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Abstract
This thesis introduces a comprehensive study for Sturm-Liouville theory of the q,{u019C}-Hahn difference operators in the regular setting. We define a hilbert space of q,{u019C}-square summable functions in terms of Jackson-Nörlund integral. The formulation of the self-adjoint operator and the properties of the eigenvalues and the eigenfunctions are discussed. The construction of green{u2019}s function is developed and study for q,{u019C}-fredholem integral operator is established. Hence, an eigenfunctions expansion theorem is derived and illustrative examples are exhibited. We also introduce a numerical simulations and illustrations. We give some comparisons between trigonometric functions and the q,{u019C}-counterparts. We also test numerically the asymptotic behaviour of the zeros of q,{u019C}-trigonometric functions. The numerical experiments precisely reflects the theoretical results with this respect