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ChebyshevU

Chebyshev function of the second kind

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ChebyshevU(n, x)

Parameters

n

-

algebraic expression (the degree)

x

-

algebraic expression

Description

• 

If the first parameter is a non-negative integer, then the ChebyshevU(n, x) function computes the nth Chebyshev polynomial of the second kind evaluated at x.

• 

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

11wtChebyshevUm,tChebyshevUn,tⅆt={0nm12πn=m

• 

Chebyshev polynomials of the second kind satisfy the following recurrence relation:

ChebyshevUn,x=2xChebyshevUn1,xChebyshevUn2,x,for n >= 2

  

where ChebyshevU(0,x) = 1 and ChebyshevU(1,x) = 2*x.

• 

This definition is analytically extended for arbitrary values of the first argument by

ChebyshevUn,x=n+1hypergeomn,n+2,32,1212x

Examples

ChebyshevU3,x

ChebyshevU3,x

(1)

simplify,'ChebyshevU'

8x34x

(2)

ChebyshevU3.2,2.1

86.44386715

(3)

See Also

ChebyshevT

GegenbauerC

HermiteH

JacobiP

LaguerreL

LegendreP

orthopoly[U]

 


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