Jacobi Polynomial

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 , reduces to a Legendre polynomial. The Gegenbauer polynomial

(1)

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

Plugging

(2)

into the Jacobi differential equation gives the recurrence relation

(3)

for , 1, ..., where

(4)

Solving the recurrence relation gives

(5)

for . They form a complete orthogonal system in the interval with respect to the weighting function

(6)

and are normalized according to

(7)

where is a binomial coefficient. Jacobi polynomials can also be written

(8)

where is the gamma function and

(9)

Jacobi polynomials are orthogonal polynomials and satisfy

(10)

The coefficient of the term in is given by

(11)

They satisfy the recurrence relation

(12)

where is a Pochhammer symbol

(13)

The derivative is given by

(14)

The orthogonal polynomials with weighting function on the closed interval can be expressed in the form

(15)

(Szegö 1975, p. 58).

Special cases with are

			(16)
			(17)
			(18)
			(19)

Further identities are

			(20)
			(21)

(22)

(Szegö 1975, p. 79).

The kernel polynomial is

(23)

(Szegö 1975, p. 71).

The polynomial discriminant is

(24)

(Szegö 1975, p. 143).

In terms of the hypergeometric function,

			(25)
			(26)
			(27)

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

Let be the number of zeros in , the number of zeros in , and the number of zeros in . Define Klein's symbol

(28)

where is the floor function, and

			(29)
			(30)
			(31)

If the cases , , ..., , , , ..., , and , , ..., are excluded, then the number of zeros of in the respective intervals are

			(32)
			(33)
			(34)

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

The first few polynomials are

			(35)
			(36)
			(37)

(Abramowitz and Stegun 1972, p. 793).

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

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