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^k_(i,n)(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^k_(i,n)(x) are orthogonal with respect to weights w_(1,k)(x)=x^k-1/√1-x^2k, w_(2,k)(x)=x^k-1√1-x^2kon the interval [-1,1], where k is positive odd integers. Spectral properties Bachok--Hasham polynomials of the first and second kind Z^k_(i,n)(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. [ABSTRACT FROM AUTHOR]