d-Analog
المؤلف:
Flajolet, P.; Labelle, G.; Laforest, L.; and Salvy, B.
المصدر:
"Hypergeometrics and the Cost Structure of Quadtrees." Random Structure Alg. 7, 117-144, 1995. http://algo.inria.fr/flajolet/Publications/publist.html.
الجزء والصفحة:
...
18-8-2019
1861
d-Analog
The 
-analog of a complex number 
is defined as
![[s]_d=1-(2^d)/(s^d)](http://mathworld.wolfram.com/images/equations/d-Analog/NumberedEquation1.gif) |
(1)
|
(Flajolet et al. 1995). For integer 
, ![[2]!=1](http://mathworld.wolfram.com/images/equations/d-Analog/Inline4.gif)
and
It can then be extended to complex values via
![[s]_d!=product_(j=1)^infty([j+2])/([j+s])](http://mathworld.wolfram.com/images/equations/d-Analog/NumberedEquation2.gif) |
(4)
|
(Flajolet et al. 1995). It satisfies the basic functional identity
![[s]_d!=[s]_d[s-1]_d!.](http://mathworld.wolfram.com/images/equations/d-Analog/NumberedEquation3.gif) |
(5)
|
The 
-analog of the polygamma function is
The first few values are
where 
is the digamma function.
The 
-analog of the Euler-Mascheroni constant 
is
(Flajolet et al. 1995). The first few values are
where 
is a harmonic number.
The 
-analog of the harmonic numbers is ![[H_2]_d=0](http://mathworld.wolfram.com/images/equations/d-Analog/Inline47.gif)
and
(Flajolet et al. 1995).
The 
-analog of infinity factorial is given by
![[infty]_d!=product_(n=3)^infty(1-(2^d)/(n^d)).](http://mathworld.wolfram.com/images/equations/d-Analog/NumberedEquation4.gif) |
(18)
|
This infinite product can be evaluated in closed form in terms of 
, the hyperbolic sine 
, and gamma functions 
involving roots of unity 
,
These are all special cases of a general result for infinite products.
REFERENCES:
Flajolet, P.; Labelle, G.; Laforest, L.; and Salvy, B. "Hypergeometrics and the Cost Structure of Quadtrees." Random Structure Alg. 7, 117-144, 1995. http://algo.inria.fr/flajolet/Publications/publist.html.
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