Certain results associated with hybrid relatives of the q-Sheffer sequences

The intended objective of this paper is to introduce a new class of the hybrid q-Sheffer polynomials by means of the generating function and series definition. The determinant definition and other striking properties of these polynomials are established. Certain results for the continuous q-Hermite-Appell polynomials are obtained. The graphical depictions are performed for certain members of the hybrid q-Sheffer family. The zeros of these members are also explored using numerical simulations. Finally, the orthogonality condition for the hybrid q-Sheffer polynomials is established.

ON A FAMILY OF (p, q)-HYBRID POLYNOMIALS

In this paper, the class of (p,q)-Bessel-Appell polynomials is introduced. The generating function, series definition and determinant definition of this class are established. Certain members of (p,q)-Bessel-Appell polynomials are considered and some properties of these members are also derived. Further, the class of 2D (p,q)-Bessel-Appell polynomials is introduced by means of the generating function and series definition. In addition, the graphical representations of some members of (p,q)-Bessel-Appell polynomials and 2D (p,q)-Bessel-Appell polynomials are plotted with the help of Matlab.

General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given.

q-Binomial Convolution and Transformations of q-Appell Polynomials

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

Differential and integral equations for Legendre-Laguerre based hybrid polynomials

UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.