convert/Bessel_related converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Bessel related functions. The Bessel related functions are

FunctionAdvisor( Bessel_related );

The 14 functions in the "Bessel_related" class are:

$\left[\mathrm{AiryAi}\,\mathrm{AiryBi}\,\mathrm{BesselI}\,\mathrm{BesselJ}\,\mathrm{BesselK}\,\mathrm{BesselY}\,\mathrm{HankelH1}\,\mathrm{HankelH2}\,\mathrm{KelvinBei}\,\mathrm{KelvinBer}\,\mathrm{KelvinHei}\,\mathrm{KelvinHer}\,\mathrm{KelvinKei}\,\mathrm{KelvinKer}\right]$

$\frac{\frac{1}{\mathrm{\Gamma }\left(1+a\right)}{z}^a}{2^a}\mathrm{hypergeom}\left(\left[\right]\,\left[1+a\right]\,-\frac{1}{4}{z}^2\right)$

$\frac{z^a\mathrm{hypergeom}\left(\left[\right]\,\left[1+a\right]\,-\frac{z^2}{4}\right)}{\mathrm{\Gamma }\left(1+a\right)2^a}$

$\mathrm{convert}\left(\,\mathrm{Bessel\_related}\right)$

$\mathrm{BesselJ}\left(a\,z\right)$

$\mathrm{LaguerreL}\left(-\frac{1}{2}-a\,2a\,2\mathrm{I}z\right)-\mathrm{WhittakerM}\left(0\,a\,2\mathrm{I}z\right)$

$\mathrm{LaguerreL}\left(-\frac{1}{2}-a\,2a\,2\mathrm{I}z\right)-\mathrm{WhittakerM}\left(0\,a\,2\mathrm{I}z\right)$

$\mathrm{convert}\left(\,\mathrm{Bessel\_related}\right)$

$\frac{\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}+a}{-\frac{1}{2}-a}\right){\ⅇ}^{\mathrm{I}z}\mathrm{\Gamma }\left(1+a\right)\mathrm{BesselJ}\left(a\,z\right)2^a}{z^a}-\frac{{\left(2\mathrm{I}z\right)}^{a+\frac{1}{2}}\mathrm{\Gamma }\left(1+a\right)\mathrm{BesselJ}\left(a\,z\right)2^a}{z^a}$

$\mathrm{KummerU}\left(a+\frac{1}{2}\,2a+1\,z\right)$

$\mathrm{KummerU}\left(a+\frac{1}{2}\,2a+1\,z\right)$

$\mathrm{convert}\left(\,\mathrm{Bessel\_related}\right)$

$\frac{\sqrt{\mathrm{\pi }}{\ⅇ}^{\frac{z}{2}}\left(-\frac{\mathrm{BesselJ}\left(-a\,\frac{\mathrm{I}}{2}z\right){\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{2a+1}}{z^{2a}{2}^a}+\frac{\mathrm{BesselJ}\left(a\,\frac{\mathrm{I}}{2}z\right)2^a}{{\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{-1+2a}}\right)}{4\sin \left(\mathrm{\pi }\left(1+a\right)\right)}$

$\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[\frac{1}{2}a\right]\,\left[-\frac{1}{2}a\right]\right]\,z\right)$

$\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[\frac{a}{2}\right]\,\left[-\frac{a}{2}\right]\right]\,z\right)$

$\mathrm{convert}\left(\,\mathrm{Bessel\_related}\right)$

$\frac{z^{\frac{a}{2}}\mathrm{BesselJ}\left(a\,2\sqrt{z}\right)2^a}{{\left(2\sqrt{z}\right)}^a}$