Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions.

2-Variable Fubini-degenerate Apostol-type polynomials

This work deals with the mathematical inspection of a hybrid family of the degenerate polynomials of the Apostol-type. The inclusion of the derivation of few series expansion formulas, explicit representations and difference equations for this hybrid family brings a novelty to the existing literature. Moreover, certain connection formulas and several novel identities for these polynomials are established and investigated. The graphical representations of certain degenerate polynomials are explored and several new interesting pattern of the zeros are observed.

New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

On the k-Mersenne–Lucas numbers

In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.

Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

The principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

A note on Pell-Padovan numbers and their connection with Fibonacci numbers

In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

Lupaş-type inequality and applications to Markov-type inequalities in weighted Sobolev spaces

Weighted Sobolev spaces play a main role in the study of Sobolev orthogonal polynomials. In particular, analytic properties of such polynomials have been extensively studied, mainly focused on their asymptotic behavior and the location of their zeros. On the other hand, the behavior of the Fourier–Sobolev projector allows to deal with very interesting approximation problems. The aim of this paper is twofold. First, we improve a well-known inequality by Lupaş by using connection formulas for Jacobi polynomials with different parameters. In the next step, we deduce Markov-type inequalities in weighted Sobolev spaces associated with generalized Laguerre and generalized Hermite weights.