GAMMA_related - Maple Help

convert/GAMMA_related

convert special functions in an expression into GAMMA related functions

 Calling Sequence convert(expr, GAMMA_related)

Parameters

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

Description

 • convert/GAMMA_related converts, when possible, the special functions in an expression into GAMMA_related functions. The GAMMA_related transcendental functions are
 The 8 functions in the "GAMMA_related" class are:
 $\left[{\mathrm{Β}}{,}{\mathrm{\Gamma }}{,}{\mathrm{binomial}}{,}{\mathrm{doublefactorial}}{,}{\mathrm{factorial}}{,}{\mathrm{lnGAMMA}}{,}{\mathrm{multinomial}}{,}{\mathrm{pochhammer}}\right]$ (1)

Examples

 > $\mathrm{WhittakerW}\left(-\frac{1}{2}+\frac{1}{2}a,\frac{1}{2}a,z\right){z}^{a}\mathrm{hypergeom}\left(\left[a\right],\left[1+a\right],-z\right)$
 ${\mathrm{WhittakerW}}{}\left({-}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{{a}}{{2}}{,}{z}\right){}{{z}}^{{a}}{}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{1}{+}{a}\right]{,}{-}{z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{GAMMA_related}\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({a}{,}{z}\right){}{{ⅇ}}^{\frac{{z}}{{2}}}{}{a}{}{\mathrm{\Gamma }}{}\left({a}\right)}{{{z}}^{{-}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}}}{-}\frac{{{\mathrm{\Gamma }}{}\left({a}{,}{z}\right)}^{{2}}{}{{ⅇ}}^{\frac{{z}}{{2}}}{}{a}}{{{z}}^{{-}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}}}$ (3)
 > $\frac{{z}^{a}}{\mathrm{exp}\left(z\right)}\mathrm{KummerU}\left(1,a+1,z\right)+z\mathrm{hypergeom}\left(\left[1,1,1,1\right],\left[2,2,2\right],z\right)$
 $\frac{{{z}}^{{a}}{}{\mathrm{KummerU}}{}\left({1}{,}{1}{+}{a}{,}{z}\right)}{{{ⅇ}}^{{z}}}{+}{z}{}{\mathrm{hypergeom}}{}\left(\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}{z}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{GAMMA_related}\right)$
 ${\mathrm{\Gamma }}{}\left({a}{,}{z}\right){+}{z}{}{\mathrm{hypergeom}}{}\left(\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}\left[{2}{,}{2}{,}{2}\right]{,}{z}\right)$ (5)
 > $\mathrm{LaguerreL}\left(-a,a,-z\right)$
 ${\mathrm{LaguerreL}}{}\left({-}{a}{,}{a}{,}{-}{z}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{GAMMA_related}\right)$
 ${-}\frac{\left(\left\{\begin{array}{cc}{1}& {-}{a}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right)}{{\mathrm{\pi }}{}{a}}& {\mathrm{otherwise}}\end{array}\right\\right){}{a}{}{\mathrm{\Gamma }}{}\left({a}{,}{z}\right)}{{{z}}^{{a}}}{+}\frac{\left(\left\{\begin{array}{cc}{1}& {-}{a}{=}{0}\\ \frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right)}{{\mathrm{\pi }}{}{a}}& {\mathrm{otherwise}}\end{array}\right\\right){}{a}{}{\mathrm{\Gamma }}{}\left({a}\right)}{{{z}}^{{a}}}$ (7)