convert special functions admitting 1F1 or 0F1 hypergeometric representation into Kummer functions - Maple Help

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convert/Kummer - convert special functions admitting 1F1 or 0F1 hypergeometric representation into Kummer functions

 Calling Sequence convert(expr, Kummer)

Parameters

 expr - Maple expression, equation, or a set or list of them

Description

 • convert/Kummer converts, when possible, special functions admitting a 1F1 or 0F1 hypergeometric representation into Kummer functions. The Kummer functions are
 > FunctionAdvisor( Kummer );
 The 2 functions in the "Kummer" class are:
 $\left[{\mathrm{KummerM}}{,}{\mathrm{KummerU}}\right]$ (1)

Examples

 > $\mathrm{AiryAi}\left(z\right)$
 ${\mathrm{AiryAi}}{}\left({z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Kummer}\right)$
 $\frac{\frac{{1}}{{3}}{}\frac{{{3}}^{{1}{/}{3}}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{6}}{,}\frac{{1}}{{3}}{,}\frac{{4}}{{3}}{}{{z}}^{{3}{/}{2}}\right)}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}{-}\frac{{1}}{{2}}{}\frac{{z}{}{{3}}^{{1}{/}{6}}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}{\mathrm{KummerM}}{}\left(\frac{{5}}{{6}}{,}\frac{{5}}{{3}}{,}\frac{{4}}{{3}}{}{{z}}^{{3}{/}{2}}\right)}{{\mathrm{π}}}}{{{ⅇ}}^{\frac{{2}}{{3}}{}{{z}}^{{3}{/}{2}}}}$ (3)
 > $\mathrm{HermiteH}\left(a,z\right)\mathrm{LaguerreL}\left(2,{ⅇ}^{z}\right)$
 ${\mathrm{HermiteH}}{}\left({a}{,}{z}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{{ⅇ}}^{{z}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Kummer}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0<\mathrm{ℜ}\left(z\right)$
 ${{2}}^{{a}}{}\sqrt{{\mathrm{π}}}{}\left(\frac{{\mathrm{KummerM}}{}\left({-}\frac{{1}}{{2}}{}{a}{,}\frac{{1}}{{2}}{,}{{z}}^{{2}}\right)}{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}\right)}{-}\frac{{2}{}{z}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}{,}\frac{{3}}{{2}}{,}{{z}}^{{2}}\right)}{{\mathrm{Γ}}{}\left({-}\frac{{1}}{{2}}{}{a}\right)}\right){}{\mathrm{KummerM}}{}\left({-}{2}{,}{1}{,}{\mathrm{KummerM}}{}\left({1}{,}{1}{,}{z}\right)\right)$ (5)
 > $\frac{\left({ⅇ}^{z}\mathrm{erf}\left({z}^{2}\right)+\mathrm{WhittakerW}\left(-1,\frac{1}{2},z\right){ⅇ}^{\frac{1z}{2}}\right)\mathrm{MeijerG}\left(\left[\left[1-a\right],\left[\right]\right],\left[\left[0,1-b\right],\left[\right]\right],\frac{1}{z}\right)}{\mathrm{BesselK}\left(-3,1-z\right)}$
 $\frac{\left({{ⅇ}}^{{z}}{}{\mathrm{erf}}{}\left({{z}}^{{2}}\right){+}{\mathrm{WhittakerW}}{}\left({-}{1}{,}\frac{{1}}{{2}}{,}{z}\right){}{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}\right){}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{-}{a}\right]{,}\left[{}\right]\right]{,}\left[\left[{0}{,}{1}{-}{b}\right]{,}\left[{}\right]\right]{,}\frac{{1}}{{z}}\right)}{{\mathrm{BesselK}}{}\left({3}{,}{1}{-}{z}\right)}$ (6)
 > $\mathrm{convert}\left(,\mathrm{Kummer}\right)$
 $\frac{\left(\frac{{2}{}{\mathrm{KummerM}}{}\left({1}{,}{1}{,}{z}\right){}{{z}}^{{2}}{}{\mathrm{KummerM}}{}\left(\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{,}{-}{{z}}^{{4}}\right)}{\sqrt{{\mathrm{π}}}}{+}\frac{{\mathrm{KummerU}}{}\left({2}{,}{2}{,}{z}\right){}{z}{}{\mathrm{KummerM}}{}\left({1}{,}{1}{,}\frac{{1}}{{2}}{}{z}\right)}{{{ⅇ}}^{\frac{{1}}{{2}}{}{z}}}\right){}\left({\mathrm{Γ}}{}\left({a}\right){}{\mathrm{Γ}}{}\left({1}{-}{b}\right){}{\mathrm{KummerM}}{}\left({a}{,}{b}{,}\frac{{1}}{{z}}\right){+}\frac{{\mathrm{KummerM}}{}\left({1}{+}{a}{-}{b}{,}{2}{-}{b}{,}\frac{{1}}{{z}}\right){}{\mathrm{Γ}}{}\left({1}{+}{a}{-}{b}\right){}{\mathrm{Γ}}{}\left({-}{1}{+}{b}\right)}{{\left(\frac{{1}}{{z}}\right)}^{{-}{1}{+}{b}}}\right)}{\sqrt{{\mathrm{π}}}{}{\left({2}{-}{2}{}{z}\right)}^{{3}}{}{\mathrm{KummerU}}{}\left(\frac{{7}}{{2}}{,}{7}{,}{2}{-}{2}{}{z}\right){}{{ⅇ}}^{{-}{1}{+}{z}}}$ (7)
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