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16 Generalized Hypergeometric Functions & Meijer G-FunctionGeneralized Hypergeometric Functions

Β§16.12 Products

16.12.1 F10⁑(βˆ’;a;z)⁒F10⁑(βˆ’;b;z)=F32⁑(12⁒(a+b),12⁒(a+bβˆ’1)a,b,a+bβˆ’1;4⁒z).

The following formula is often referred to as Clausen’s formula

16.12.2 (F12⁑(a,ba+b+12;z))2=F23⁑(2⁒a,2⁒b,a+ba+b+12,2⁒a+2⁒b;z).

More generally,

16.12.3 (F12⁑(a,bc;z))2=βˆ‘k=0∞(2⁒a)k⁒(2⁒b)k⁒(cβˆ’12)k(c)k⁒(2⁒cβˆ’1)k⁒k!⁒F34⁑(βˆ’12⁒k,12⁒(1βˆ’k),a+bβˆ’c+12,12a+12,b+12,32βˆ’kβˆ’c;1)⁒zk,
|z|<1.

For further identities see Goursat (1883) and ErdΓ©lyi et al. (1953a, Β§4.3).