Victor H.Hugo Moll. |title=A Generalized polygamma function. |journal=Integral Transforms and Special Functions Vol. |volume=15, No. |issue=2, April|date=Apr 2004, pp|pages=101–115|doi=10.1080/10652460310001600573 101–115]}}{{open access}}</ref> It generalizes the [[polygamma function]] to negative and fractional order, but remains equal to it for integer positive orders.

It generalizes the [[polygamma function]] to negative and fractional order, but remains equal to it for integer positive orders.

It is defined as follows:

==Definition==

: <math>\psi(z,q)=\frac{\zeta'(z+1,q)+(\psi(-z)+\gamma ) \zeta (z+1,q)}{\Gamma (-z)} \, </math>

ItThe generalized polygamma function is defined as follows:

: <math>\psi(z,q)=\frac{\zeta'(z+1,q)+\bigl(\psi(-z)+\gamma \bigr) \zeta (z+1,q)}{\Gamma (-z)} \, </math>

or alternatively,

: <math>\psi(z,q)=e^{- \gamma z}\frac{\partial}{\partial z}\left(e^{\gamma z}\frac{\zeta(z+1,q)}{\Gamma(-z)}\right),</math>

where {{math|''ψ''(''z'')}} is the [[polygamma function]] and {{math|''ζ''(''z'',''q'')}}, is the [[Hurwitz zeta function]].

Several special functions can be expressed in terms of generalized polygamma function.

The function is balanced, in that it satisfies the conditions

* <math>\psi(x)=\psi(0,x)\,</math>

:<math>f(0)=f(1) \quad \text{and} \quad \int_0^1 f(x)\, dx = 0</math>.

==Relations==

* <math>\psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})</math>

Several special functions can be expressed in terms of generalized polygamma function.

* <math>\Gamma(x)=e^{\psi(-1,x)+\frac 12 \ln(2\pi)}\,\,\,</math>

* <math>\zeta(z,q)=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,\frac{q}{2}+\frac{1}{2}\right)+\psi \left(z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln(2)}</math>

:where <math>\zeta(z,q),</math> is the [[Hurwitz zeta function]]

* <math>B_n(q) = -\frac{\Gamma (n+1) \left(2^{n-1} \left(\psi\left(-n,\frac{q}{2}+\frac{1}{2}\right)+\psi\left(-n,\frac{q}{2}\right)\right)-\psi(-n,q)\right)}{\ln (2)}</math>

:<math>\begin{align}

:where <math>B_n(q)</math> are [[Bernoulli polynomials]]

* <math>\psi(x) &= \psi(0,x)\,</math>\

* <math>\psi^{(n)}(x)&=\psi(n,x) \,\,\,(qquad n\in\mathbb{N})</math> \\

* <math>\Gamma(x)&=e^{\exp\left( \psi(-1,x)+\frac 12tfrac12 \ln( 2\pi \right)}\,\,\,</math>

\zeta(z, q)&=\frac{(-1)^z}{\Gamma(z)} \psi(z - 1, q)\\

\zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \\

\end{align}</math>

* :<math>K(z)=A e^{\exp\left(\psi(-2,z)+\frac{z^2-z}{2}}\right)</math>

:where {{math|''K''(''z'')}} is the [[K-function|{{mvar|K}}-function]] and {{mvar|A}} is the [[Glaisher constant]], which itself can be expressed in terms of generalized polygamma function:.

==Special values==

*<math>A =\frac{\sqrt[36]{128{\pi}^{30}}}{\pi}e^{\frac{1}{3}+\frac{2}{3}\psi(-2,\frac 12)-\frac 13\ln(2\pi)}</math>

The balanced polygamma function can be expressed in a closed form at certain points (where {{mvar|A}} is the [[Glaisher constant]] and {{mvar|G}} is the [[Catalan constant]]):

:<math>\begin{align}

\psi\left(-2,\tfrac14\right)&=\tfrac18\ln A+\frac{G}{4\pi} && \\

\psi\left(-2,\tfrac12\right)&=\tfrac12\ln A-\tfrac{1}{24}\ln 2 & \\

\psi\left(-3,\tfrac12\right)&=\frac{3\zeta(3)}{32\pi^2}\\

\psi(-2,1)&=-\ln A &\\

\psi(-3,1)&=\frac{-\zeta(3)}{8\pi^2}\\

\psi(-2,2)&=-\ln A-1 &\\

\psi(-3,2)&=\frac{-\zeta(3)}{8\pi^2}-\tfrac34 \\\end{align}</math>

==References==