d-Analog

Author

Eric W. Weisstein
January 16, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/d-Analog.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/d-Analog.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

DFactorial[n_,d_]:=Module[{k},Product[1-2^d/k^d,{k,3,n}]]
DEulerGamma[n_,d_]:=Module[{m},d 2^d Sum[1/m/(m^d-2^d),{m,3,n}]]
DPolyGamma[s_,d_]:=Module[{m},-d 2^d Sum[1/(m+s-1)/((m+s-1)^d-2^d),{m,Infinity}]]
DHarmonicNumber[n_,d_]:=DPolyGamma[n+1,d]+DEulerGamma[d,d]

d-Factorial

DFactorial[n,d]

V5

FullSimplify[Table[DFactorial[n],{n,10}]]//TableForm//TraditionalForm

N[%,20]

V7

FullSimplify[Table[DFactorial[n],{n,10}]]//TableForm//TraditionalForm

(Subscript[d, 5])

{1/1240/Abs[Gamma[2 (((-1))^(1/5))]Gamma[-2 (((-1))^(2/5))]]^2,(1/(1240 Gamma[2 (((-1))^(1/5))] Gamma[-2 (((-1))^(2/5))] Gamma[2 (((-1))^(3/5))] Gamma[-2 (((-1))^(4/5))]))}//N

(Subscript[d, 7])

N[Times@@{Gamma[2 (((-1))^(1/7))], Gamma[-2 (((-1))^(2/7))] ,Gamma[2 (((-1))^(3/7))], Gamma[-2 (((-1))^(4/7))], Gamma[2 (((-1))^(5/7))], Gamma[-2 (((-1))^(6/7))]},20]

N[Abs[Gamma[2 (((-1))^(1/7))]Gamma[-2 (((-1))^(2/7))] Gamma[2 (((-1))^(3/7))]]^2,20]

(Subscript[d, 8])

{Sinh[2 (((-1))^(1/4)) π] Sinh[2 (((-1))^(3/4)) π]}//N

-Abs[Sinh[2 (((-1))^(1/4)) π]]^2//N

(Subscript[d, 9])

N[{Sinh[(Sqrt[3]) π]/(588672 (Sqrt[3]) π Gamma[2 (((-1))^(1/9))] Gamma[-2 (((-1))^(2/9))] Gamma[-2 (((-1))^(4/9))] Gamma[2 (((-1))^(5/9))] Gamma[2 (((-1))^(7/9))] Gamma[-2 (((-1))^(8/9))]),Sinh[(Sqrt[3]) π]/(588672 (Sqrt[3]) π Abs[ Gamma[2 (((-1))^(1/9))] Gamma[-2 (((-1))^(2/9))] Gamma[-2 (((-1))^(4/9))]]^2)},20]

Euler - Mascheroni

Simplify[Table[DEulerGamma[d],{d,5}]]

FullSimplify[Table[DEulerGamma[d],{d,5}]]

N[%]

PolyGamma

-d (2^d)Sum[(1/((m+s)((((m+s))^d)-(2^d)))),{m,1,∞}]

V5

FullSimplify[Table[DPolyGamma[s,d],{d,5}]]

V7

FullSimplify[Table[DPolyGamma[s,d],{d,5}]]//Column//TraditionalForm

Harmonic Number

FullSimplify[Table[DHarmonicNumber[n,d],{d,5}]]