GegenbauerC - Maple Help

GegenbauerC

Gegenbauer (ultraspherical) function

 Calling Sequence GegenbauerC(n, a, x)

Parameters

 n - algebraic expression a - algebraic expression x - algebraic expression

Description

 • The GegenbauerC(n, a, x) function computes the nth Gegenbauer polynomial - see Abramowitz and Stegun, Handbook of Mathematical Functions, Chap. 22.
 • When all of $\left\{2a,1+n,n+2a\right\}$ are not a negative integer or zero, the Gegenbauer polynomials satisfy:
 > GegenbauerC(n,a,z) = 'piecewise'(n::negint,0, n=0, 1,convert(GegenbauerC(n,a,z),hypergeom));
 ${\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{z}\right){=}\left\{\begin{array}{cc}{0}& {n}{::}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{negint}}\right]\right)\\ {1}& {n}{=}{0}\\ \frac{{\mathrm{\Gamma }}{}\left({n}{+}{2}{}{a}\right){}{\mathrm{hypergeom}}{}\left(\left[{-}{n}{,}{n}{+}{2}{}{a}\right]{,}\left[\frac{{1}}{{2}}{+}{a}\right]{,}{-}\frac{{z}}{{2}}{+}\frac{{1}}{{2}}\right)}{{\mathrm{\Gamma }}{}\left({1}{+}{n}\right){}{\mathrm{\Gamma }}{}\left({2}{}{a}\right)}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 and are orthogonal on the interval $\left[-1,1\right]$ with respect to the weight function $w\left(z\right)={\left(-{z}^{2}+1\right)}^{a-\frac{1}{2}}$:
 > Int(w(z)* GegenbauerC(m, a, z) * GegenbauerC(n, a, z), z=-1..1) = 'piecewise'(n=m, Pi*2^(1-2*a)*GAMMA(n+2*a)/(n!*(n+a)*GAMMA(a)^2),0);
 ${{\int }}_{{-1}}^{{1}}{w}{}\left({z}\right){}{\mathrm{GegenbauerC}}{}\left({m}{,}{a}{,}{z}\right){}{\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{z}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}{=}\left\{\begin{array}{cc}\frac{{\mathrm{\pi }}{}{{2}}^{{1}{-}{2}{}{a}}{}{\mathrm{\Gamma }}{}\left({n}{+}{2}{}{a}\right)}{{n}{!}{}\left({n}{+}{a}\right){}{{\mathrm{\Gamma }}{}\left({a}\right)}^{{2}}}& {n}{=}{m}\\ {0}& {\mathrm{otherwise}}\end{array}\right\$ (2)
 • When any of $\left\{2a,1+n,n+2a\right\}$ is a negative integer or zero, the Gegenbauer polynomials are computed using the following identity:
 > GegenbauerC(n,a,z) = (2*a*z*(1+2*a)*GegenbauerC(n-1,1+a,z) + 4*(-1+z^2)*a*(1+a)*GegenbauerC(n-2,a+2,z)) / ((n+2*a)*n);
 ${\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{z}\right){=}\frac{{2}{}{a}{}{z}{}\left({1}{+}{2}{}{a}\right){}{\mathrm{GegenbauerC}}{}\left({n}{-}{1}{,}{1}{+}{a}{,}{z}\right){+}{4}{}\left({{z}}^{{2}}{-}{1}\right){}{a}{}\left({1}{+}{a}\right){}{\mathrm{GegenbauerC}}{}\left({n}{-}{2}{,}{a}{+}{2}{,}{z}\right)}{\left({n}{+}{2}{}{a}\right){}{n}}$ (3)
 which in turn can be derived from the differential equation with respect to z satisfied by this function:
 > f(z) = GegenbauerC(a,b,z);
 ${f}{}\left({z}\right){=}{\mathrm{GegenbauerC}}{}\left({a}{,}{b}{,}{z}\right)$ (4)
 > diff(f(z),z,z) = (-1-2*b)*z/(-1+z^2)*diff(f(z),z)+a*(2*b+a)/(-1+z^2)*f(z);
 $\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{z}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right){=}\frac{\left({-}{1}{-}{2}{}{b}\right){}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({z}\right)\right)}{{{z}}^{{2}}{-}{1}}{+}\frac{{a}{}\left({2}{}{b}{+}{a}\right){}{f}{}\left({z}\right)}{{{z}}^{{2}}{-}{1}}$ (5)
 • For n::posint and n > 1 and a <> 0, the Gegenbauer polynomials satisfy the following recurrence relations:
 > GegenbauerC(0,a,z) = 1:
 > GegenbauerC(1,a,z) = 2*a*z:
 > GegenbauerC(n,a,z) = 2*(n+a-1)/n*z*GegenbauerC(n-1,a,z) - (n+2*a-2)/n*GegenbauerC(n-2,a,z):
 and for a = 0, they are related to the ChebyshevT polynomials:
 > GegenbauerC(n,0,z) = 2/n*ChebyshevT(n,z):

Examples

Special values with respect to n:

 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n::\mathrm{negint}$
 ${0}$ (6)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n=0$
 ${1}$ (7)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(3,a,z\right),'\mathrm{GegenbauerC}'\right)$
 $\frac{{4}{}\left({1}{+}{a}\right){}{a}{}\left({{z}}^{{2}}{}{a}{+}{2}{}{{z}}^{{2}}{-}\frac{{3}}{{2}}\right){}{z}}{{3}}$ (8)

Special values with respect to a:

 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a::\mathrm{negint}$
 ${0}$ (9)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(2,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a=0$
 ${2}{}{{z}}^{{2}}{-}{1}$ (10)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,-z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}a=0,n::\mathrm{posint}$
 ${\left({-1}\right)}^{{n}}{}{\mathrm{GegenbauerC}}{}\left({n}{,}{a}{,}{z}\right)$ (11)

Special values with respect to z:

 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}z=0$
 $\frac{{{2}}^{{n}}{}{\mathrm{\Gamma }}{}\left({a}{+}\frac{{n}}{{2}}\right){}\sqrt{{\mathrm{\pi }}}}{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{-}\frac{{n}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({1}{+}{n}\right)}$ (12)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}z=1,n::\mathrm{nonnegint}$
 $\frac{{\mathrm{\Gamma }}{}\left({n}{+}{2}{}{a}\right)}{{\mathrm{\Gamma }}{}\left({1}{+}{n}\right){}{\mathrm{\Gamma }}{}\left({2}{}{a}\right)}$ (13)
 > $\mathrm{simplify}\left(\mathrm{GegenbauerC}\left(n,a,z\right),'\mathrm{GegenbauerC}'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}z=-1,n::\mathrm{nonnegint}$
 $\frac{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}{2}{}{a}\right)}{{\mathrm{\Gamma }}{}\left({2}{}{a}\right){}{n}{!}}$ (14)