الفهرس 
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Abstract
The aim of this thesis is to introduce new unified families of polynomials including unified Apostol-Bernoulli, Euler and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities. Also, a new generalization of Apostol Hermite-Genocchi polynomials and some statistical applications of the new family in probability distribution theory and reliability are considered. Moreover, we introduce a new gen- eralized Hermite-Genocchi distribution and its applications in reliability. The thesis is organized as follows : Chapter 1: In this chapter, we introduce a review on some types of im- portant numbers and polynomials such as Hermite polynomials, Bernoulli, Euler and Genocchi. Also, the reliability concepts for discrete life times are considered. In addition, some different goodness of fit statistical tests such as Akaike information criterion, Correct Akaike information criterion and Bayesian information criterion are discussed. Chapter 2: In this chapter, we introduced and investigated the new gen- eralization of the Apostol Hermite-Genocchi polynomials and some basic properties is derived. Also, we give implicit summation formulas for this generalization and consider some statistical applications of the new family in probability distribution theory and reliability. The results of this chapter are contained in a paper with a title ”An extension of Apostol type of Hermite-Genocchi polynomials and their probabilistic representation”. Chapter 3: In this chapter, the two-variable unified family of generalized Apostol-Euler, Bernoulli and Genocchi polynomials is introduced. Also, some important summation formulas and general symmetric identity are derived. The results give an extension of some known summations and identities of generalized Bernoulli, Euler and Genocchi numbers and poly- nomials. Moreover, some relationships between our polynomials and some types of generalized polynomials are also obtained. The results of this chapter are contained in a paper with a title ”Unification of Two-Variable Family of Apostol-Type Polynomi- als”. Chapter 4: In this chapter, a new family of multi-variable Apostol- type polynomials is investigate. This family is related to Apostol-Euler, Apostol-Bernoulli, Apostol-Genocchi and Apostol-laguerre polynomials. Moreover, we derive some implicit summation formulae and general sym- metry identities involving the new numbers and polynomials. The new family of polynomials introduced here, gives many interesting special cases. The results of this chapter are contained in a paper with a title ”The Multi-Variable Unified Family of Generalized Apostol-Type Polynomials”. Chapter 5: In this chapter, we derive a new generalized Hermite-Genocchi distribution. Also, various properties including cumulative distribution function, moments, moment generating function and probability generating function are investigated. Moreover, some reliability studies are presented and the maximum likelihood estimates of the parameters are discussed. Finally, the profiles of the log-likelihood function of parameters of GHD are plotted and the procedure is illustrated by real data set It is shown that the introduced models are more competitive than other models. The results of this chapter are contained in a paper with a title ”New Discrete Lifetime Distribution with Applications to Count Data.