Abstract
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.
Highlights
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Based on the quantum calculus, The family of q-Appell polynomials [3] were introduced by Al-Salam in 1967
Polynomials play an important role in approximation theory
Summary
Let Aq(y) = Aμ,q(y) μ≥0 be a sequence of polynomials such that Aq(0) = Aμ,q(0) μ≥0 ∈ Gq. Recall that Aq(y) is called a q-Appell sequence if one of the following equivalent conditions is satisfied: Dq,y Aμ,q(y) = [μ]q Aμ−1,q(y), μ ∈ N,. 2. The Abelian Group Structure of Aq Let Aq(y), Cq(y) ∈ Aq. The q-binomial convolution of Aq(y) and Cq(y), denoted by (Aq ×q Cq)(y) = (Aq ×q Cq)μ(y) μ≥0 and is defined as (Aq ×q Cq)(y) = Aq(y) ×q Cq(0) = Aq(0) ×q Cq(y) = Aq(0) ×q Cq(0) ×q Iq(y), (11). We will show that Cq(y) ∈ Aq will be the inverse of Aq(y) ∈ Aq. Similar to the method used in Corollary 1, let Cq(0) = (Cμ,q(0))μ≥0 ∈ Gq be the real sequence having a generating function as. Note that Theorem 1 is equivalent to q-Appell polynomials determinantal approach, we state the following: Corollary 2.
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