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 [n, a, b, z].

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

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

Plugging

into the Jacobi differential equation gives the recurrence relation

for 👁 nu=0
, 1, ..., where

Solving the recurrence relation gives

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

and are normalized according to

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

where 👁 Gamma(z)
is the gamma function and

Jacobi polynomials are orthogonal polynomials and satisfy

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

They satisfy the recurrence relation

where 👁 (m)_n
is a Pochhammer symbol

The derivative is given by

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

(Szegö 1975, p. 58).

Special cases with 👁 alpha=beta
are

Further identities are

(Szegö 1975, p. 79).

The kernel polynomial is

(Szegö 1975, p. 71).

The polynomial discriminant is

(Szegö 1975, p. 143).

In terms of the hypergeometric function,

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

where 👁 |_x_|
is the floor function, and

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

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

The first few polynomials are

(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, 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 Resource. https://mathworld.wolfram.com/JacobiPolynomial.html

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