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JacobiP

Jacobi function

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

JacobiP(n, a, b, x)

Parameters

n

-

algebraic expression

a

-

nonrational algebraic expression or rational number greater than -1

b

-

nonrational algebraic expression or rational number greater than -1

x

-

algebraic expression

Description

• 

If the first parameter is a non-negative integer, the JacobiP(n, a, b, x) function computes the nth Jacobi polynomial with parameters a and b evaluated at x.

• 

These polynomials are orthogonal on the interval 1,1 with respect to the weight function wx=1xa1+xb when a and b are greater than -1. They satisfy the following:

11wtJacobiPm,a,b,tJacobiPn,a,b,tⅆt={0nm2a+b+1Γn+a+1Γn+b+12n+a+b+1n!Γn+a+b+1n=m

  

The Jacobi polynomials are undefined for negative integer values of a or b.

• 

The polynomials satisfy the following recurrence relation:

JacobiP0,a,b,x=1

JacobiP1,a,b,x=12a12b+1+12a+12bx

JacobiPn,a,b,x=122n+a+b1a2b2+2n+a+b22n+a+bxJacobiPn1,a,b,xnn+a+b2n+a+b2n+a1n+b12n+a+bJacobiPn2,a,b,xnn+a+b2n+a+b2,for n > 1.

• 

For n not equal to a non-negative integer, the analytic extension of the Jacobi polynomial is given by the following:

JacobiPn,a,b,x=binomiala+n,ahypergeomn,a+b+n+1,a+1,1212x

Examples

JacobiP4,1,34,x

JacobiP4,1,34,x

(1)

simplify,'JacobiP'

1154+1354x+418564x12+488251024x13+38083532768x14

(2)

JacobiP2.2,1,23,0.4

0.1993478307

(3)

See Also

ChebyshevT

ChebyshevU

GAMMA

GegenbauerC

HermiteH

LaguerreL

LegendreP

numtheory[jacobi]

numtheory[legendre]

orthopoly[P]

 


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