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. Moll]].<ref>[{{cite journal|url=http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf |first1=Olivier |last1=Espinosa |first2=Victor H. |last2=Moll. |title=A Generalized polygamma function. |journal=Integral Transforms and Special Functions Vol. |volume=15, No. |issue=2, April|date=Apr 2004,|page=101–115}}{{open pp. 101–115]access}}</ref>

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

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The generalized polygamma function is defined as follows:

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

* <math>\psi^{(n)}(x) &= \psi(n0,x) \,\,\,(n\in\mathbb{N})</math>[8px]

* <math>\Gammapsi^{(n)}(x)&=e^{\psi(-1n,x)+\frac 12\qquad n\ln(2in\pi)mathbb{N} \,\,\,</math>[8px]

*\Gamma(x)&=\exp\left( <math>\psi(-1,x)+\tfrac12 \ln 2\pi \right)\\[8px]
\zeta(z,q)&=\frac{\Gamma (1-z) \left(2^{-z} \left(\psi \left(z-1,\frac{q}{2}+\frac{1}{2}\right)+2^{-z} \psi \left(z-1,\frac{q}{2}\right)\right)-\psi(z-1,q)\right)}{\ln( 2)}</math> \\[8px]

* <math>\zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}</math> \\[8px]

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

\end{align}</math>

where {{math|''B<mathsub>B_nn</sub>''(''q'')</math>}} are [[Bernoulli polynomials]]

* :<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]].

==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\left(-2,\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]].

* <math>\psi\left(-2, \frac12tfrac14\right)&=\frac14tfrac18\ln 2\pi+\frac32tfrac98\ln A+\frac5frac{24G}{4\pi} &&\\ln2</math>[8px]

\psi\left(-2,\tfrac12\right)&=\tfrac14\ln\pi+\tfrac32\ln A+\tfrac5{24}\ln2 &

* <math>\psi(-2,1)=\frac12\ln(2\pi)</math>

* <math>\psi\left(-3,\frac12tfrac12\right)&=\frac1tfrac1{16}\ln( 2\pi)+\frac12tfrac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}</math> \\[8px]

* <math>\psi(-2,2)=\ln(2\pi)-1</math>

* <math>\psi(-2,1)&=\frac12tfrac12\ln( 2\pi)</math> &

* <math>\psi\left(-3,\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}</math>

* <math>\psi(-3,1)&=\frac14tfrac14\ln( 2\pi)+\ln A</math> \\[8px]

* <math>\psi(-32,2)&=\ln( 2\pi)+2\ln-1 A-\frac34</math>&

\psi(-3,2)&=\ln 2\pi+2\ln A-\tfrac34 \end{align}</math>

==References==