hypergeom - Maple Help

convert/hypergeom

convert functions in an expression to hypergeometric form

 Calling Sequence convert( expr, hypergeom, opt_1, opt_2, $...$ )

Parameters

 expr - expression opt_i - optional arguments, see convert/to_special_function

Description

 • Converts the functions found in expr, including sum and Sum, to their hypergeometric forms when possible.
 • For sums, no attempt is made to ensure that the resulting hypergeometric function is convergent or terminates.
 • convert/hypergeom reacts to the setting of the environment variable _EnvFormal (for more information, see sum/details). When that variable is set to $\mathrm{true}$, the conversion will be attempted regardless of whether the original sum may be divergent or not.

Examples

 > ${\sum }_{k=1}^{\mathrm{∞}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{1}{k\mathrm{binomial}\left(k+n,k\right)}$
 ${\sum }_{{k}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{1}}{{k}{}\left(\genfrac{}{}{0}{}{{k}{+}{n}}{{k}}\right)}$ (1)
 > $\mathrm{convert}\left(,\mathrm{hypergeom}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}0
 $\frac{{1}}{{n}}$ (2)
 > $\mathrm{BesselJ}\left(a,z\right)$
 ${\mathrm{BesselJ}}{}\left({a}{,}{z}\right)$ (3)
 > $=\mathrm{convert}\left(,\mathrm{hypergeom}\right)$
 ${\mathrm{BesselJ}}{}\left({a}{,}{z}\right){=}\frac{{{z}}^{{a}}{}{\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[{a}{+}{1}\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right)}{{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{{2}}^{{a}}}$ (4)
 > $\mathrm{LegendreP}\left(a,b,z\right)$
 ${\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right)$ (5)
 > $=\mathrm{convert}\left(,\mathrm{hypergeom}\right)$
 ${\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{\left({z}{+}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{a}{,}{a}{+}{1}\right]{,}\left[{1}{-}{b}\right]{,}\frac{{1}}{{2}}{-}\frac{{z}}{{2}}\right)}{{\left({z}{-}{1}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{\Gamma }}{}\left({1}{-}{b}\right)}$ (6)
 > $\mathrm{KummerU}\left(a,b,z\right)$
 ${\mathrm{KummerU}}{}\left({a}{,}{b}{,}{z}\right)$ (7)
 > $=\mathrm{convert}\left(,\mathrm{hypergeom}\right)$
 ${\mathrm{KummerU}}{}\left({a}{,}{b}{,}{z}\right){=}\frac{{\mathrm{\Gamma }}{}\left({-}{1}{+}{b}\right){}{\mathrm{hypergeom}}{}\left(\left[{a}{-}{b}{+}{1}\right]{,}\left[{2}{-}{b}\right]{,}{z}\right)}{{{z}}^{{-}{1}{+}{b}}{}{\mathrm{\Gamma }}{}\left({a}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left({1}{-}{b}\right){}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{-}{b}{+}{1}\right)}$ (8)

For negative a we have

 > $=\mathrm{convert}\left(,\mathrm{hypergeom}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{negint}$
 ${\mathrm{KummerU}}{}\left({a}{,}{b}{,}{z}\right){=}{\mathrm{pochhammer}}{}\left({a}{-}{b}{+}{1}{,}{-}{a}\right){}{\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{b}\right]{,}{z}\right)$ (9)

Elementary functions are not converted by default (see convert/to_special_function)

 > $\mathrm{convert}\left(\mathrm{sin}\left(z\right),\mathrm{hypergeom}\right)$
 ${\mathrm{sin}}{}\left({z}\right)$ (10)

To convert them use either of the optional arguments: include=elementary, or include=all

 > $\mathrm{sin}\left(z\right)$
 ${\mathrm{sin}}{}\left({z}\right)$ (11)
 > $=\mathrm{convert}\left(,\mathrm{hypergeom},\mathrm{include}=\mathrm{all}\right)$
 ${\mathrm{sin}}{}\left({z}\right){=}{z}{}{\mathrm{hypergeom}}{}\left(\left[\right]{,}\left[\frac{{3}}{{2}}\right]{,}{-}\frac{{{z}}^{{2}}}{{4}}\right)$ (12)

Parametric or divergents sums may not be converted by default. Set _EnvFormal to $\mathrm{true}$, or use appropriate assumptions, to obtain the desired conversion:

 > $S≔{\sum }_{n=1}^{\mathrm{∞}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{x}^{n}}{n}$
 ${S}{≔}{\sum }_{{n}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{{x}}^{{n}}}{{n}}$ (13)
 > $\mathrm{convert}\left(S,\mathrm{hypergeom}\right)$
 ${\sum }_{{n}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{{x}}^{{n}}}{{n}}$ (14)
 > $\mathrm{convert}\left(S,\mathrm{hypergeom}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left|x\right|<1$
 ${-}{\mathrm{ln}}{}\left({1}{-}{x}\right)$ (15)
 > $\mathrm{_EnvFormal}≔\mathrm{true}$
 ${\mathrm{_EnvFormal}}{≔}{\mathrm{true}}$ (16)
 > $\mathrm{convert}\left(S,\mathrm{hypergeom}\right)$
 ${-}{\mathrm{ln}}{}\left({1}{-}{x}\right)$ (17)