Lanczos Approximation

Author

Eric W. Weisstein
August 6, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/LanczosApproximation.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Coefficients

Table[(1/(Pochhammer[z+1,k]Pochhammer[z+1,-k])),{k,0,3}]//Factor

Table[(((((-1))^k)Pochhammer[-z,k])/(Pochhammer[z+1,k])),{k,0,3}]//Factor

Lanczos and Related

Gamma[z+1]/(z+EulerGamma+1/2)^(z+1/2)/Exp[z+EulerGamma+1/2]/Sqrt[2Pi]

Lanczos

H[0,z]:=1
H[k_,z_]:=1/Pochhammer[z+1,k]/Pochhammer[z+1,-k]

eps[0]:=1
eps[n_]:=2
g[k_]:=1/Sqrt[2Pi] Exp[s]eps[k](-1)^k Sum[(-1)^r  Binomial[k,r]Pochhammer[k,r](E/(r+s+1/2))^(r+1/2),{r,0,k}]

Table[H[k,z],{k,0,3}]

FullSimplify[g[0]]

Simplify[Table[g[k],{k,0,3}]]

Simplify[Table[H[k,z]g[k],{k,0,3}]]

Sums

(LogGamma[z])==(1/2) (∑_n^∞)((z c(n))/(n (Pochhammer[z, n+1])))-z+(z-(1/2)) log(z)+(1/2) log(2 π)

Whittaker and Watson 1990, p. 253

FullSimplify[Pochhammer[x,n+1]/x]//FunctionExpand//FullSimplify

Table[((Subscript[c,n])/( n ((Gamma[1+n+x])/(Gamma[1+x])))),{n,3}]//FunctionExpand

Table[((x!Subscript[c,n])/( n (n+x)!)),{n,3}]//FunctionExpand

Table[((z Subscript[c,n])/( n Pochhammer[z,n+1])),{n,3}]

Table[((z Integrate[Pochhammer[x,n](2x-1),{x,0,1}])/( n Pochhammer[z,n+1])),{n,3}]

{LogGamma[z],(1/2)Sum[((z Integrate[Pochhammer[x,n](2x-1),{x,0,1}])/( n Pochhammer[z,n+1])),{n,20}]-z+(z-(1/2))Log[z]+(1/2)Log[2π]}/.z->3.

{LogGamma[z],(1/2)Sum[((z Integrate[Pochhammer[x,n](2x-1),{x,0,1}])/( n Pochhammer[z,n+1])),{n,20}]-z+(z-(1/2))Log[z]+(1/2)Log[2π]}/.z->99.

Table[Integrate[Pochhammer[x,n](2x-1),{x,0,1}],{n,10}]

V5

Integrate[Pochhammer[x,n](2x-1),{x,0,1}]

V7

Integrate[Pochhammer[x,n](2x-1),{x,0,1}]

FunctionExpand[Pochhammer[x,n](2x-1)]//FullSimplify

Integrate[(((-1+2 x) Gamma[n+x])/(Gamma[x])),{x,0,1}]

V8

Integrate[Pochhammer[x,n](2x-1),{x,0,1}]

(LogGamma[z])==(1/2) (∑_(n=2)^∞)(((n-1) (HurwitzZeta[n, z+1]))/(n (n+1)))-z+(z-(1/2)) ln(z)+(1/2) ln(2 π)

Sum[(1/(((z+r))^n)),{r,∞}]

V??

(z-(1/2))Log[z]-z+(1/2)Log[2π]+(1/2)Sum[((n-1)/(n(n+1)))HurwitzZeta[n,1+z],{n,2,∞}]//Timing//InputForm

V8

(z-(1/2))Log[z]-z+(1/2)Log[2π]+(1/2)Sum[((n-1)/(n(n+1)))HurwitzZeta[n,1+z],{n,2,∞}]

...crash...

Numeric

Subtract@@@Table[{LogGamma[z],(z-(1/2))Log[z]-z+(1/2)Log[2π]+(1/2)Sum[((n-1)/(n(n+1)))HurwitzZeta[n,1+z],{n,2,100}]}/.z->RandomComplex[5(I+1){1},WorkingPrecision->20],{10}]//Chop//Union

(LogGamma[z])==(1/2) (∑_(n=2)^∞)(((((-1))^n) (n-1) (PolyGamma2[n-1, z+1]))/((n+1)!))-z+(z-(1/2)) z ln+(1/2) (2 π) ln

(1/2)Sum[((n-1)/(n(n+1)))HurwitzZeta[n,1+z],{n,2,4}]//FunctionExpand

FunctionExpand[HurwitzZeta[n,1+z],n∈Integers&&n>1]

FullSimplify[((n-1)/(n(n+1)))FunctionExpand[HurwitzZeta[n,1+z],n∈Integers&&n>1],n∈Integers&&n>1]

(((((-1))^n) (-1+n) PolyGamma[-1+n,1+z])/((n+1)!))

Gamma[2+n]-(n+1)!//FullSimplify

Sum[(((((-1))^n) (-1+n) PolyGamma[-1+n,1+z])/((n+1)!)),{n,2,∞}]