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Jacobi Polynomial

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The Jacobi polynomials, also known as hypergeometric polynomials, occur in the study of rotation groups and in the solution to the equations of motion of the symmetric top. They are solutions to the Jacobi differential equation, and give some other special named polynomials as special cases. They are implemented in the Wolfram Language as JacobiP[n, a, b, z].

For alpha=beta=0, P_n^((0,0))(x) reduces to a Legendre polynomial. The Gegenbauer polynomial

G_n(p,q,x)=(n!Gamma(n+p))/(Gamma(2n+p))P_n^((p-q,q-1))(2x-1)
(1)

and Chebyshev polynomial of the first kind can also be viewed as special cases of the Jacobi polynomials.

Plugging

y=sum_(nu=0)^inftya_nu(x-1)^nu
(2)

into the Jacobi differential equation gives the recurrence relation

[gamma-nu(nu+alpha+beta+1)]a_nu-2(nu+1)(nu+alpha+1)a_(nu+1)=0
(3)

for nu=0, 1, ..., where

gamma=n(n+alpha+beta+1).
(4)

Solving the recurrence relation gives

P_n^((alpha,beta))(x)=((-1)^n)/(2^nn!)(1-x)^(-alpha)(1+x)^(-beta)(d^n)/(dx^n)[(1-x)^(alpha+n)(1+x)^(beta+n)]
(5)

for alpha,beta>-1. They form a complete orthogonal system in the interval [-1,1] with respect to the weighting function

w_n(x)=(1-x)^alpha(1+x)^beta,
(6)

and are normalized according to

P_n^((alpha,beta))(1)=(n+alpha; n),
(7)

where (n; k) is a binomial coefficient. Jacobi polynomials can also be written

P_n^((alpha,beta))=(Gamma(2n+alpha+beta+1))/(n!Gamma(n+alpha+beta+1))G_n(alpha+beta+1,beta+1,1/2(x+1)),
(8)

where Gamma(z) is the gamma function and

G_n(p,q,x)=(n!Gamma(n+p))/(Gamma(2n+p))P_n^((p-q,q-1))(2x-1).
(9)

Jacobi polynomials are orthogonal polynomials and satisfy

int_(-1)^1P_m^((alpha,beta))P_n^((alpha,beta))(1-x)^alpha(1+x)^betadx =(2^(alpha+beta+1))/(2n+alpha+beta+1)(Gamma(n+alpha+1)Gamma(n+beta+1))/(n!Gamma(n+alpha+beta+1))delta_(mn).
(10)

The coefficient of the term x^n in P_n^((alpha,beta))(x) is given by

A_n=(Gamma(2n+alpha+beta+1))/(2^nn!Gamma(n+alpha+beta+1)).
(11)

They satisfy the recurrence relation

2(n+1)(n+alpha+beta+1)(2n+alpha+beta)P_(n+1)^((alpha,beta))(x) =[(2n+alpha+beta+1)(alpha^2-beta^2)+(2n+alpha+beta)_3x]P_n^((alpha,beta))(x)-2(n+alpha)(n+beta)(2n+alpha+beta+2)P_(n-1)^((alpha,beta))(x),
(12)

where (m)_n is a Pochhammer symbol

(m)_n=m(m+1)...(m+n-1)=((m+n-1)!)/((m-1)!).
(13)

The derivative is given by

d/(dx)[P_n^((alpha,beta))(x)]=1/2(n+alpha+beta+1)P_(n-1)^((alpha+1,beta+1))(x).
(14)

The orthogonal polynomials with weighting function (b-x)^alpha(x-a)^beta on the closed interval [a,b] can be expressed in the form

[const]P_n^((alpha,beta))(2(x-a)/(b-a)-1)
(15)

(Szegö 1975, p. 58).

Special cases with alpha=beta are

P_(2nu)^((alpha,alpha))(x)=(Gamma(2nu+alpha+1)Gamma(nu+1))/(Gamma(nu+alpha+1)Gamma(2nu+1))P_nu^((alpha,-1/2))(2x^2-1)
(16)
=(-1)^nu(Gamma(2nu+alpha+1)Gamma(nu+1))/(Gamma(nu+alpha+1)Gamma(2nu+1))P_nu^((-1/2,alpha))(1-2x^2)
(17)
P_(2nu+1)^((alpha,alpha))(x)=(Gamma(2nu+alpha+2)Gamma(nu+1))/(Gamma(nu+alpha+1)Gamma(2nu+2))xP_nu^((alpha,1/2))(2x^2-1)
(18)
=(-1)^nu(Gamma(2nu+alpha+2)Gamma(nu+1))/(Gamma(nu+alpha+1)Gamma(2nu+2))xP_nu^((1/2,alpha))(1-2x^2).
(19)

Further identities are

P_n^((alpha+1,beta))(x)=2/(2n+alpha+beta+2)((n+alpha+1)P_n^((alpha,beta))-(n+1)P_(n+1)^((alpha,beta))(x))/(1-x)
(20)
P_n^((alpha,beta+1))(x)=2/(2n+alpha+beta+2)((n+beta+1)P_n^((alpha,beta))(x)+(n+1)P_(n+1)^((alpha,beta))(x))/(1+x)
(21)
sum_(nu=0)^n(2nu+alpha+beta+1)/(2^(alpha+beta+1))(Gamma(nu+1)Gamma(nu+alpha+beta+1))/(Gamma(nu+alpha+1)Gamma(nu+beta+1))P_nu^((alpha,beta))(x)Q_nu^((alpha,beta))(y) =1/2((y-1)^(-alpha)(y+1)^(-beta))/(y-x)+(2^(-alpha-beta))/(2n+alpha+beta+2)(Gamma(n+2)Gamma(n+alpha+beta+2))/(Gamma(n+alpha+1)Gamma(n+beta+1))(P_(n+1)^((alpha,beta))(x)Q_n^((alpha,beta))(y)-P_n^((alpha,beta))(x)Q_(n+1)^(alpha,beta)(y))/(x-y)
(22)

(Szegö 1975, p. 79).

