Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stir...

from the early 1950s, estimating the autocorrelations of polynomials with coefficients on the unit circle has found applications in ising spin systems and in surface acoustic wave designs. in this paper, a technique is introduced that not only estimates the autocorrelations, but for some special types of such polynomials, it locates the frequencies at which maximum autocorrelation occurs.

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider two new q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci polynomials and the recent works on the combinatorics of the Matrix Ansatz of the PASEP.

A probabilistic interpretation of a modified Gegenbauer polynomial is furnished by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinat...

An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed.

with given a, b, t0, t1 and n ≥ 0. This sequence was introduced by Horadam [3] in 1965, and it generalizes many sequences (see [1, 4]). Examples of such sequences are Fibonacci polynomials sequence (Fn(x))n≥0, Lucas polynomials sequence (Ln(x))n≥0, and Pell polynomials sequence (Pn(x))n≥0, when one has a = x, b = t1 = 1, t0 = 0; a = t1 = x, b = 1, t0 = 2; and a = 2x, b = t1 = 1, t0 = 0; respect...

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard...

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.