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GegenbauerC

Gegenbauer (ultraspherical) function

 

Calling Sequence

Parameters

Description

Examples

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 2a,1+n,n+2a 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));

GegenbauerCn,a,z={0n::negint1n=0Γn+2ahypergeomn,n+2a,12+a,12z+12Γ1+nΓ2aotherwise

(1)
  

and are orthogonal on the interval 1,1 with respect to the weight function wz=z2+1a12:

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);

∫11wzGegenbauerCm,a,zGegenbauerCn,a,zⅆz={π212aΓn+2an!n+aΓa2n=m0otherwise

(2)
• 

When any of 2a,1+n,n+2a 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);

GegenbauerCn,a,z=2az1+2aGegenbauerCn1,1+a,z+4z21a1+aGegenbauerCn2,a+2,zn+2an

(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);

fz=GegenbauerCa,b,z

(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);

ⅆ2ⅆz2fz=12bzⅆⅆzfzz21+a2b+afzz21

(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:

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assumingn::negint

0

(6)

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assumingn&equals;0

1

(7)

simplifyGegenbauerC3&comma;a&comma;z&comma;&apos;GegenbauerC&apos;

232az2&plus;4z23z1&plus;aa

(8)

Special values with respect to a:

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assuminga::negint

0

(9)

simplifyGegenbauerC2&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assuminga&equals;0

2z21

(10)

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assuminga&equals;0&comma;n::posint

1nGegenbauerCn&comma;a&comma;z

(11)

Special values with respect to z:

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assumingz&equals;0

2n&Gamma;a&plus;12n&pi;&Gamma;a&Gamma;1212n&Gamma;1&plus;n

(12)

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assumingz&equals;1&comma;n::nonnegint

&Gamma;n&plus;2a&Gamma;1&plus;n&Gamma;2a

(13)

simplifyGegenbauerCn&comma;a&comma;z&comma;&apos;GegenbauerC&apos;assumingz&equals;1&comma;n::nonnegint

1n&Gamma;n&plus;2a&Gamma;2an&excl;

(14)

See Also

ChebyshevT

ChebyshevU

GAMMA

HermiteH

JacobiP

LaguerreL

LegendreP

orthopoly[G]

simplify

 


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