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 $w(1,k)(x)=xk−11−x2k,w(2,k)(x)=xk−11−x2k$ 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.