# .

In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) $$P_{\lambda}(kt)$$ are orthogonal polynomials in several variables associated to root systems. They were introduced by Heckman and Opdam (1987).

They generalize Jack polynomials when the roots system is of type A, and are limits of Macdonald polynomials $$P_{\lambda}(q, t)$$ as q tends to 1 and (1 − t)/(1 − q) tends to k. Main properties of the Heckman–Opdam polynomials have been detailed by Siddhartha Sahi [1]

References

A new formula for weight multiplicities and characters, Theorem 1.3. about Heckman–Opdam polynomials, Siddhartha Sahi arXiv:math/9802127 [1]

Heckman, G. J.; Opdam, E. M. (1987), "Root systems and hypergeometric functions. I", Compositio Mathematica 64 (3): 329–352, MR 0918416
Heckman, G. J.; Opdam, E. M. (1987b), "Root systems and hypergeometric functions. II", Compositio Mathematica 64 (3): 353–373, MR 0918417
Opdam, E. M. (1988), "Root systems and hypergeometric functions. III", Compositio Mathematica 67 (1): 21–49, MR 0949270
Opdam, E. M. (1988b), "Root systems and hypergeometric functions. IV", Compositio Mathematica 67 (2): 191–209., MR 0951750