Characteristic Equations of Chebyshev Polynomials of Third and Fourth Kinds and Their Generating Matrices

The main goal of the article is to obtain matrix representation for the third and fourth kinds of Chebyshev polynomials by using a tridiagonal matrix. We present a connection between the determinant of the tridiagonal matrix and the third and fourth kinds of Chebyshev polynomials. We also determine the characteristic equations for the third and fourth kinds of the Chebyshev polynomials up to degree three. We also prove some properties relating to matrix representation. We obtain a connection between the second, third kind and fourth kinds of Chebyshev polynomials and matrix power. It elaborates the theorem to validate through examples. The applications of the Chebyshev polynomials is also discussed. The practical application of the Chebyshev polynomials of the third kind in approximation theory is also detailed.