Q-umbral Calculus Research Articles

A note on infinite series whose terms involve truncated degenerate exponentials

The degenerate exponentials are degenerate versions of the ordinary exponential and the truncated degenerate exponentials are obtained from the Taylor expansions of them by truncating the first finitely many terms. The degenerate exponentials play an important role in recent studies on degenerate versions of many special numbers and polynomials, the degenerate gamma function, the degenerate umbral calculus and the degenerate q-umbral calculus. The aim of this note is to consider infinite series whose terms involve truncated degenerate exponentials together with binomial coefficients, the generalized falling factorials and the Stirling numbers of the second kind, and to find either their values or some other expressions of them as finite sums.

Some recurrence formulas for the q-Bernoulli and q-Euler polynomials

The recurrence relations have a very important place for the special polynomials such as q-Appell polynomials. In this paper, we give some recurrence formulas that allow us a better understanding of q-Appell polynomials. We investigate the q-Bernoulli polynomials and q-Euler polynomials, which are q-Appell polynomials, and we obtain their recurrence formulas by using the methods of the q-umbral calculus and the quantum calculus. Our methods include some operators which are quite handy for obtaining relations for the q-Appell polynomials. Especially, some applications of q-derivative operator are used in this work.

Applications of q-Umbral Calculus to Modified Apostol Type q-Bernoulli Polynomials

This article aims to identify the generating function of modified Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modified Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modified q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.

A novel approach for obtaining new identities for the lambda extension of q-Euler polynomials arising from the q-umbral calculus

A novel approach for obtaining new identities for the lambda extension of q-Euler polynomials arising from the q-umbral calculus

Some Hermite base polynomials on q-umbral algebra

The aim of this paper is to investigate the q-Hermite type polynomials by using umbral calculus methods. Using this method, we derive new type polynomials which are related to the q-Bernoulli polynomials and the q-Hermite type polynomials. Furthermore, we also derive some new identities of those polynomials which are derived from q-umbral calculus.

Q-Bernoulli polynomials and q-umbral calculus

In this paper, we investigate some properties of q-Bernoulli polynomials arising from q-umbral calculus. We find a formula for expressing any polynomial as a linear combination of q-Bernoulli polynomials with explicit coefficients. Also, we establish some connections between q-Bernoulli polynomials and higher-order q-Bernoulli polynomials.

Some identities of q-Euler polynomials arising from q-umbral calculus

Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (see (Araci et al. in J. Number Theory 133(10):3348-3361, 2013)). In this paper, we develop the new method of q-umbral calculus due to Roman, and we study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.

Extended finite operator calculus—an example of algebraization of analysis

Abstract “A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota—Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form—a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota—Mullin or equivalently—of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]).

Discreteq-derivatives and symmetries ofq-difference equations

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation

More on the umbral calculus, with emphasis on the q-umbral calculus

The interrelationship between distinct umbral calculi is studied. These ideas are applied in particular to q -umbral calculus, which shows how Andrews' q -theory relates to the q -theory discussed in the previous paper by the author. The q -Hermite polynomials and basic hypergeometric series are briefly discussed.

A q-umbral calculus

An algebraic setting for the Roman-Rota umbral calculus is introduced. It is shown how many of the umbral calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbral calculus. Our umbral calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included.