Abstract
In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated.
Highlights
During the last two decades, much research work has been done for different members of the family of the q-Appell polynomials and the 2D q-Appell polynomials
By making suitable selections for the functions Ȧq (t) and Äq (t), the members belonging to the family of the twice-iterated 2D q-Appell polynomials Ak,q ( x1, x2 ) can be obtained
Eq ( t ) − 1 the 2I2DqAP reduce to 2DqEBP E Bm,q ( x1, x2 ) and are defined by means of generating functions as follows: 2t eq ( t ) + 1 eq ( t ) − 1
Summary
In order to introduce the twice-iterated 2D q-Appell polynomials (2I2DqAP), we consider two different sets of the 2D q-Appell polynomials Ȧm,q ( x1 , x2 ) and Äm,q ( x1 , x2 ) such that. The twice-iterated 2D q-Appell polynomials Am,q ( x1 , x2 ) satisfy the following operational relations: Dq,x1 Am,q ( x1 , x2 ) = [m]q Am−1,q ( x1 , x2 ),. The twice-iterated 2D q-Appell polynomials Am,q ( x1 , x2 ) satisfy the following linear homogeneous recurrence relation: Am,q (qx , x2 ). After some simplification, by equating the coefficients of like powers of t on both sides of the resulting equation, we arrive at the assertion (52) of Theorem 5. The twice-iterated 2D q-Appell polynomials Am,q ( x1 , x2 ) are the solutions of the following q-difference equations: qm−s s. The proof of the assertions (53) and (54) of Theorem 6 would follow directly upon using the Equations (43) and (44), respectively, in the recurrence relation (45). (Section 3 below), the determinant forms for the 2I2DqAP are established
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