Let K be a finite field and Q ∈ K[T ] a polynomial of positive degree. A function f on K[T ] is called (completely) Q-additive if f(A + BQ) = f(A) + f(B), where A,B ∈ K[T ] and deg(A) < deg(Q). We prove that the values (f1(A), ..., fd(A)) are asymptotically equidistributed on the (finite) image set {(f1(A), ..., fd(A)) : A ∈ K[T ]} if Qj are pairwise coprime and fj : K[T ] → K[T ] are Qj-additi...

Here expounded is a kind of symbolic operator method by making use of the defined Sheffer-type polynomial sequences and their generalization, which can be used to construct many power series transformation and expansion formulas. The convergence of the expansions are also discussed. AMS Subject Classification: 41A58, 41A80, 65B10, 05A15, 33C45, 39A70.

Let ν be either ω ∈ C \ {0} or q ∈ C \ {0, 1}, and let Dν be the corresponding difference operator defined in the usual way either by Dωp(x) = p(x+ω)−p(x) ω or Dqp(x) = p(qx)−p(x) (q−1)x . Let U and V be two moment regular linear functionals and let {Pn(x)}n≥0 and {Qn(x)}n≥0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete ort...

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the LDU decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition...

We say that the polynomial sequence (Q (λ) n ) is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product 〈p, r〉S = 〈u, p r〉+ λ 〈u,DpDr〉 , where u is a semiclassical linear functional, D is the differential, the difference or the q–difference operator, and λ is a positive constant. In this paper we get algebraic and differential/difference properties ...

We recount previous development of d-fold doubling of orthogonal polynomial sequences and give new results on rational function coefficients, recurrence formulas, continued fractions, Rodrigues’ type formulas, and differential equations, for the general case and, in particular, for the d-fold Hermite orthogonal rational functions.

We introduce an infinite class of polynomial sequences at(n; z) with integer parameter t > 1, which reduce to the well-known Stern (diatomic) sequence when z = 1 and are (0, 1)-polynomials when t > 2. Using these polynomial sequences, we derive two different characterizations of all hyperbinary expansions of an integer n > 1. Furthermore, we study the polynomials at(n; z) as objects in their ow...

The application of Clifford Analysis methods in Combinatorics has some peculiarities due to the use of noncommutative algebras. But it seems natural to expect from here some new results different from those obtained by using approaches based on several complex variable. For instances, the fact that in Clifford Analysis the point-wise multiplication of monogenic functions as well as their compos...

For a hypergeometric series ∑ k f(k, a, b, . . . , c) with parameters a, b, . . . , c, Paule has found a variation of Zeilberger’s algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general problem concerning several similar hypergeometric terms f1(k, a, b, . . . , c), f2(k, a, b, . . . , c), . . ., fm(k, a, b, . . . , c). We present an algorithm ...

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The Volume Conjecture for small angles states that the value of the n-th colored Jones polynomial at eα/n is a sequence of complex numbers that grows subexponentially, for a fixed small complex angle α. In an earlier publication, the authors proved the Volume Conjecture...