Time series problems involve analysis of periodic functions for predicting the future. A flexible regression method should be able to dynamically select the appropriate model to fit the available data. In this paper, we present a function approximation scheme that can be used for modeling periodic functions using a series of orthogonal polynomials, named Chebychev polynomials. In our approach, ...

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,

A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre–Gauss–Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and ...

We consider the space Pn of orthogonal polynomials of degree n on the unit disc for a general radially symmetricweight function.We show that there exists a single orthogonal polynomialwhose rotations through the angles j n+1 , j = 0, 1, . . . , n forms an orthonormal basis for Pn, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Lege...

The principle result of this paper is the following operational Tau method based upon Müntz-Legendre polynomials. This methodology provides a computational technique for numerical solution of fractional differential equations by using a sequence of matrix operations. The main property of Müntz polynomials is that fractional derivatives of these polynomials can be expressed in terms of the same ...

We consider a fourth-order eigenvalue problem on a semi-infinite strip which arises in the study of viscoelastic shear flow. The eigenvalues and eigenfunctions are computed by a spectral method involving Laguerre functions and Legendre polynomials.

Biological cells can be treated as an inhomogeneous particle. In addition to biomaterials, inhomogeneous particles are also important in more traditional colloidal science. By using two energy methods that are based on Legendre polynomials and Green’s function, respectively, we investigate the interaction between biological cells or colloidal particles in the presence of an external electric fi...

Although general methods led me to a complete solution, I soon saw that the result is obtained faster when the general procedure is left, and when one follows the path suggested by the particular problem at hand. S. Bernstein first lines of [6] Abstract. One establishes inequalities for the coefficients of orthogonal polynomials Φn(z) = z n + ξnz n−1 + · · · + Φn(0), n = 0, 1, . . . which are o...

In 1951, F. Brafman derived several “unusual” generating functions of classical orthogonal polynomials, in particular, of Legendre polynomials Pn(x). His result was a consequence of Bailey’s identity for a special case of Appell’s hypergeometric function of the fourth type. In this paper, we present a generalization of Bailey’s identity and its implication to generating functions of Legendre po...

These are notes on a preliminary follow-up to a question of Jim Propp, about cyclic sieving of cyclic codes. We show that two of the Mahonian polynomials are cyclic sieving polynomials for certain Dual Hamming Codes: X and X inv for q = 2, 3 and q = 2, respectively.