الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
In this thesis, we suggest new approaches to studying the relation between quantum groups and q_special functions. The _rst approach is by considering a new nonassociative algebra generated by four operators J1; J2; J3 and J4, we call it Dq_algebra. On the Dq_algebra we de_ne an operator M = J1J2_J3J4, which gives us the di_erence equations for all q_hypergeometric polynomials Yn(x), such that x 6= cos(_). In the second approach, we used quantum groups to de_ne q_special functions by using the quantum di_erential operator [@x]q = q@x_q_@x q_q_1 . In this case, q_special functions arise as eigenfunctions of the q_di_erential representation of some types of quantum groups and hence we obtained a di_erential equations for the _rst time in studying the relation between quantum groups and q_special functions. This thesis contains 7 chapters, in chapters one and two we reviewed the basic de_nitions and theorems of q_Special Functions, Lie algebra, Hoph algebra and Quantum groups. In chapter 3, we de_ned a new q_Special function A_ n (x; b; c; q) . This new function is a generalization of the q_Laguerre function and the StieltjesWigert function. Also, limq_!1 A_ n ((1 _ q)x;__; 1; q) gives L(_;_) n (x; q) which is a __modi_cation of the ordinary Laguerre function. In chapter 4, we de_ned a new algebra generated by the di_erence operators Dq and Dq_1 with two analytic functions _(x) and _(x). Also,we de_ned an operator M = J1J2 _ J3J4 s.t. all q_Hypergeometric orthogonal polynomials Yn(x), x 6= cos(_), are eigenfunctions of the operator M with eigenvalues _q[n]q. The choice of _(x) and _(x) depend on the weight function of Yn(x). In chapter 5, we studied various functional equations for q_exponentials and we deduced some identities for q_special functions involving q_commuting variables x and y satisfying xy = qyx + (q _ 1)y2, where 0 <j q j< 1, q 2 C. In chapter 6, we used quantum groups to de_ne q_special functions. The Casimir operators of a variation of SUq(2) and Eq(2) are derived. The proposed q_associated Legendre and q_Bessel functions are the eigenfunctions of the Casimirs. The results di_er from ordinary q_special functions, but this is expected since the q_generalization is not unique. In chapter 7, a basis for an irreducible representation of the quantum algebra Summary 109 Eq(2) is given, consisting of eigenfunctions of the q_di_erential representation of the Casimir operator of the quantum algebra Eq(2).