Bivariate Gončarov polynomials and integer sequences
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Univariate Gončarov polynomials arose from the Gončarov interpolation problem in numerical analysis. They provide a natural basis of polynomials for working with u-parking functions, which are integer sequences whose order statistics are bounded by a given sequence u. In this paper, we study multivariate Gončarov polynomials, which form a basis of solutions for multivariate Gončarov interpolation problem. We present algebraic and analytic properties of multivariate Gončarov polynomials and establish a combinatorial relation with integer sequences. Explicitly, we prove that multivariate Gončarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in ℕ k . It leads to a higher-dimensional generalization of parking functions, for which many enumerative results can be derived from the theory of multivariate Gončarov polynomials.
KeywordsGončarov polynomials interpolation parking functions order statistics
MSC(2010)05A15 05A10 41A05 41A63
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