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

FunctionAdvisor( Kelvin );

The 6 functions in the "Kelvin" class are:

$\left[\mathrm{KelvinBei}\,\mathrm{KelvinBer}\,\mathrm{KelvinHei}\,\mathrm{KelvinHer}\,\mathrm{KelvinKei}\,\mathrm{KelvinKer}\right]$

$\mathrm{AiryAi}\left(z\right)$

$\mathrm{AiryAi}\left(z\right)$

$\mathrm{convert}\left(\&comma;\mathrm{Kelvin}\right)\mathbf{assuming}0<\mathrm{\Re }\left(z\right)$

$\frac{\sqrt{3}\sqrt{z}\left(\mathrm{KelvinKer}\left(\frac{1}{3}\&comma;\left(\frac{1}{3}-\frac{\mathrm{I}}{3}\right)z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2}\right)+\mathrm{I}\mathrm{KelvinKei}\left(\frac{1}{3}\&comma;\left(\frac{1}{3}-\frac{\mathrm{I}}{3}\right)z^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2}\right)\right)\left(\sqrt{3}+\mathrm{I}\right)}{6\mathrm{\pi }}$

$\mathrm{sol}≔y\left(z\right)=\mathrm{\_C1}\mathrm{KummerM}\left(\frac{1}{2}+a\&comma;1+2a\&comma;z\right)+\mathrm{\_C2}\mathrm{KummerU}\left(\frac{1}{2}+a\&comma;1+2a\&comma;z\right)$

$\mathrm{sol}≔y\left(z\right)=\mathrm{\_C1}\mathrm{KummerM}\left(\frac{1}{2}+a\&comma;1+2a\&comma;z\right)+\mathrm{\_C2}\mathrm{KummerU}\left(\frac{1}{2}+a\&comma;1+2a\&comma;z\right)$

$\mathrm{PDEtools}\left[\mathrm{dpolyform}\right]\left(\mathrm{sol}\&comma;\mathrm{no\_Fn}\right)$

$\left[\frac{{\&DifferentialD;}^2}{\&DifferentialD;z^2}y\left(z\right)=\frac{\left(-2a+z-1\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;z}y\left(z\right)\right)}{z}+\frac{\left(1+2a\right)y\left(z\right)}{2z}\right]\mathbf{where}\left[\right]$

$\mathrm{convert}\left(\mathrm{sol}\&comma;\mathrm{Kelvin}\right)$

$y\left(z\right)=\frac{\mathrm{c\_\_1}{\&ExponentialE;}^{\frac{z}{2}}\mathrm{\Gamma }\left(1+a\right)\left(\mathrm{KelvinBer}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)+\mathrm{I}\mathrm{KelvinBei}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)\right)2^a}{{\left(\frac{\mathrm{I}}{2}z\right)}^a}+\frac{\mathrm{c\_\_2}\sqrt{\mathrm{\pi }}{\&ExponentialE;}^{\frac{z}{2}}\left(\frac{2^a\mathrm{KelvinBer}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)}{{\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{-1+2a}}-\frac{{\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{1+2a}\mathrm{KelvinBer}\left(-a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)}{z^{2a}{2}^a}-\frac{\mathrm{I}\mathrm{KelvinBei}\left(-a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right){\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{1+2a}}{z^{2a}{2}^a}+\frac{\mathrm{I}\mathrm{KelvinBei}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)2^a}{{\left(\frac{\mathrm{I}}{2}z\right)}^a{2}^{-1+2a}}\right)}{4\sin \left(\mathrm{\pi }\left(1+a\right)\right)}$

$\mathrm{collect}\left(\&comma;\left[\mathrm{KelvinBer}\&comma;\mathrm{KelvinBei}\right]\&comma;\mathrm{simplify}\right)$

$y\left(z\right)={\left(\mathrm{I}z\right)}^{-a}{\&ExponentialE;}^{\frac{z}{2}}\left(4^a\mathrm{c\_\_1}\mathrm{\Gamma }\left(1+a\right)-\frac{\mathrm{c\_\_2}\sqrt{\mathrm{\pi }}\csc \left(\mathrm{\pi }a\right)}{2}\right)\mathrm{KelvinBer}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)+\frac{\mathrm{c\_\_2}{z}^{-2a}\sqrt{\mathrm{\pi }}{\left(\mathrm{I}z\right)}^a{\&ExponentialE;}^{\frac{z}{2}}\csc \left(\mathrm{\pi }a\right)\mathrm{KelvinBer}\left(-a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)}{2}+\mathrm{I}{\left(\mathrm{I}z\right)}^{-a}{\&ExponentialE;}^{\frac{z}{2}}\left(4^a\mathrm{c\_\_1}\mathrm{\Gamma }\left(1+a\right)-\frac{\mathrm{c\_\_2}\sqrt{\mathrm{\pi }}\csc \left(\mathrm{\pi }a\right)}{2}\right)\mathrm{KelvinBei}\left(a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)+\frac{\mathrm{I}\mathrm{c\_\_2}{z}^{-2a}\sqrt{\mathrm{\pi }}{\left(\mathrm{I}z\right)}^a{\&ExponentialE;}^{\frac{z}{2}}\csc \left(\mathrm{\pi }a\right)\mathrm{KelvinBei}\left(-a\&comma;\left(\frac{1}{4}-\frac{\mathrm{I}}{4}\right)z\sqrt{2}\right)}{2}$