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15 Hypergeometric FunctionProperties

§15.16 Products

15.16.1 F(a,bc12;z)F(ca,cbc+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 (1z)a+bcF(2a,2b;2c1;z)=s=0Aszs,
|z|<1.

Also,

15.16.3 F(a,bc;z)F(a,bc;ζ)=s=0(a)s(b)s(ca)s(cb)s(c)s(c)2ss!(zζ)sF(a+s,b+sc+2s;z+ζzζ),
|z|<1, |ζ|<1, |z+ζzζ|<1.
15.16.4 F(a,bc;z)F(a,bc;z)+ab(ac)(bc)c2(1c2)z2F(1+a,1+b2+c;z)F(1a,1b2c;z)=1.

Generalized Legendre’s Relation

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

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