Eshkuvatov Z.Alhawamda H.Taib B.M.Ibrahim R.I.2024-05-292024-05-29201897807400000000094243X10.1063/1.50415372-s2.0-85049783031https://www.scopus.com/inward/record.uri?eid=2-s2.0-85049783031&doi=10.1063%2f1.5041537&partnerID=40&md5=bc0ac308ccfbcbfc1861354c951eed71https://oarep.usim.edu.my/handle/123456789/9638In 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).en-USBachok-Hasham polynomialssingular integral equationsSpectral propertiesweightsConference Paper197420006