En matemáticas , la función polygamma generalizada es una función introducida por Olivier Espinosa y Victor H. Moll.[ 1]
Consiste en una generalización de la función polygamma a orden negativo y fraccionario, permaneciendo igual a ésta para órdenes enteros positivos.
La función polygamma generalizada está definida como sigue:
ψ
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z
,
q
)
=
ζ
′
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z
+
1
,
q
)
+
(
ψ
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−
z
)
+
γ
)
ζ
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z
+
1
,
q
)
Γ
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−
z
)
{\displaystyle \psi (z,q)={\frac {\zeta '(z+1,q)+{\big (}\psi (-z)+\gamma {\big )}\zeta (z+1,q)}{\Gamma (-z)}}}
o alternativamente,
ψ
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z
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q
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=
e
−
γ
z
∂
∂
z
(
e
γ
z
ζ
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z
+
1
,
q
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Γ
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−
z
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)
,
{\displaystyle \psi (z,q)=e^{-\gamma z}{\frac {\partial }{\partial z}}\left(e^{\gamma z}{\frac {\zeta (z+1,q)}{\Gamma (-z)}}\right),}
donde ψ(z) es la función polygamma y ζ(z,q) , es la función zeta de Hurwitz .
La función está balanceada si satisface las condiciones
f
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0
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=
f
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1
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y
∫
0
1
f
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x
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d
x
=
0
{\displaystyle f(0)=f(1)\quad {\text{y}}\quad \int _{0}^{1}f(x)\,dx=0}
Varias funciones especiales pueden ser expresadas en términos de función polygamma generalizada:
ψ
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x
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=
ψ
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0
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x
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ψ
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n
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x
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=
ψ
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n
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x
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n
∈
N
Γ
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x
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=
exp
(
ψ
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−
1
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x
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+
1
2
ln
2
π
)
ζ
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z
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q
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=
Γ
(
1
−
z
)
ln
2
(
2
−
z
ψ
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z
−
1
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q
+
1
2
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+
2
−
z
ψ
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z
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1
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q
2
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−
ψ
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z
−
1
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)
ζ
′
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−
1
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x
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=
ψ
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−
2
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x
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+
x
2
2
−
x
2
+
1
12
B
n
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q
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=
−
Γ
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n
+
1
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ln
2
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2
n
−
1
ψ
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−
n
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q
+
1
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+
2
n
−
1
ψ
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2
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−
ψ
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−
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q
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)
{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\[8px]\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\[8px]\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\ln 2\pi \right)\\[8px]\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)\\[8px]\zeta '(-1,x)&=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}\\[8px]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}}}
donde Bn (q )polinomios de Bernoulli
K
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z
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=
A
exp
(
ψ
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−
2
,
z
)
+
z
2
−
z
2
)
{\displaystyle K(z)=A\exp \left(\psi (-2,z)+{\frac {z^{2}-z}{2}}\right)}
donde K(z) es la función K y A es la constante de Glaisher .
La función polygamma generalizada puede ser expresada en forma compacta en ciertos puntos (donde A es la constante de Glaisher y G es la constante de Catalan ):
ψ
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−
2
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1
4
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=
1
8
ln
2
π
+
9
8
ln
A
+
G
4
π
ψ
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−
2
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1
2
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=
1
4
ln
π
+
3
2
ln
A
+
5
24
ln
2
ψ
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−
3
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1
2
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=
1
16
ln
2
π
+
1
2
ln
A
+
7
ζ
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3
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32
π
2
ψ
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−
2
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1
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=
1
2
ln
2
π
ψ
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−
3
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1
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=
1
4
ln
2
π
+
ln
A
ψ
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−
2
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2
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=
ln
2
π
−
1
ψ
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−
3
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2
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=
ln
2
π
+
2
ln
A
−
3
4
{\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 }}&&\\[8px]\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}}}\\[8px]\psi (-2,1)&={\tfrac {1}{2}}\ln 2\pi &\psi (-3,1)&={\tfrac {1}{4}}\ln 2\pi +\ln A\\[8px]\psi (-2,2)&=\ln 2\pi -1&\psi (-3,2)&=\ln 2\pi +2\ln A-{\tfrac {3}{4}}\end{aligned}}}
![{\displaystyle {\begin{aligned}\psi (x)&=\psi (0,x)\\[8px]\psi ^{(n)}(x)&=\psi (n,x)\qquad n\in \mathbb {N} \\[8px]\Gamma (x)&=\exp \left(\psi (-1,x)+{\tfrac {1}{2}}\ln 2\pi \right)\\[8px]\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)\\[8px]\zeta '(-1,x)&=\psi (-2,x)+{\frac {x^{2}}{2}}-{\frac {x}{2}}+{\frac {1}{12}}\\[8px]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/1c40f6972a408c32a54e0f7550ae3f53eecaa129)
![{\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 }}&&\\[8px]\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}}}\\[8px]\psi (-2,1)&={\tfrac {1}{2}}\ln 2\pi &\psi (-3,1)&={\tfrac {1}{4}}\ln 2\pi +\ln A\\[8px]\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/bdb50b9b62941c5c121c924af190a8b1bf5bc9f9)
Datos: Q4849995