Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces are investigated. Some results on orthogonal projections and interpolations are established. Explicit expressions describing the dependence of approximation results on the parameters of Jacobi polynomials are given. These results serve as an important tool in the analysis of numerous quadratures and numerical methods for diff...

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of second kind with a weakly singular kernel. We use some function transformation and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi or...

We investigate the interlacing of the zeros of linear combinations pn+ aqm with the zeros of the components pn and qm, where {pn} ∞ n=0 and {qm} ∞ m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials P (α,β) n (x) and P (α,β) m (x) for certain values of α and β with m = n or m = n−1. Numerical counterexamples are given in situations ...

Using the Nikiforov-Uvarov method, the bound state energy eigenvalues and eigenfunctions of the PT -/non-PT -symmetric and non-Hermitian generalized Woods-Saxon (WS) potential with the real and complex-valued energy levels are obtained in terms of the Jacobi polynomials. According to the PT -symmetric quantum mechanics, we exactly solved the time-independent Schrödinger equation with same poten...

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the ...