“Jacobi Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Jacobi Polynomial

释义

Jacobi Polynomial

Also known as the Hypergeometric Polynomials, they occur in the study ofRotation Groups and in the solution to the equations of motion of the symmetrictop. They are solutions to the Jacobi Differential Equation. Plugging

(1)

into the differential equation gives the Recurrence Relation

(2)

for

, 1, ..., where

(3)

Solving the Recurrence Relation gives

(4)

for

. They form a complete orthogonal system in the interval

with respect to the weightingfunction

(5)

and are normalized according to

(6)

where

is a Binomial Coefficient. Jacobi polynomials can also be written

(7)

where

is the Gamma Function and

(8)

Jacobi polynomials are Orthogonal satisfying

(9)

The Coefficient of the term

is given by

(10)

They satisfy the Recurrence Relation



	(11)

where

is the Rising Factorial

(12)

The Derivative is given by

(13)

The Orthogonal Polynomials with Weighting Function on the Closed Interval can be expressed in the form

(14)

(Szegö 1975, p. 58).

Special cases with are

			(15)
			(16)
			(17)
			(18)

Further identities are

(19)

(20)


	(21)

(Szegö 1975, p. 79).

The Kernel Polynomial is


	(22)

(Szegö 1975, p. 71).

The Discriminant is

(23)

(Szegö 1975, p. 143).

For , reduces to a Legendre Polynomial. The Gegenbauer Polynomial

(24)

and Chebyshev Polynomial of the First Kind can also be viewed as special cases of the JacobiPolynomials. In terms of the Hypergeometric Function,

(25)

(26)

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

(27)

where

is the Floor Function, and

			(28)
			(29)
			(30)

If the cases

, ...,

, and

, ...,

are excluded, then the number of zeros of

in the respective intervals are

			(31)
			(32)
			(33)

(Szegö 1975, pp. 144-146).

The first few Polynomials are

			(34)
			(35)
			(36)

where

is a Rising Factorial. See Abramowitz and Stegun (1972, pp. 782-793) and Szegö (1975, Ch. 4) foradditional identities.

See also Chebyshev Polynomial of the First Kind, Gegenbauer Polynomial, Jacobi Function of the SecondKind, Rising Factorial, Zernike Polynomial

References

Abramowitz, M. and Stegun, C. 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.

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.

Szegö, G. ``Jacobi Polynomials.'' Ch. 4 in Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.

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