We present several new and compact formulas for the modified integral form of Macdonald polynomials, building on “multiline queue” formula polynomials due to Corteel, Mandelshtam, Williams. also introduce a quasisymmetric analogue polynomials. These “quasisymmetric polynomials" refine (symmetric) specialize Schur defined by Haglund, Luoto, Mason, van Willigenburg.

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy”; these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the o...

We give a bijection between N/S/E/W walks and N/S walks that remain on one side of the origin and also a simple derivation of the enumeration of the former, based on Touchard’s identity.

We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with ...

Adomian polynomials are constituted of reduced polynomials and derivatives of nonlinear operator. The reduced polynomials are independent of the form of the nonlinear operator. A recursive algorithm of the reduced polynomials is discovered and its symbolic implementation by the software Mathematica is given. As a result, a new and convenient algorithm for the Adomian polynomials is obtained.

We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection...

Multisymmetric polynomials are the r-fold diagonal invariants of the symmetric group Sn. Elementary multisymmetric polynomials are analogues of the elementary symmetric polynomials, in the multisymmetric setting. In this paper, we give a necessary and sufficient condition on a ring A for the algebra of multisymmetric polynomials with coefficients in A to be generated by the elementary multisymm...