The kernel polynomial is

K_n^((alpha,beta))(x,y)=(2^(-alpha-beta))/(2n+alpha+beta+2)(Gamma(n+2)Gamma(n+alpha+beta+2))/(Gamma(n+alpha+1)Gamma(n+beta+1))(P_(n+1)^((alpha,beta))(x)P_n^((alpha,beta))(y)-P_n^((alpha,beta))(x)P_(n+1)^((alpha,beta))(y))/(x-y)
(23)

(Szegö 1975, p. 71).

The polynomial discriminant is

D_n^((alpha,beta))=2^(-n(n-1))product_(nu=1)^nnu^(nu-2n+2)(nu+alpha)^(nu-1)(nu+beta)^(nu-1) ×(n+nu+alpha+beta)^(n-nu)
(24)

(Szegö 1975, p. 143).

In terms of the hypergeometric function,

P_n^((alpha,beta))(x)=(n+alpha; n)_2F_1(-n,n+alpha+beta+1;alpha+1;1/2(1-x))
(25)
=((alpha+1)_n)/(n!)_2F_1(-n,n+alpha+beta+1;alpha+1;1/2(1-x))
(26)
=(n+alpha; n)((x+1)/2)^n_2F_1(-n,-n-beta;alpha+1;(x-1)/(x+1)),
(27)

where (alpha)_n is the Pochhammer symbol (Abramowitz and Stegun 1972, p. 561; Koekoek and Swarttouw 1998).

Let N_1 be the number of zeros in x in (-1,1), N_2 the number of zeros in x in (-infty,-1), and N_3 the number of zeros in x in (1,infty). Define Klein's symbol

E(u)={0 if u<=0; |_u_| if u positive and nonintegral; u-1 if u=1, 2, ...,
(28)

where |_x_| is the floor function, and

X(alpha,beta)=E[1/2(|2n+alpha+beta+1|-|alpha|-|beta|+1)]
(29)
Y(alpha,beta)=E[1/2(-|2n+alpha+beta+1|+|alpha|-|beta|+1)]
(30)
Z(alpha,beta)=E[1/2(-|2n+alpha+beta+1|-|alpha|+|beta|+1)].
(31)

If the cases alpha=-1, -2, ..., -n, beta=-1, -2, ..., -n, and n+alpha+beta=-1, -2, ..., -n are excluded, then the number of zeros of P_n^((alpha,beta)) in the respective intervals are

N_1(alpha,beta)={2|_1/2(X+1)_| for (-1)^n(n+alpha; n)(n+beta; n)>0; 2|_1/2X_|+1 for (-1)^n(n+alpha; n)(n+beta; n)<0
(32)
N_2(alpha,beta)={2|_1/2(Y+1)_| for (2n+alpha+beta; n)(n+beta; n)>0; 2|_1/2Y_|+1 for (2n+alpha+beta; n)(n+beta; n)<0
(33)
N_3(alpha,beta)={2|_1/2(Z+1)_| for (2n+alpha+beta; n)(n+alpha; n)>0; 2|_1/2Z_|+1 for (2n+alpha+beta; n)(n+alpha; n)<0
(34)

(Szegö 1975, pp. 144-146), where |_x_| is again the floor function.

The first few polynomials are

P_0^((alpha,beta))(x)=1
(35)
P_1^((alpha,beta))(x)=1/2[2(alpha+1)+(alpha+beta+2)(x-1)]
(36)
P_2^((alpha,beta))(x)=1/8[4(alpha+1)(alpha+2)+4(alpha+beta+3)(alpha+2)(x-1)+(alpha+beta+3)(alpha+beta+4)(x-1)^2]
(37)

(Abramowitz and Stegun 1972, p. 793).

See Abramowitz and Stegun (1972, pp. 782-793) and Szegö (1975, Ch. 4) for additional identities.

See also

Chebyshev Polynomial of the First Kind, Gegenbauer Polynomial, Jacobi Function of the Second Kind, Multivariate Jacobi Polynomial, Rising Factorial, Zernike Polynomial

Related Wolfram sites

http://functions.wolfram.com/Polynomials/JacobiP/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.Andrews, G. E.; Askey, R.; and Roy, R. "Jacobi Polynomials and Gram Determinants" and "Generating Functions for Jacobi Polynomials." §6.3 and 6.4 in Special Functions. Cambridge, England: Cambridge University Press, pp. 293-306, 1999.Iyanaga, S. and Kawada, Y. (Eds.). "Jacobi Polynomials." Appendix A, Table 20.V in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980.Koekoek, R. and Swarttouw, R. F. "Jacobi." §1.8 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 38-44, 1998.Roman, S. "The Theory of the Umbral Calculus I." J. Math. Anal. Appl. 87, 58-115, 1982.Szegö, G. "Jacobi Polynomials." Ch. 4 in Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.

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Jacobi Polynomial

Cite this as:

Weisstein, Eric W. "Jacobi Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiPolynomial.html

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