Jacobi Polynomials

JACOBI_POLYNOMIAL is a C++ library which evaluates the Jacobi polynomial.

For a given choice of the parameters a and b, both greater than -1, the Jacobi polynomials are a set of polynomials which are pairwise orthogonal with respect to the integral:

        integral (-1<=x<=+1) J(i,a,b,x) J(j,a,b,x) (1-x)^a (1+x)^b dx
That is, this integral is 0 unless i = j. J(i,a,b,x) indicates the Jacobi polynomial of degree i.

The standard Jacobi polynomials can be defined by a three term recurrence formula that is a bit too ugly to quote here.

It is worth noting that the definition of the Jacobi polynomials is general enough that it includes some familiar families as special cases:


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


JACOBI_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C++ library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a C++ library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

GEGENBAUER_POLYNOMIAL, a C++ library which evaluates the Gegenbauer polynomial and associated functions.

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ library which evaluates the shifted Legendre polynomial, with domain [0,1].

LOBATTO_POLYNOMIAL, a C++ library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a C++ library which evaluates a variety of mathematical functions.

TEST_VALUES, a C++ library which supplies test values of various mathematical functions.


  1. Theodore Chihara,
    An Introduction to Orthogonal Polynomials,
    Gordon and Breach, 1978,
    ISBN: 0677041500,
    LC: QA404.5 C44.
  2. Walter Gautschi,
    Orthogonal Polynomials: Computation and Approximation,
    Oxford, 2004,
    ISBN: 0-19-850672-4,
    LC: QA404.5 G3555.
  3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
    NIST Handbook of Mathematical Functions,
    Cambridge University Press, 2010,
    ISBN: 978-0521192255,
    LC: QA331.N57.
  4. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the C++ source codes.

Last modified on 19 April 2012.