Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials

Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

Motzkin Paths, Motzkin Polynomials and Recurrence Relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial “counting” all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials)...

Linear Recurrence Relations for Graph Polynomials

A sequence of graphs Gn is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let Gn be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coe...

Determinantal Expression and Recursion for Jack Polynomials

We give matrices of which their determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of this matrix is a list of monomial functions, the entries of the sub-diagonal are of the form −(rα + s), with r and s ∈ +, the entries above the sub-diagonal are nonnegative integers, and below all entries are 0. The quasi-triangular nature of this matrix gives a recursio...

Asymptotics of Orthogonal Polynomials via Recurrence Relations