q-Multiple Appell Polynomials

In 1880, Paul Emile Appell introduced a certain kind of sequence which is named Appell polynomials in the literature. Besides the trivial examples, the most famous Appell polynomials are the Hermite, Bernoulli, and Euler polynomials. An interesting generalization of Appell polynomials, namely q-Appell polynomials were introduced by Walled A. Al-Salam in 1967. The multiple Appell polynomials have recently introduced and investigated in 2011 by D.W.Lee. Also, 2 iterated Appell polynomials defined by Subuhi Khan and Nusrat Raza in 2013. The main purpose of this thesis is to define and investigate univariate q-multiple Appell polynomials, bivariate q-multiple Appell polynomials and 2 iterated q-multiple Appell polynomials. This thesis consist of 5 chapters. In Chapter 1, we recalled the main definitions and properties of the Appell polynomials, the 2 iterated Appell polynomials, the multiple Appell polynomials, the q-Calculus, and the q-Appell polynomials. Chapters 2,3,4 and 5 are original. In chapter 2 we define univariate q-multiple Appell polynomials and obtain equivalence theorem and recurrence relations for them. In chapter 3, we introduce bivariate q-multiple Appellpolynomials via the concept of univariate q-multiple Appell polynomials and obtain explicit representation, equivalence theorem, and recurrence relations for them. In chapter 4, we provide some examples for the polynomials that we define in chapters 2 and 3 such as q-multiple power polynomials, bivariate q-multiple Bernoulli polynomials, bivariate q-multiple Euler polynomials, bivariate q-multiple Bernoulli-Euler polynomials, and q-multiple Hermite polynomials. In the last chapter, we define 2-iterated q-multiple Appell polynomials and we show how we can obtain q-analogue of multiple Hermite polynomials from this definition. We further obtain reccurencen relation for 2-iterated q-mutliple Appell polynomials. Keywords: q-Calculus or Quantum Calculus, Appell Polynomials, q-Appell Polynomials, Two Iterated Multiple Appell Polynomials.