تابع بسل-کلیفورد

${\displaystyle xy''+(n+1)y'=y.\qquad }$

${\displaystyle {\mathcal {C}}_{n}(z)=\pi (n)\ _{0}F_{1}(;n+1;z).}$${\displaystyle {\displaystyle {\mathcal {C}}_{n}(z)=\pi (n)\ _{0}F_{1}(;n+1;z).}}$

${\displaystyle J_{n}(z)=\left({\frac {z}{2}}\right)^{n}{\mathcal {C}}_{n}\left(-{\frac {z^{2}}{4}}\right);}$

${\displaystyle I_{n}(z)=\left({\frac {z}{2}}\right)^{n}{\mathcal {C}}_{n}\left({\frac {z^{2}}{4}}\right).}$${\displaystyle {\displaystyle I_{n}(z)=\left({\frac {z}{2}}\right)^{n}{\mathcal {C}}_{n}\left({\frac {z^{2}}{4}}\right).}}$

${\displaystyle {\mathcal {C}}_{n}(z)=z^{-n/2}I_{n}(2{\sqrt {z}});}$

${\displaystyle {\displaystyle {\frac {d}{dx}}{\mathcal {C}}_{n}(x)={\mathcal {C}}_{n+1}(x).}}$

${\displaystyle {\displaystyle x{\mathcal {C}}_{n+2}(x)+(n+1){\mathcal {C}}_{n+1}(x)={\mathcal {C}}_{n}(x),}}$

${\displaystyle {\displaystyle {\frac {{\mathcal {C}}_{n+1}(x)}{{\mathcal {C}}_{n}(x)}}={\cfrac {1}{n+1+{\cfrac {x}{n+2+{\cfrac {x}{n+3+{\cfrac {x}{\ddots }}}}}}}}.}}$

${\displaystyle {\displaystyle \exp \left(t+{\frac {z}{t}}\right)=\sum _{n=-\infty }^{\infty }t^{n}{\mathcal {C}}_{n}(z).}}$${\displaystyle \exp \left(t+{\frac {z}{t}}\right)=\sum _{n=-\infty }^{\infty }t^{n}{\mathcal {C}}_{n}(z).}$

${\displaystyle {\mathcal {C}}_{n}(z)={\frac {1}{2\pi i}}\oint _{C}{\frac {\exp(t+z/t)}{t^{n+1}}}\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\exp(z\exp(-i\theta )+\exp(i\theta )-ni\theta )\,d\theta .}$