CITE THIS NOTEBOOK: Hypergeometric function in terms of the digamma function by Robert Reynolds. Wolfram Community DEC 13 2022.
ORIGINAL ARTICLE: Robert Reynolds, Allan Stauffer. Extended Prudnikov sum[J]. AIMS Mathematics, 2022, 7(10): 18576-18586. https://doi.org/10.3934/math.20221021.
ORIGINAL ARTICLE: Robert Reynolds, Allan Stauffer. Extended Prudnikov sum[J]. AIMS Mathematics, 2022, 7(10): 18576-18586. https://doi.org/10.3934/math.20221021.
In this Notebook we look at a special case of the Hurwitz-Lerch Zeta function expressed in terms of the Log-gamma function. Some derived formulae are given by:
Start with equation (6.7) in [1] to get
[,0,u],Log+Log
(0,1,0)
HurwitzLerchPhi
Gamma
u
4
2Gamma
2+u
4
Gamma
1+u
4
2Gamma
3+u
4
Next rewrite the left-hand side in terms of the logarithm function and take the first partial derivative with respect to a to get
DExpLog[a+k],-Log-Log,a//FullSimplify
kπ
2
Gamma
a
4
2Gamma
2+a
4
Gamma
1+a
4
2Gamma
3+a
4
Next take the infinite sum as per the Lerch function to get and simplify using equation (9.559) in [2] to get
Sum,{k,0,∞},-PolyGamma0,-PolyGamma0,+PolyGamma0,+PolyGamma0,/.{a->s}
kπ
2
a+k
1
4
a
4
1+a
4
2+a
4
3+a
4
Hypergeometric2F1[1,s,1+s,],-PolyGamma0,-PolyGamma0,+PolyGamma0,+PolyGamma0,
s
4
s
4
1+s
4
2+s
4
3+s
4
Next take the definite integral over s to get
Integrate,-PolyGamma0,-PolyGamma0,+PolyGamma0,+PolyGamma0,,{a,α,β}
Hypergeometric2F1[1,a,1+a,]
a
1
4
a
4
1+a
4
2+a
4
3+a
4
Plots of the right-hand side in terms of the digamma function
In[]:=
s
4
s
4
1+s
4
2+s
4
3+s
4
In[]:=
Plot3DRe(a+b)-PolyGamma0,(a+b)-PolyGamma0,(1+a+b)+PolyGamma0,(2+a+b)+PolyGamma0,(3+a+b),{a,-7,1},{b,-2,2},AxesLabelAutomatic,ClippingStyleNone,PlotTheme"Web"
1
4
1
4
1
4
1
4
1
4
Out[]=
In[]:=
Plot3DIm(a+b)-PolyGamma0,(a+b)-PolyGamma0,(1+a+b)+PolyGamma0,(2+a+b)+PolyGamma0,(3+a+b),{a,-7,1},{b,-2,2},AxesLabelAutomatic,ClippingStyleNone,PlotTheme"Web"
1
4
1
4
1
4
1
4
1
4
Out[]=
In[]:=
Plot3DAbs(a+b)-PolyGamma0,(a+b)-PolyGamma0,(1+a+b)+PolyGamma0,(2+a+b)+PolyGamma0,(3+a+b),{a,-7,1},{b,-2,2},AxesLabelAutomatic,ClippingStyleNone,PlotTheme"Web"
1
4
1
4
1
4
1
4
1
4
Out[]=
References
[1] Robert Reynolds, Allan Stauffer. Extended Prudnikov sum, [J]. AIMS Mathematics, 2022, 7(10): 18576-18586. doi: 10.3934/math.20221021
[2] Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integrals, Series and Products, 6th ed.; Academic Press: Cambridge, MA, USA, 2000.
[1] Robert Reynolds, Allan Stauffer. Extended Prudnikov sum, [J]. AIMS Mathematics, 2022, 7(10): 18576-18586. doi: 10.3934/math.20221021
[2] Gradshteyn, I.S.; Ryzhik, I.M. Tables of Integrals, Series and Products, 6th ed.; Academic Press: Cambridge, MA, USA, 2000.