Английская Википедия:Balanced polygamma function

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In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.

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

Definition

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,

<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 Шаблон:Math is the polygamma function and Шаблон:Math, is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions

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

Relations

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

<math>\begin{align}

\psi(x) &= \psi(0,x)\\ \psi^{(n)}(x)&=\psi(n,x) \qquad n\in\mathbb{N} \\ \Gamma(x)&=\exp\left( \psi(-1,x)+\tfrac12 \ln 2\pi \right)\\ \zeta(z,q)&=\frac{\Gamma (1-z)}{\ln 2} \left(2^{-z} \psi \left(z-1,\frac{q+1}{2}\right)+2^{-z} \psi \left(z-1,\frac{q}{2}\right)-\psi(z-1,q)\right)\\ \zeta'(-1,x)&=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12} \\ B_n(q) &= -\frac{\Gamma (n+1)}{\ln 2} \left(2^{n-1} \psi\left(-n,\frac{q+1}{2}\right)+2^{n-1} \psi\left(-n,\frac{q}{2}\right)-\psi(-n,q)\right) \end{align}</math>

where Шаблон:Math are the Bernoulli polynomials

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

where Шаблон:Math is the [[K-function|Шаблон:Mvar-function]] and Шаблон:Mvar is the Glaisher constant.

Special values

The balanced polygamma function can be expressed in a closed form at certain points (where Шаблон:Mvar is the Glaisher constant and Шаблон:Mvar is the Catalan constant):

<math>\begin{align}

\psi\left(-2,\tfrac14\right)&=\tfrac18\ln 2\pi+\tfrac98\ln A+\frac{G}{4\pi} && \\ \psi\left(-2,\tfrac12\right)&=\tfrac14\ln\pi+\tfrac32\ln A+\tfrac5{24}\ln2 & \\ \psi\left(-3,\tfrac12\right)&=\tfrac1{16}\ln 2\pi+\tfrac12\ln A+\frac{7\zeta(3)}{32\pi^2}\\ \psi(-2,1)&=\tfrac12\ln 2\pi &\\ \psi(-3,1)&=\tfrac14\ln 2\pi+\ln A\\ \psi(-2,2)&=\ln 2\pi-1 &\\ \psi(-3,2)&=\ln 2\pi+2\ln A-\tfrac34 \\\end{align}</math>