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Macdonald Polynomial


The Macdonald polynomials P_lambda(x;q,t) are a two-parameter family of symmetric orthogonal polynomials indexed by partitions. They are triangular with respect to the monomial symmetric functions and orthogonal for the scalar product determined on power sums by

 <p_lambda,p_mu>_(q,t)=delta_(lambdamu)z_lambdaproduct_(i)(1-q^(lambda_i))/(1-t^(lambda_i)).

Here z_lambda=product_(i)i^(m_i)m_i! when lambda has m_i parts equal to i. Specializations include the Schur polynomial, Hall-Littlewood polynomial, and Jack polynomial families.


See also

Jack Polynomial, Partition, Schur Polynomial, Symmetric Polynomial

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References

Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University Press, 1995.

Cite this as:

Weisstein, Eric W. "Macdonald Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MacdonaldPolynomial.html

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