Chebyshev function of the first kind - Maple Help

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ChebyshevT - Chebyshev function of the first kind

Calling Sequence

ChebyshevT(n, x)

Parameters

n

-

algebraic expression (the degree)

x

-

algebraic expression

Description

• 

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

• 

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

11wtChebyshevTm,tChebyshevTn,tⅆt=0nmπn=m=012πn=m0

• 

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

ChebyshevTn,x=2xChebyshevTn1,xChebyshevTn2,x,for n >= 2

  

where ChebyshevT(0,x) = 1 and ChebyshevT(1,x) = x.

• 

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

ChebyshevTa,x=hypergeoma,a,12,1212x

Examples

ChebyshevT3,x

ChebyshevT3,x

(1)

simplify,'ChebyshevT'

4x33x

(2)

ChebyshevT2.2,0.5

0.6691306064

(3)

ChebyshevT13,x

ChebyshevT13,x

(4)

series,'ChebyshevT'

cos13arccosx

(5)

ⅆⅆxChebyshevT1,x

xChebyshevT1,xx2+1+ChebyshevT0,xx2+1

(6)

See Also

ChebyshevU, GegenbauerC, HermiteH, JacobiP, LaguerreL, LambertW, LegendreP, numapprox[chebyshev], orthopoly[T]


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