In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.[1]
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:
![{\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\bigl (}\psi (-z)+\gamma {\bigr )}\zeta (z+1,q)}{\Gamma (-z)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1564abeebefc95af3cdebde3e6f44643fafdccc8)
or alternatively,
![{\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5443d79f5b0b01a3a50fbde7d7a6b196862c5463)
where ψ(z) is the polygamma function and ζ(z,q), is the Hurwitz zeta function.
The function is balanced, in that it satisfies the conditions
.
Relations
Several special functions can be expressed in terms of generalized polygamma function.
![{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\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}}+{\frac {1}{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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e052418d781970fa0f2f34a6ef7c77f9c2918d2)
where Bn(q) are the Bernoulli polynomials
![{\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1086fcc0c1677f7765280a066d63d1ae9994b7d9)
where K(z) is the 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 A is the Glaisher constant and G is the Catalan constant):
![{\displaystyle {\begin{aligned}\psi \left(-2,{\tfrac {1}{4}}\right)&={\tfrac {1}{8}}\ln 2\pi +{\tfrac {9}{8}}\ln A+{\frac {G}{4\pi }}&&\\\psi \left(-2,{\tfrac {1}{2}}\right)&={\tfrac {1}{4}}\ln \pi +{\tfrac {3}{2}}\ln A+{\tfrac {5}{24}}\ln 2&\\\psi \left(-3,{\tfrac {1}{2}}\right)&={\tfrac {1}{16}}\ln 2\pi +{\tfrac {1}{2}}\ln A+{\frac {7\zeta (3)}{32\pi ^{2}}}\\\psi (-2,1)&={\tfrac {1}{2}}\ln 2\pi &\\\psi (-3,1)&={\tfrac {1}{4}}\ln 2\pi +\ln A\\\psi (-2,2)&=\ln 2\pi -1&\\\psi (-3,2)&=\ln 2\pi +2\ln A-{\tfrac {3}{4}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3cdcb91dffc303530868ab9699585605f28a82)
References
- ^ Espinosa, Olivier; Moll, Victor Hugo (Apr 2004). "A Generalized polygamma function" (PDF). Integral Transforms and Special Functions. 15 (2): 101–115. doi:10.1080/10652460310001600573.