ガウスの連分数
cosh  ( z ) = 0 F 1 ( 1 2 ; z 2 4 ) {\displaystyle \cosh(z)=\,_{0}F_{1}({\tfrac {1}{2}};{\tfrac {z^{2}}{4}})}
sinh  ( z ) = z 0 F 1 ( 3 2 ; z 2 4 ) {\displaystyle \sinh(z)=z\,_{0}F_{1}({\tfrac {3}{2}};{\tfrac {z^{2}}{4}})}
より、
tanh  ( z ) = z 0 F 1 ( 3 2 ; z 2 4 ) 0 F 1 ( 1 2 ; z 2 4 ) = z / 2 1 2 + z 2 4 3 2 + z 2 4 5 2 + z 2 4 7 2 + ⋱ = z 1 + z 2 3 + z 2 5 + z 2 7 + ⋱ {\displaystyle \tanh(z)={\frac {z\,_{0}F_{1}({\tfrac {3}{2}};{\tfrac {z^{2}}{4}})}{\,_{0}F_{1}({\tfrac {1}{2}};{\tfrac {z^{2}}{4}})}}={\cfrac {z/2}{{\tfrac {1}{2}}+{\cfrac {\tfrac {z^{2}}{4}}{{\tfrac {3}{2}}+{\cfrac {\tfrac {z^{2}}{4}}{{\tfrac {5}{2}}+{\cfrac {\tfrac {z^{2}}{4}}{{\tfrac {7}{2}}+{}\ddots }}}}}}}}={\cfrac {z}{1+{\cfrac {z^{2}}{3+{\cfrac {z^{2}}{5+{\cfrac {z^{2}}{7+{}\ddots }}}}}}}}}
次は、関数
1 F 1 ( a ; b ; z ) = 1 + a b 1 ! z + a ( a + 1 ) b ( b + 1 ) 2 ! z 2 + a ( a + 1 ) ( a + 2 ) b ( b + 1 ) ( b + 2 ) 3 ! z 3 + ⋯ {\displaystyle {}_{1}F_{1}(a;b;z)=1+{\frac {a}{b\,1!}}z+{\frac {a(a+1)}{b(b+1)\,2!}}z^{2}+{\frac {a(a+1)(a+2)}{b(b+1)(b+2)\,3!}}z^{3}+\cdots }
から生じる2通りの恒等式
1 F 1 ( a ; b − 1 ; z ) − 1 F 1 ( a + 1 ; b ; z ) = ( a − b + 1 ) z b ( b − 1 ) 1 F 1 ( a + 1 ; b + 1 ; z ) {\displaystyle \,_{1}F_{1}(a;b-1;z)-\,_{1}F_{1}(a+1;b;z)={\frac {(a-b+1)z}{b(b-1)}}\,_{1}F_{1}(a+1;b+1;z)}
1 F 1 ( a ; b − 1 ; z ) − 1 F 1 ( a ; b ; z ) = a z b ( b − 1 ) 1 F 1 ( a + 1 ; b + 1 ; z ) {\displaystyle \,_{1}F_{1}(a;b-1;z)-\,_{1}F_{1}(a;b;z)={\frac {az}{b(b-1)}}\,_{1}F_{1}(a+1;b+1;z)}
を交互に用いて、