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15 Hypergeometric FunctionProperties

§15.7 Continued Fractions

If |ph(1-z)|<π, then

15.7.1 F(a,b;c;z)F(a,b+1;c+1;z)=t0-u1zt1-u2zt2-u3zt3-,

where

15.7.2 tn =c+n,
u2n+1 =(a+n)(c-b+n),
u2n =(b+n)(c-a+n).

If |z|<1, then

15.7.3 F(a,b;c;z)F(a,b+1;c+1;z)=v0-w1v1-w2v2-w3v3-,

where

15.7.4 vn =c+n+(b-a+n+1)z,
wn =(b+n)(c-a+n)z.

If z<12, then

15.7.5 F(a,b;c;z)F(a+1,b+1;c+1;z)=x0+y1x1+y2x2+y3x3+,

where

15.7.6 xn =c+n-(a+b+2n+1)z,
yn =(a+n)(b+n)z(1-z).

See also Cuyt et al. (2008, pp. 295–309).