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 H. Moll.[1]

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


The generalized polygamma function is defined as follows:

\psi(z,q)=\frac{\zeta'(z+1,q)+(\psi(-z)+\gamma ) \zeta (z+1,q)}{\Gamma (-z)} \,

or alternatively,

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

where \psi(z) is the Polygamma function and \zeta(z,q), is the Hurwitz zeta function.

The function is balanced, in that it satisfies the conditions f(0)=f(1) and \int_0^1 f(x) dx = 0.


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

  • \psi(x)=\psi(0,x)\,
  • \psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})
  • \Gamma(x)=e^{\psi(-1,x)+\frac 12 \ln(2\pi)}\,\,\,
  • \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)}
  • \zeta'(-1,x)=\psi(-2, x) + \frac{x^2}2 - \frac{x}2 + \frac1{12}
  • 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)}

where B_n(q) are Bernoulli polynomials

  • K(z)=A e^{\psi(-2,z)+\frac{z^2-z}{2}}

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

Special values[edit]

The balanced polygamma function can be expressed in a closed form at certain points:

  • \psi\left(-2, \frac12\right)=\frac14\ln\pi+\frac32\ln A+\frac5{24}\ln2
  • \psi(-2,1)=\frac12\ln(2\pi)
  • \psi(-2,2)=\ln(2\pi)-1
  • \psi\left(-3,\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}
  • \psi(-3,1)=\frac14\ln(2\pi)+\ln A
  • \psi(-3,2)=\ln(2\pi)+2\ln A-\frac34


  1. ^ Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115