We show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second-order linear recurrences.Also considering how to the negatively subscripted terms of second-order linear recurrences can be connected to Chebyshev polynomials by determinants of these matrices, we give factorizations and representations of these number...

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the theoretical results.

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1223 (there is no occurrence πi < πj < πj+1 such that 1 ≤ i ≤ j − 2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. ...

Babson and Steingrimsson (2000, Séminaire Lotharingien de Combinatoire, B44b, 18) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Let fτ ;r (n) be the number of 1-3-2-avoiding permutations on n letters that contain exactly r occurrences of τ , where τ is a generalized pattern on k letters. Let Fτ ...

A unified approach is presented for determining all the constants Ym.n (m > 0, n > 0) which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that Ym,m+2 = 1/3 (m > 0), a problem which had remained open.

The connection between weighted counts of Motzkin paths and moments of orthogonal polynomials is well known. We look at the inverse generating function of Motzkin paths with weighted horizontal steps, and relate it to Chebyshev polynomials of the second kind. The inverse can be used to express the number of paths ending at a certain height in terms of those ending at height 0. Paths of a more g...

By using Dickson polynomials in several variables and Chebyshev polynomials of the second kind, we derive the explicit expression of the entries in the array defining the sequence A185905. As a result, we obtain a straightforward proof of some conjectures of Jeffery concerning this sequence and other related ones.

Algebraic properties of Chebyshev polynomials are presented. The complete factorization of Chebyshev polynomials of the rst kind (Tn(x)) and second kind (Un(x)) over the integers are linked directly to divisors of n and n + 1 respectively. For any odd integer n, it is shown that the polynomial Tn(x)=x is irreducible over the integers i n is prime. The result leads to a generalization of Fermat'...

We present approximation kernels for orthogonal expansions with respect to Bernstein-Szegö polynomials. The construction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein-Szegö polynomials.

In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.