Chapter 4: A Sturm–Liouville Theory for Hahn Difference Operator

This chapter introduces a comprehensive study for Sturm–Liouville theory of the q, ω-Hahn difference operators in the regular setting. We define a Hilbert space of q, ω-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’s function is developed and a study for q, ω-Fredholem integral operator is established. Hence, an eigenfunctions expansion theorem is derived and illustrative examples are exhibited. We also introduce a separate section for numerical simulations and illustrations. We give some comparisons between trigonometric functions and the q and q, ω counterparts. We also test numerically the asymptotic behavior of the zeros of q and q, ω trigonometric functions. The numerical experiments precisely reflect the theoretical results with this respect.