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.

==Definition==

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The 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,

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: <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>\psi|''ψ''(''z'')</math>}} is the [[Polygammapolygamma function]] and <{{math>\zeta|''ζ''(''z'',''q'')}},</math> is the [[Hurwitz zeta function]].

The function is balanced, in that it satisfies the conditions
:<math>f(0)=f(1)</math> \quad \text{and} \quad <math>\int_0^1 f(x)\, dx = 0</math>.

==Relations==

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

* :<math>\psi(x)=\psi(0,x)\,</math>begin{align}

\psi(x) &= \psi(0,x)\\

\psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\

* <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)\\

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

\end{align}</math>

* :<math>\psi^{(n)}K(xz)=A \exp\left(\psi(n-2,xz)+\,\,\,(n\in\mathbbfrac{Nz^2-z}{2}\right)</math>

where {{math|''K''(''z'')}} is the [[K-function|{{mvar|K}}-function]] and {{mvar|A}} is the [[Glaisher constant]].

* <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>

* <math>\zeta'(-1,x)=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}</math>

* <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>

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

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

where ''K''(''z'') is [[K-function]] and A is the [[Glaisher constant]].

==Special values==

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}

* <math>\psi^{(-2)}\left(\frac14\right)=\frac18\ln(2\pi)+\frac98\ln A+\frac{G}{4\pi},</math> where <math>A</math> is the [[Glaisher constant]] and <math>G</math> is the [[Catalan constant]].

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

* <math>\psi^{\left(-2)},\left(\frac12tfrac12\right)&=\frac14\ln\pi+\frac32tfrac12\ln A+-\frac5tfrac{1}{24}\ln2</math>ln 2 & \\

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

* <math>\psi^{(-2)}(,1)&=\frac12-\ln(2 A &\\pi)</math>

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

* <math>\psi^{(-2)}(,2)&=-\ln(2\pi) A-1</math> &\\

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

==References==