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16 Generalized Hypergeometric Functions and Meijer G-FunctionGeneralized Hypergeometric Functions

§16.12 Products

16.12.1 F10(-;a;z)F10(-;b;z)=F32(12(a+b),12(a+b-1)a,b,a+b-1;4z).
16.12.2 (F12(a,ba+b+12;z))2=F23(2a,2b,a+ba+b+12,2a+2b;z).

More generally,

16.12.3 (F12(a,bc;z))2=k=0(2a)k(2b)k(c-12)k(c)k(2c-1)kk!F34(-12k,12(1-k),a+b-c+12,12a+12,b+12,32-k-c;1)zk,
|z|<1.

For further identities see Goursat (1883) and Erdélyi et al. (1953a, §4.3).