Generalized Goncarov polynomials

Abstract

We introduce the sequence of generalized Gončarov polynomials, which is a basis for the solutions to the Gončarov interpolation problem with respect to a delta operator. Explicitly, a generalized Gončarov basis is a sequence $(t_n(x){)}_{n\ge 0}$ of polynomials defined by the biorthogonality relation ${\epsilon }_{z_i}({\mathfrak{d}}^i(t_n(x)))=n!{\delta }_{i,n}$ for all $i,n\in \mathbf{N}$, where $\mathfrak{d}$ is a delta operator, $\mathcal{Z}=(z_i{)}_{i\ge 0}$ a sequence of scalars, and ${\epsilon }_{z_i}$ the evaluation at $z_i$. We present algebraic and analytic properties of generalized Gončarov polynomials and show that such polynomial sequences provide a natural algebraic tool for enumerating combinatorial structures with a linear constraint on their order statistics.

24 pp., 2 figures

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