ガウスの連分数
2 F 1 ( a + 1 , b ; c + 1 ; z ) 2 F 1 ( a , b ; c ; z ) = 1 1 + ( a − c ) b c ( c + 1 ) z 1 + ( b − c − 1 ) ( a + 1 ) ( c + 1 ) ( c + 2 ) z 1 + ( a − c − 1 ) ( b + 1 ) ( c + 2 ) ( c + 3 ) z 1 + ( b − c − 2 ) ( a + 2 ) ( c + 3 ) ( c + 4 ) z 1 + ⋱ {\displaystyle {\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}}={\cfrac {1}{1+{\cfrac {{\frac {(a-c)b}{c(c+1)}}z}{1+{\cfrac {{\frac {(b-c-1)(a+1)}{(c+1)(c+2)}}z}{1+{\cfrac {{\frac {(a-c-1)(b+1)}{(c+2)(c+3)}}z}{1+{\cfrac {{\frac {(b-c-2)(a+2)}{(c+3)(c+4)}}z}{1+{}\ddots }}}}}}}}}}}
または
2 F 1 ( a + 1 , b ; c + 1 ; z ) c 2 F 1 ( a , b ; c ; z ) = 1 c + ( a − c ) b z ( c + 1 ) + ( b − c − 1 ) ( a + 1 ) z ( c + 2 ) + ( a − c − 1 ) ( b + 1 ) z ( c + 3 ) + ( b − c − 2 ) ( a + 2 ) z ( c + 4 ) + ⋱ {\displaystyle {\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{c{}_{2}F_{1}(a,b;c;z)}}={\cfrac {1}{c+{\cfrac {(a-c)bz}{(c+1)+{\cfrac {(b-c-1)(a+1)z}{(c+2)+{\cfrac {(a-c-1)(b+1)z}{(c+3)+{\cfrac {(b-c-2)(a+2)z}{(c+4)+{}\ddots }}}}}}}}}}}