Bachok-Hasham Polynomials for Solving a Special Class of Singular Integral Equations

In this note, we propose a new class of orthogonal polynomials (named Bachok-Hasham polynomials of the first and second kind for order k, denote it as Z((i,n))(k)(x), i = {1, 2}, which is extension of the Chebyshev polynomials of the first and second kind respectively. It is found that Bachok-Hasham polynomials of first and second kind Z((i,n))(k)(x) are orthogonal with respect to weights w ((1,k)) (x)=-x(k-1)/root 1-x(2k), w((2,k)) (x) = x(k-1) root 1-x(2k) on the interval [-1,1], where k is positive odd integers. Spectral properties Bachok--Hasham polynomials of the first and second kind Z((i,n))(k)(x), i = {1,2} are proved. These properties are used to solve a special class of singular integral equations. Finally, numerical examples and comparison results with other methods are provided to illustrate the effectiveness and accuracy of the proposed method.