Related papers: A revisit to periodic continuants

We give a short proof of the trace formula for Hecke operators on modular forms for the modular group, using the action of Hecke operators on the space of period polynomials.

We construct the differential operators for which bivariate Chebyshev polynomials of the first kind, associated with simple Lie algebras $C_2$ and $G_2$, are eigenfunctions.

Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are…

By using purely algebraic tools, we establish well-known properties of roots of Chebyshev polynomials. Especially, we show that these zeros are simple and lie in $(-1,1)$ and we prove in two ways that they are mostly irrational.

We give description of rational solutions of polynomial-equations.

We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials.…

We consider polynomials of the form $\operatorname{s}_\lambda(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\lambda$ is an integer partition, $\operatorname{s}_\lambda$ is the Schur polynomial associated to $\lambda$, and…

We obtain exact, simple and very compact expressions for the linearization coefficients of the products of orthogonal polynomials; both the conventional Clebsch-Gordan-type and the modified version. The expressions are general depending…

We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's…

Some identities of Chebyshev polynomials are deduced from Waring's formula on symmetric functions. In particular, these formulae generalize some recent results of Grabner and Prodinger.

We obtain some new inequalities of Chebyshev Type.

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions…

A new recursion in only one variable allows very simple verifications of Bressoud's polynomial identities, which lead to the Rogers-Ramanujan identities. This approach might be compared with an earlier approach due to Chapman. Applying the…

We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations.

In this paper we evaluate Chebyshev polynomials of the second-kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly…

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix.…

In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.

We extend the close interplay between continued fractions, orthogonal polynomials, and Gaussian quadrature rules to several variables in a special but natural setting which we characterize in terms of moment sequences. The crucial condition…