A Comprehensive Study on the Class of q-Appell Polynomials

Abstract

This thesis is aimed to study the q-analogue of the class of so called Appell polynomials from different aspects and using various algebraic as well as analytic approaches. To achieve this aim, not only many new results are found based on a proposed general generating function for all members belonging to the aforementioned family of polynomials, but also various relations between famous members of this family are derived. 2D q-Appell polynomials as the q-Appell polynomials in two variables can be considered as another new achievement of this thesis. In addition to the definition of the class of q-Appell polynomials by means of their generating function, a determinantal representation, for the first time, is proposed for indicating different members of the class of q-Appell polynomials. Moreover, it is shown that how easy some results can be proved by using the new proposed linear algebraic indication and applying basic properties of determinant. In the sequel, this family of q-polynomials are studied also from q-umbral point of view and many interesting results are found based on this algebraic approach. Keywords: q-Appell, q-Calculus, Determinatal, q-Umbral, q-Polynomilas, q-Apostol, q- Bernoulli, q-Euler, q-Genocchi, q-Hermite.

Öz:Bu tez farklı açılardan ve çeşitli cebirsel yanı sıra analitik yaklaşımlar kullanarak q-Appell polinomların sınıfının incelenmesini amaçlanmaktadır. Bu amaca ulaşmak için, yukarıda belirtilen q-Appell polinomlar ailesine ait tüm üyeler üyeleri arasında çeşitli ilişkiler elde edilmektedir. İki değişkenli q-Appell polinomları olarak 2D q-Appell polinomları bu tezin yeni bir başarı olarak kabul edilebilir. Ayrıca, bazı sonuçlar yeni önerilen lineer cebirsel gösterge kullanılarak ve determinantın temel özelliklerini uygulanarak ispat edilebilir. Ayrıca, bu tezde q-polinomların birçok ilginç özellikleri q-umbral açısından da incelenmiştir. Anahtar Kelimeler: q-Appell, q-Matematik, q-Umbral, q-Polynomlar, q-Apostol, q- Bernoulli, q-Euler, q-Genocchi, q-Hermite.