0F1 - Maple Help

convert/0F1

convert to special functions admitting a 0F1 hypergeometric representation

 Calling Sequence convert(expr, 0F1)

Parameters

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

Description

 • convert/0F1 converts, when possible, hypergeometric, MeijerG, and special functions admitting a 1F1 hypergeometric representation into special functions admitting a 0F1 hypergeometric representation; that is, into one of
 > FunctionAdvisor( 0F1 );
 The 26 functions in the "0F1" class are particular cases of the hypergeometric function and are given by:
 $\left[{\mathrm{AiryAi}}{,}{\mathrm{AiryBi}}{,}{\mathrm{BesselI}}{,}{\mathrm{BesselJ}}{,}{\mathrm{BesselK}}{,}{\mathrm{BesselY}}{,}{\mathrm{HankelH1}}{,}{\mathrm{HankelH2}}{,}{\mathrm{KelvinBei}}{,}{\mathrm{KelvinBer}}{,}{\mathrm{KelvinHei}}{,}{\mathrm{KelvinHer}}{,}{\mathrm{KelvinKei}}{,}{\mathrm{KelvinKer}}{,}{\mathrm{cos}}{,}{\mathrm{cosh}}{,}{\mathrm{cot}}{,}{\mathrm{coth}}{,}{\mathrm{csc}}{,}{\mathrm{csch}}{,}{\mathrm{sec}}{,}{\mathrm{sech}}{,}{\mathrm{sin}}{,}{\mathrm{sinh}}{,}{\mathrm{tan}}{,}{\mathrm{tanh}}\right]$ (1)
 • convert/0F1 accepts as optional arguments all those described in convert/to_special_function.

Examples

 > $\frac{\frac{\frac{1}{\mathrm{\Gamma }\left(1+a\right)}{z}^{a}}{{2}^{a}}\mathrm{WhittakerM}\left(0,a,2Iz\right)}{{\left(2Iz\right)}^{\frac{1}{2}+a}}$
 $\frac{{{z}}^{{a}}{}{\mathrm{WhittakerM}}{}\left({0}{,}{a}{,}{2}{}{I}{}{z}\right)}{{\mathrm{\Gamma }}{}\left({1}{+}{a}\right){}{{2}}^{{a}}{}{\left({2}{}{I}{}{z}\right)}^{\frac{{1}}{{2}}{+}{a}}}$ (2)
 > $\mathrm{convert}\left(,\mathrm{0F1}\right)$
 $\frac{{{z}}^{{a}}{}{\mathrm{BesselI}}{}\left({a}{,}{I}{}{z}\right)}{{\left({I}{}{z}\right)}^{{a}}}$ (3)
 > $\frac{{2}^{-a-1}\mathrm{\pi }{z}^{a}}{\mathrm{sin}\left(\mathrm{\pi }\left(1+a\right)\right)}\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[0\right],\left[-a\right]\right],-\frac{1}{4}{z}^{2}\right)+\frac{{2}^{-1+a}\mathrm{\pi }}{\mathrm{sin}\left(\mathrm{\pi }a\right)}{z}^{-a}\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[0\right],\left[a\right]\right],-\frac{1}{4}{z}^{2}\right)$
 $\frac{{{2}}^{{-}{a}{-}{1}}{}{\mathrm{\pi }}{}{{z}}^{{a}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}{a}\right]\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right)}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}\left({1}{+}{a}\right)\right)}{+}\frac{{{2}}^{{-}{1}{+}{a}}{}{\mathrm{\pi }}{}{{z}}^{{-}{a}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{a}\right]\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right)}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right)}$ (4)
 > $\mathrm{convert}\left(,\mathrm{0F1}\right)$
 $\frac{{{2}}^{{-}{a}{-}{1}}{}{\mathrm{\pi }}{}{\mathrm{BesselI}}{}\left({a}{,}{z}\right){}{{2}}^{{a}}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}\left({1}{+}{a}\right)\right)}{+}\frac{{{2}}^{{-}{1}{+}{a}}{}{\mathrm{\pi }}{}{{z}}^{{-}{a}}{}{\mathrm{BesselI}}{}\left({-}{a}{,}{z}\right){}{{z}}^{{a}}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right){}{{2}}^{{a}}}$ (5)
 > $\mathrm{LaguerreL}\left(-\frac{1}{2}-a,2a,2Iz\right)-\mathrm{WhittakerM}\left(0,a,2Iz\right)$
 ${\mathrm{LaguerreL}}{}\left({-}\frac{{1}}{{2}}{-}{a}{,}{2}{}{a}{,}{2}{}{I}{}{z}\right){-}{\mathrm{WhittakerM}}{}\left({0}{,}{a}{,}{2}{}{I}{}{z}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{0F1}\right)$
 $\frac{{{2}}^{{a}}{}{\mathrm{\Gamma }}{}\left({1}{+}{a}\right){}\left(\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{-}\frac{{1}}{{2}}{-}{a}}\right){}{{ⅇ}}^{{I}{}{z}}{-}{\left({2}{}{I}{}{z}\right)}^{\frac{{1}}{{2}}{+}{a}}\right){}{\mathrm{BesselI}}{}\left({a}{,}{I}{}{z}\right)}{{\left({I}{}{z}\right)}^{{a}}}$ (7)