Projection pursuit regression: Difference between revisions

Revision as of 16:07, 15 January 2014

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.

Definition

The generalized polygamma function is defined as follows:

ψ (z, q) = \frac{ζ^{'} (z + 1, q) + (ψ (- z) + γ) ζ (z + 1, q)}{Γ (- z)}

or alternatively,

ψ (z, q) = e^{- γ z} \frac{\partial}{\partial z} (e^{γ z} \frac{ζ (z + 1, q)}{Γ (- z)}),

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 $f (0) = f (1)$ and $\int_{0}^{1} f (x) d x = 0$ .

Relations

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

$ψ (x) = ψ (0, x)$

$ψ^{(n)} (x) = ψ (n, x) (n \in ℕ)$

$Γ (x) = e^{ψ (- 1, x) + \frac{1}{2} \ln (2 π)}$

$ζ (z, q) = \frac{Γ (1 - z) (2^{- z} (ψ (z - 1, \frac{q}{2} + \frac{1}{2}) + ψ (z - 1, \frac{q}{2})) - ψ (z - 1, q))}{\ln (2)}$

$ζ^{'} (- 1, x) = ψ (- 2, x) + \frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{12}$

$B_{n} (q) = - \frac{Γ (n + 1) (2^{n - 1} (ψ (- n, \frac{q}{2} + \frac{1}{2}) + ψ (- n, \frac{q}{2})) - ψ (- n, q))}{\ln (2)}$

where $B_{n} (q)$ are Bernoulli polynomials

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

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:

$ψ^{(- 2)} (\frac{1}{4}) = \frac{1}{8} \ln (2 π) + \frac{9}{8} \ln A + \frac{G}{4 π},$ where $A$ is the Glaisher constant and $G$ is the Catalan constant.

$ψ^{(- 2)} (\frac{1}{2}) = \frac{1}{4} \ln π + \frac{3}{2} \ln A + \frac{5}{24} \ln 2$

$ψ^{(- 2)} (1) = \frac{1}{2} \ln (2 π)$

$ψ^{(- 2)} (2) = \ln (2 π) - 1$

$ψ^{(- 3)} (\frac{1}{2}) = \frac{1}{16} \ln (2 π) + \frac{1}{2} \ln A + \frac{7 ζ (3)}{32 π^{2}}$

$ψ^{(- 3)} (1) = \frac{1}{4} \ln (2 π) + \ln A$

$ψ^{(- 3)} (2) = \ln (2 π) + 2 \ln A - \frac{3}{4}$

References

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

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

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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>[http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115]</ref>
+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)+(\psi(-z)+\gamma ) \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 <math>\psi(z)</math> is the [[Polygamma function]] and <math>\zeta(z,q),</math> is the [[Hurwitz zeta function]].
+The function is balanced, in that it satisfies the conditions <math>f(0)=f(1)</math> and <math>\int_0^1 f(x) dx = 0</math>.
+==Relations==
+Several special functions can be expressed in terms of generalized polygamma function.
+* <math>\psi(x)=\psi(0,x)\,</math>
+* <math>\psi^{(n)}(x)=\psi(n,x)\,\,\,(n\in\mathbb{N})</math>
+* <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:
+* <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]].
+* <math>\psi^{(-2)}\left(\frac12\right)=\frac14\ln\pi+\frac32\ln A+\frac5{24}\ln2</math>
+* <math>\psi^{(-2)}(1)=\frac12\ln(2\pi)</math>
+* <math>\psi^{(-2)}(2)=\ln(2\pi)-1</math>
+* <math>\psi^{(-3)}\left(\frac12\right)=\frac1{16}\ln(2\pi)+\frac12\ln A+\frac{7\,\zeta(3)}{32\,\pi^2}</math>
+* <math>\psi^{(-3)}(1)=\frac14\ln(2\pi)+\ln A</math>
+* <math>\psi^{(-3)}(2)=\ln(2\pi)+2\ln A-\frac34</math>
+==References==
+<references />
+{{DEFAULTSORT:Generalized Polygamma Function}}
+[[Category:Gamma and related functions]]

Projection pursuit regression: Difference between revisions

Revision as of 16:07, 15 January 2014

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