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Abstract
The thesis is organized as follows:
Chapter 1: In this chapter we give a review on some applications in quantum mechanics, which were previously studied, such as: Boson operators, Normal ordering, X and D representation, Coherent States, Ordered Boson operators and Algebraic properties of boson operators
Chapter 2 : This chapter is mainly concerned with a review on Stirling numbers of the first and second kinds and their generalization. Also, we introduce q-stirling numbers, and a generalization of ordinary Bell and Stirling numbers which arise in the solution of the general normal ordering problem for a boson string (monomial). Chapter 3 : In this chapter we derive a new family of generalized Stirling numbers of the first kind. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers are derived. Finally, some applications using matrix representation and a Maple program are given.
Chapter 4 : In this chapter we investigate another new family of generalized Stirling numbers of the first kind. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, a recurrence relation and an explicit formula of these numbers are derived. With some applications in Quantum Mechanics. Some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Finally, some applications using a Maple program are given.
In the appendix we give some tables calculated by using Maple for some results obtained throughout the thesis
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