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. � 2018 Author(s).

In this note, we have developed new classes of one dimensional orthogonal polynomials Zk(i,n)(x),i = {1,2},n = 0,1,2,..., namely extended Chebyshev polynomials (ECPs) of the first and second kinds, which are an extension of the Chebyshev polynomials of the first and second kinds respectively. For non-homogeneous SIEs (bounded and unbounded case) truncated series of the first and second kind of ECPs are used to find approximate solution. It is found that first and second kinds of ECPs Zk(in)(x),i = {1,2} are orthogonal with weights w(1,k)(x)= xk-1/1-x2k and w(2,k)(x) = xk-11-x2k, where k is positive odd integer. Spectral properties of first and second kind of ECPs are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method.

In this note, we propose a new class of orthogonal polynomials (named Bachok–Hasham polynomials H1nk(x)), which is an extension of the Chebyshev polynomials. Eigenfunctions and corresponding eigenvalues are found for the homogeneous second kind of Logarithmic Singular Integral Equations (LogSIEs). For non-homogeneous LogSIEs truncated series of the first kind Bachok–Hasham polynomials are used to find approximate solution. It is found that first kind of Bachok–Hasham polynomials (H1nk(x)) are orthogonal with weight [Formula presented], where k is positive odd integer. Properties of first kind of Bachok–Hasham polynomials are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method. © 2019 THE AUTHORS