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Chebyshev Polynomials in the Numerical Solution of Differential Equations By A. G. Morris and T. S. Homer Abstract. Amongst satisfactory techniques for the numerical solution of differential equations, the use of Chebyshev series is often avoided because of the tedious nature of the calculations. A systematic application of the Chebyshev method The first few Chebyshev polynomials are as follows: T 0(z) = 1, Ti(z) = z, T 2{z) = 2 2 2-l, T 3(z) = iz 3-3z. In general, the nth Chebyshev polynomial has leading coefficient 2n_1. The nth Chebyshev polynomial T n has n real zeros. The next proposition gives more specific information: 483 Numerical Analysis for Applied Science 'Chebyshev polynomials of the third and fourth kinds'; in addition we cover the shifted polynomials T ∗ n (x), Un (x), Vn∗(x)andW∗ n (x). We shall, however, only make a passing reference to 'Chebyshev's polynomial of a discrete vari-able', referred to for example in Erd´elyi et al. (1953, Section 10.23), since this last polynomial has somewhat different properties from the Applications to generalized Chebyshev polynomials 13 5. Positivity coming from quasi-composition factors 17 6. Positivit,y homogeneous tubes and bases 20 Acknowledgements 23 References 23 1. Introduction The aim of this article is triple. Firstly, it intends to provide results con-cerning the positivity of certain bases in acyclic cluster algebras constructed in [SZ04, CZ06, Cer09, Dup08 CHEBYSHEV POLYNOMIALS 557 1 1 1 1 Figure C.1 The Chebyshev polynomial T 8(z). Theorem C.2 (minimax property of Chebyshev polynomials). Of all polynomials phaving degree exactly nand leading coefficient 2n 1, T npossesses the smallest value of kpk 1:= sup z2[ ;1] jp(z)j. (As we have seen, kT nk 1= 1.) Proof: We argue by contradiction. Assume ON CHEBYSHEV POLYNOMIALS OF MATRICES 3 (1.1) for a nonnormal matrix A we are aware of have been given in [13, Theorem 3.4]. It is shown there that TA m(z) = (z ¡ ‚)m if A = J‚, a Jordan block with eigenvalue ‚ 2 C. Note that in this case the Chebyshev polynomials of A are independent of the size of A. The above remarks show that the problem (1.1) is a mathematically interesting The generalized complex numbers can be realized in terms of 2 × 2 or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers.We also briefly q-Chebyshev polynomials Johann Cigler Fakultät für Mathematik, Universität Wien johann.cigler@univie.ac.at Abstract In this overview paper a direct approach to q Chebyshev polynomials and their elementary properties is given. Special emphasis is placed on analogies with the classical case. There are also some connections with q tangent and q Genocchi numbers. 0. Introduction Waleed A. Al The motivation for the term Chebyshev polynomial of a matrix" is as follows. All readers will be familiar with the class