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orthopoly

  

G

  

Gegenbauer (ultraspherical) polynomial

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

G(n, a, x)

Parameters

n

-

non-negative integer

a

-

rational number greater than 12 or nonrational algebraic expression

x

-

algebraic expression

Description

• 

The G(n, a, x) function computes the nth ultraspherical (Gegenbauer) polynomial with parameter a evaluated at x.

• 

These polynomials are orthogonal on the interval 1,1 with respect to the weight function wx=x2+1a12. For a0 they satisfy:

−11wtGm,a,tGn,a,tⅆt=0nmπ212aΓn+2an!n+aΓa2n=m

• 

When a&equals;0, G0&comma;0&comma;x&equals;1, and Gn&comma;0&comma;x&equals;2Tn&comma;xn for 0<n, where Tn&comma;x is the nth Chebyshev polynomial of the first kind.

• 

The ultraspherical polynomials satisfy the following recurrence relation, for a0.

G0&comma;a&comma;x&equals;1

G1&comma;a&comma;x&equals;2ax

Gn&comma;a&comma;x&equals;2n&plus;a1xGn1&comma;a&comma;xnn&plus;2a2Gn2&comma;a&comma;xn&comma;for n>1.

Examples

withorthopoly&colon;

G2&comma;5&comma;x

60x25

(1)

G3&comma;2&comma;x

32x312x

(2)

G17&comma;13&comma;23

6498374549594246012157665459056928801

(3)

See Also

GAMMA

GegenbauerC

 


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