convert/1F1 converts, when possible, hypergeometric, MeijerG, and special functions admitting a 0F1 hypergeometric representation into special functions admitting a 1F1 hypergeometric representation; that is, into one of

FunctionAdvisor( `1F1` );

The 20 functions in the "1F1" class are particular cases of the hypergeometric function and are given by:

$\left[\mathrm{CoulombF}\,\mathrm{CylinderD}\,\mathrm{CylinderU}\,\mathrm{CylinderV}\,\mathrm{Ei}\,\mathrm{FresnelC}\,\mathrm{FresnelS}\,\mathrm{Fresnelf}\,\mathrm{Fresnelg}\,\mathrm{\Gamma }\,\mathrm{HermiteH}\,\mathrm{KummerM}\,\mathrm{KummerU}\,\mathrm{LaguerreL}\,\mathrm{WhittakerM}\,\mathrm{WhittakerW}\,\mathrm{dawson}\,\mathrm{erf}\,\mathrm{erfc}\,\mathrm{erfi}\right]$

convert/1F1 accepts as optional arguments all those described in convert/to_special_function.

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

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

$\mathrm{convert}\left(\,\mathrm{`1F1`}\right)$

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

$\mathrm{hypergeom}\left(\left[\right]\,\left[c\right]\,z\right)$

$\mathrm{hypergeom}\left(\left[\right]\,\left[c\right]\,z\right)$

$\mathrm{convert}\left(\,\mathrm{`1F1`}\right)$

$\frac{\mathrm{LaguerreL}\left(\frac{1}{2}-c\,-2+2c\,4\sqrt{z}\right)}{\left(\genfrac{}{}{0ex}{}{c-\frac{3}{2}}{\frac{1}{2}-c}\right){\ⅇ}^{2\sqrt{z}}}$

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

$\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[\frac{a}{2}\,-\frac{a}{2}\right]\,\left[\right]\right]\,\frac{z^2}{4}\right)$

$\mathrm{convert}\left(\,\mathrm{`1F1`}\right)$

$\frac{\mathrm{\Gamma }\left(a\right)\mathrm{LaguerreL}\left(-\frac{1}{2}+a\,-2a\,2z\right)}{{\left(\frac{z^2}{4}\right)}^{\frac{a}{2}}\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}-a}{-\frac{1}{2}+a}\right){\ⅇ}^z}+\frac{\mathrm{\Gamma }\left(-a\right){\left(\frac{z^2}{4}\right)}^{\frac{a}{2}}\mathrm{LaguerreL}\left(-\frac{1}{2}-a\,2a\,2z\right)}{\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}+a}{-\frac{1}{2}-a}\right){\ⅇ}^z}$

$\mathrm{BesselJ}\left(a\,z\right)+\mathrm{KelvinBei}\left(a-1\,z-1\right)$

$\mathrm{BesselJ}\left(a\,z\right)+\mathrm{KelvinBei}\left(a-1\,z-1\right)$

$\mathrm{convert}\left(\,\mathrm{`1F1`}\right)$

$\frac{z^a\mathrm{LaguerreL}\left(-\frac{1}{2}-a\,2a\,2\mathrm{I}z\right)}{\mathrm{\Gamma }\left(a+1\right)2^a\left(\genfrac{}{}{0ex}{}{-\frac{1}{2}+a}{-\frac{1}{2}-a}\right){\ⅇ}^{\mathrm{I}z}}-\frac{\mathrm{I}{\ⅇ}^{\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)\left(z-1\right)\sqrt{2}}{\left(\left(-\frac{1}{2}+\frac{\mathrm{I}}{2}\right)\left(z-1\right)\sqrt{2}\right)}^{a-1}\mathrm{LaguerreL}\left(\frac{1}{2}-a\,-2+2a\,\left(\mathrm{-1}-\mathrm{I}\right)\left(z-1\right)\sqrt{2}\right)}{\mathrm{\Gamma }\left(a\right)\left(\genfrac{}{}{0ex}{}{-\frac{3}{2}+a}{\frac{1}{2}-a}\right)2^a}+\frac{\mathrm{I}{\left(\left(-\frac{1}{2}-\frac{\mathrm{I}}{2}\right)\left(z-1\right)\sqrt{2}\right)}^{a-1}\mathrm{LaguerreL}\left(\frac{1}{2}-a\,-2+2a\,\left(\mathrm{-1}+\mathrm{I}\right)\left(z-1\right)\sqrt{2}\right){\ⅇ}^{\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)\left(z-1\right)\sqrt{2}}}{\mathrm{\Gamma }\left(a\right)\left(\genfrac{}{}{0ex}{}{-\frac{3}{2}+a}{\frac{1}{2}-a}\right)2^a}$