الفهرس 
Basic conceptsOnly 14 pages are availabe for public view
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Abstract
The study of methods for obtaining generating functions for special polynomials as the following form:G(x,t)=∑_(n=0)^∞▒〖c_n f_n (x) 〗 t^n \Where, c_n is a function of n that may contain the parameter of the set {f_n (x)} and is independent of x and t. G(x,t) is called generating function of the set {f_n (x)}. For example:The Bessel functions {J_n (x)} satisfy the generating relation exp{x/2 (1-t^(-1))}=∑_(n=-∞)^∞▒〖J_n (x) t^n 〗 and the generating relation of Laguerre polynomials {L_n^α (x)} is < (1-t)^(-1-α) exp{(-xt)/(1t)}=∑_(n=0)^∞▒〖L_n^α (x) 〗 t^n In thesis we state how to arranged and to formulate the generating functions for special functions by two methods: Operational Method. (Rodrigue᾽s formulistic method) of obtaining generating functions for classical polynomials require a suitable transformation of variable by means of which the function under the sign of differentiation in the Rodrigue᾽s formula can be transformed into a function which independent of n (i. e. the degree of the polynomial) because, in general Rodrigue᾽s formulas of most of the classical polynomials contain functions (under the sign of differentiation) which depend on n. group Theoretic Method. (weisner᾽s group-Theoretic method) consist in constructing a partial differential equation from ordinary differential equation satisfied by a particular set of special functions by giving a suitable interpretation to the index of the function and then in finding a non-trivial continuous transformation group which is admitted by the partial differential equations.Thus in a particular set of special functions if we can find two generators say B and C of two one –parameter groups of continuous transformation (C raises and B lowers the index of the function by unity) by means of suitable differential difference relations.These operators B and C are considered as generators of Lie algebra where multiplication of such operators is defined by:[B,C]=BC-CB Known as a Lie product or commutator.Also we can generate two finite operators exp(αB) and exp(βC) from the two operators B and C.This thesis consists of three chapters and finished with bibliography Chapter 1 it is an introductory chapter and contains basic concepts, definition of classical polynomials as well as Laguerre, Hermite, Bessel, charlier and Gegenbauer polynomials which used in subsequent chapters.Chapter 2 in this chapter we study obtaining of generating functions for Laguerre polynomial by the Weisner᾽s group-theotric method which using a suitable replacing to the parameter α, in the differential equation of order two in order to derive Lie element for the Lie algebra under consideration and demonstration the differential operators by Taylor᾽s formula:e^bB F(x)=F(x+b) Chapter 3 in this chapter we study obtaining some unusual generating functions for some special functions like Laguerre, charlier and Gegenbauer polynomials with the help of integral formula of Hermite polynomial:〖 H〗_n (x)= 1/√π ∫_(-∞)^∞▒〖exp(-v^2 ) [2(x+iv)]^n 〗 dv Contents of chapter 3 are involved in the paper published in Al-Azhar Bulletin of Science Magazine.