Gauss-Kummer Series

Author

Eric W. Weisstein
September 6, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Gauss-KummerSeries.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sum

V6

Sum[Binomial[1/2,n]^2(h^(2n)),{n,0,∞}]

FullSimplify[%]

V7

Sum[Binomial[1/2,n]^2(h^(2n)),{n,0,∞}]

FullSimplify[%]

FunctionExpand[%]

FullSimplify[%]

Series

Series[Hypergeometric2F1[-(1/2),-(1/2),1,(x^2)],{x,0,12}]

1/FunctionExpand[ArithmeticGeometricMean[1+x,1-x],0<x<1]

V7

Series[Hypergeometric2F1[-(1/2),-(1/2),1,x],{x,0,10}]//Normal

SeriesCoefficient[Hypergeometric2F1[-(1/2),-(1/2),1,x],{x,0,n}]//InputForm

...wrong...

Table[% x^n,{n,0,10}]

FullSimplify[(((((-(3/2)+n)!))^2)/(4 π (((n!))^3))),n∈Integers&&n>0]

Table[(((Gamma[-(1/2)+n])^2)/(4 π (((n!))^3)))(x^(2n)),{n,0,12}]