Kummer's Formulas

Author

Eric W. Weisstein
September 21, 2008

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

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

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

First

(Subscript[, 2])(Subscript[F, 1])(-k+m+(1/2),-n;2 m+1;1)==((Γ(2 m+1) Γ(k+m+n+(1/2)))/(Γ(k+m+(1/2)) Γ(2 m+n+1)))

FunctionExpand[Hypergeometric2F1[(1/2)+m-k,-n,2m+1,1]]

expr=FullSimplify[Subtract@@{Hypergeometric2F1[(1/2)-k+m,-n,1+2 m,1],((Gamma[2m+1]Gamma[m+(1/2)+k+n])/(Gamma[m+(1/2)+k]Gamma[2m+1+n]))}]

Table[expr/.{k->Random[Complex],m->Random[Complex],n->Random[Complex]},{100}]//Chop//Union

(Subscript[, 2])(Subscript[F, 1])(a,b;c;-1)==((Γ((b/2)+1) Γ(-a+b+1))/(Γ(-a+(b/2)+1) Γ(b+1))), b positive integer

expr=FullSimplify[Subtract@@{Hypergeometric2F1[a,b,c,-1],((Gamma[(1/2)b+1]Gamma[b-a+1])/(Gamma[b+1]Gamma[(1/2)b-a+1]))}/.c->1+b-a]

(Subscript[, 2])(Subscript[F, 1])(a,b;c;-1)==((2 cos(((π b)/2)) (Γ(|b|) Γ(-a+b+1)))/(Γ(-a+(b/2)+1)))

expr=FullSimplify[Subtract@@{Hypergeometric2F1[a,b,c,-1],2Cos[(1/2)π b]((Gamma[Abs[b]]Gamma[b-a+1])/(Gamma[(1/2)b-a+1]))}/.c->1+b-a]

Table[expr/.{a->Random[Complex],b->-Random[Integer,{1,20}]},{20}]

Second Formula

(Subscript[M, 0,m])(z)==(z^(m+(1/2))) (e^(-(z/2))) (Subscript[, 1])(Subscript[F, 1])(m+(1/2);2 m+1;z)==(4^m) (Sqrt[z]) (Subscript[I, m])((z/2)) Γ(m+1)

exp(-(z/2)) (z^(m+(1/2))) (Subscript[, 1])(Subscript[F, 1])(m+(1/2);2 m+1;z)==(4^m) (Sqrt[z]) Γ(m+1) (Subscript[I, m])((z/2))

(z^(m+1/2))Exp[-z/2]Hypergeometric1F1[(1/2)+m,2m+1,z]//Simplify

FunctionExpand[WhittakerM[0,m,z]]//Simplify

V5

(z^(m+1/2))(1+Sum[((z^(2p))/((2^(4p))p! Pochhammer[m+1,p])),{p,∞}])//Simplify

V7

(z^(m+1/2))(1+Sum[((z^(2p))/((2^(4p))p! Pochhammer[m+1,p])),{p,∞}])

FunctionExpand[%]