Digital Library of Mathematical Functions
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15 Hypergeometric FunctionProperties

§15.16 Products

15.16.1 F(a,bc-12;z)F(c-a,c-bc+12;z)=s=0(c)s(c+12)sAszs,
|z|<1,

where A0=1 and As, s=1,2,, are defined by the generating function

15.16.2 (1-z)a+b-cF(2a,2b;2c-1;z)=s=0Aszs,
|z|<1.

Also,

15.16.3 F(a,bc;z)F(a,bc;ζ)=s=0(a)s(b)s(c-a)s(c-b)s(c)s(c)2ss!(zζ)sF(a+s,b+sc+2s;z+ζ-zζ),
|z|<1, |ζ|<1, |z+ζ-zζ|<1.
15.16.4 F(a,bc;z)F(-a,-b-c;z)+ab(a-c)(b-c)c2(1-c2)z2F(1+a,1+b2+c;z)F(1-a,1-b2-c;z)=1.

Generalized Legendre’s Relation

15.16.5 F(12+λ,-12-ν1+λ+μ;z)F(12-λ,12+ν1+ν+μ;1-z)+F(12+λ,12-ν1+λ+μ;z)F(-12-λ,12+ν1+ν+μ;1-z)-F(12+λ,12-ν1+λ+μ;z)F(12-λ,12+ν1+ν+μ;1-z)=Γ(1+λ+μ)Γ(1+ν+μ)Γ(λ+μ+ν+32)Γ(12+ν),
|phz|<π, |ph(1-z)|<π.

For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).