convert/Legendre - Maple Help

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convert/Legendre

convert special functions admitting 2F1 hypergeometric representation into Legendre functions

 Calling Sequence convert(expr, Legendre)

Parameters

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

Description

 • convert/Legendre converts, when possible, special functions admitting a 2F1 hypergeometric representation into Legendre functions. The Legendre functions are
 The 2 functions in the "Legendre" class are:
 $\left[{\mathrm{LegendreP}}{,}{\mathrm{LegendreQ}}\right]$ (1)

Examples

 > $\mathrm{arccoth}\left(z\right)$
 ${\mathrm{arccoth}}{}\left({z}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 ${\mathrm{LegendreQ}}{}\left({0}{,}\frac{{1}}{{z}}\right){+}\frac{{1}}{{2}}{}\frac{{\mathrm{π}}{}\sqrt{{-}{\left({z}{-}{1}\right)}^{{2}}}}{{z}{-}{1}}$ (3)
 > $\frac{{\left(z+1\right)}^{\frac{1b}{2}}\mathrm{Γ}\left(a+1\right)\mathrm{JacobiP}\left(a,-b,b,z\right)}{{\left(z-1\right)}^{\frac{1b}{2}}\mathrm{Γ}\left(1-b+a\right)}$
 $\frac{{\left({z}{+}{1}\right)}^{\frac{{1}}{{2}}{}{b}}{}{\mathrm{Γ}}{}\left({a}{+}{1}\right){}{\mathrm{JacobiP}}{}\left({a}{,}{-}{b}{,}{b}{,}{z}\right)}{{\left({z}{-}{1}\right)}^{\frac{{1}}{{2}}{}{b}}{}{\mathrm{Γ}}{}\left({1}{-}{b}{+}{a}\right)}$ (4)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 $\frac{{\mathrm{Γ}}{}\left({a}{+}{1}\right){}{\mathrm{binomial}}{}\left({-}{b}{+}{a}{,}{-}{b}\right){}{\mathrm{Γ}}{}\left({-}{b}{+}{1}\right){}{\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right)}{{\mathrm{Γ}}{}\left({1}{-}{b}{+}{a}\right)}$ (5)
 > $\mathrm{MeijerG}\left(\left[\left[-\frac{1a}{2}-\frac{1b}{2},\frac{1}{2}-\frac{1a}{2}-\frac{1b}{2}\right],\left[\right]\right],\left[\left[0\right],\left[-\frac{1}{2}-a\right]\right],-\frac{1}{{z}^{2}}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{-}\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{2}}{}{b}{,}\frac{{1}}{{2}}{-}\frac{{1}}{{2}}{}{a}{-}\frac{{1}}{{2}}{}{b}\right]{,}\left[{}\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}{-}{a}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 $\frac{{2}{}{{z}}^{{1}{+}{a}{+}{b}}{}{\mathrm{LegendreQ}}{}\left({a}{,}{b}{,}{z}\right)}{{{2}}^{{b}}{}{\left({z}{-}{1}\right)}^{\frac{{1}}{{2}}{}{b}}{}{\left({z}{+}{1}\right)}^{\frac{{1}}{{2}}{}{b}}{}{{ⅇ}}^{{I}{}{b}{}{\mathrm{π}}}}$ (7)

When converting to a function class, for example, Legendre, it is possible to request additional conversion rules to be performed. For instance, compare these two different outputs:

 > $\mathrm{GegenbauerC}\left(a,\frac{1}{2},z\right)$
 ${\mathrm{GegenbauerC}}{}\left({a}{,}\frac{{1}}{{2}}{,}{z}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{Legendre}\right)$
 ${\mathrm{LegendreP}}{}\left({a}{,}{z}\right)$ (9)
 > $\mathrm{convert}\left(,\mathrm{Legendre},"raise a"\right)$
 $\frac{\left({3}{+}{2}{}{a}\right){}{z}{}{\mathrm{LegendreP}}{}\left({a}{+}{1}{,}{z}\right)}{{a}{+}{1}}{-}\frac{\left({a}{+}{2}\right){}{\mathrm{LegendreP}}{}\left({a}{+}{2}{,}{z}\right)}{{a}{+}{1}}$ (10)