تابع بسل-کلیفورد

${\displaystyle x{y}^{″}+(n+1)y^{\prime }=y.}$

${\displaystyle {\mathcal{C}}_n(z)=\pi (n){}_0{F}_1(;n+1;z).}$

${\displaystyle {\displaystyle {\mathcal{C}}_n(z)=\pi (n){}_0{F}_1(;n+1;z).}}$

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

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

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

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

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

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

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

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

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

${\displaystyle {\mathcal{C}}_n(z)=\frac{1}{2\pi i}{\oint }_C\frac{\exp (t+z/t)}{t^{n+1}}dt=\frac{1}{2\pi }{\int }_0^{2\pi }\exp (z\exp (-i\theta )+\exp (i\theta )-ni\theta )d\theta .}$