Abstract: Presenting a comprehensive theory of orthogonal polynomials in two real variables and properties of Fourier series in these polynomials, this volume also gives cases of orthogonality over a region and on a contour. The text includes the classification of differential equations which admits orthogonal polynomials as eigenfunctions and several two-dimensional analogies of classical orthogonal polynomials.

Abstract: The approximate solutions of optimal control problems are investigated in this article with Duffing oscillator model. Duffing oscillator models have lots of interesting engineering application. The orthogonal Boubaker-Turki polynomials first are presented with some new interesting properties. Then, an algorithm based on iterative technique is considered with orthogonal Boubaker-Turki polynomials as a basis functions. By utilizing OBTPs, the proposed technique produces a simple procedure to get an optimal approximate solution that can use to problems that are more complex. Convergence of the algorithm is discussed and numerical examples are solved. The results obtained illustrate and emphasize the efficiency of the suggested algorithm when comparing the results with results obtained in previous works.

Abstract: We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

Abstract: In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and we deal with, in detail, two special cases which we call them $(r,s)$-Fibonacci and $(r,s)$-Lucas polynomials. We present Binet's formulas, generating functions, Simson's formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices associated with these sequences. Finally, we present several expressions and combinatorial results of the generalized Fibonacci polynomials.

Abstract: This paper is an improvement of a previous work on the problem recovering a function or probability density function from a finite number of its geometric moments, [1]. The previous worked solved the problem with the help of the B-Spline theory which is a great approach as long as the resulting linear system is not very large. In this work, two solution algorithms based on the approximate representation of the target probability distribution function via an orthogonal expansion are provided. One primary application of this theory is the reconstruction of the Particle Size Distribution (PSD), occurring in chemical engineering applications. Another application of this theory is the reconstruction of the Radon transform of an image at an unknown angle using the moments of the transform at known angles which leads to the reconstruction of the image form limited data. The aim is to recover a probability density function from a finite number of its geometric moments. The tool is the orthogonal expansion approach. The Shifted-Legendre Polynomials and the Chebyshev Polynomials as bases for the orthogonal expansion are used in this study. A high degree of accuracy has been obtained in recovering a function without facing a possible ill-conditioned linear system, which is the case with many typical approaches of solving the problem. In fact, for a normalized template function f on the interval [0, 1], and a reconstructed function ; the reconstruction accuracy is measured in two domains. One is the moment domain, in which the error (difference between the moments of f and the moments of ) is zero. The other measure is the standard difference in the norm -space ||f- || which can be ≈ 10-6 or less. This paper discusses the problem of recovering a function from a finite number of its geometric moments for the PSD application. Linear transformations were used, as needed, so that the function is supported on the unit interval [0, 1], or on [0, α] for some choice of α. This transformation forces the sequence of moments to vanish. Then, an orthogonal expansion of the Scaled Shifted Legendre Polynomials, as well as the Chebyshev Polynomials, are developed. The result shows good accuracy in recovering different types of synthetic functions. It is believed that up to fifteen moments, this approach is safe and reliable.

Abstract: This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved that the diagonal recurrence coefficient $\alpha_n(t)$ satisfies the fourth Painlev\'{e} equation. In this paper we show that the off-diagonal recurrence coefficient $\beta_n(t)$ fulfills the first member of Chazy II system. We also prove that the sub-leading coefficient of the monic semi-classical Laguerre polynomials satisfies both the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'{e} IV. By using Dyson's Coulomb fluid approach together with the discrete system for $\alpha_n(t)$ and $\beta_n(t)$, we obtain the large $n$ asymptotic expansions of the recurrence coefficients and the sub-leading coefficient. The large $n$ asymptotics of the associate Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub-leading coefficient.

Abstract: In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.

Abstract: This chapter deals with extremal problems of Markov–Bernstein type for polynomials in integral norms. Orthogonal polynomials on the real line, especially those with respect to the classical weight functions, as one of the basic tools in the study of extremal problems of this type, are treated in Section 3.1 . Special attention is paid to the recurrence relations and to formulas for differentiation of the classical orthogonal polynomials (Jacobi, generalized Laguerre, and Hermite polynomials). Besides the standard extremal problems of Markov's type in L 2 norm for the classical weight functions, in this chapter we consider different modifications of the weighted L 2 Markov–Bernstein extremal problems and the corresponding inequalities, different generalizations in L r norm, and the extremal problems on some restricted classes of polynomials.

Abstract: In this chapter, we give an overview of the links between Riordan arrays and orthogonal polynomials, and then we study some specialized areas including classical and semi-classical orthogonal polynomials defined by Riordan arrays, orthogonal polynomials that can be described as the moment sequences of Riordan arrays, applications of exponential Riordan arrays to the Toda lattice equations, and combinatorial polynomials that are moments of Riordan arrays. Orthogonal polynomials enjoy a special place both in pure mathematics and in applied mathematics. Stieltjes, in studying the moment sequences associated to families of orthogonal polynomials, defined the integral that now bears his name. Chebyshev and others, by putting the theory of orthogonal polynomials on a firm basis, provided a tool that has proved invaluable to mathematicians working in the area of approximation of functions, in the area of differential equations, and in many branches of mathematical physics. Traditional orthogonal polynomials are studied on the real line and on the circle. More sophisticated approaches study orthogonal polynomials defined on curves. We shall see that the orthogonal polynomials that can be defined by Riordan arrays are defined either over finite intervals on the real line, or intervals and some discrete points, or in the case of exponential Riordan arrays, over intervals of infinite extent, such as $$[0,\infty )$$ or even $$(-\infty , \infty )$$ . Families of orthogonal polynomials over the real line are typically associated with a measure, which is often realized through a density or weight function. In the case of orthogonal polynomials defined by ordinary Riordan arrays, this weight function can be determined. However, in the case of orthogonal polynomials determined by exponential Riordan arrays, such weight functions are only known in special cases. The production matrix plays a vital role in this theory, as it is precisely when the production matrix is tri-diagonal in form that the corresponding Riordan array (either ordinary or exponential) defines a family of orthogonal polynomials. In the theory of orthogonal polynomials, a distinction is made between so-called “classical” orthogonal polynomials and those that are not “classical”. Such a distinction can also be made for those orthogonal polynomials that can be defined by Riordan arrays.

Abstract: The main purpose of this paper is to investigate various formulas, identities and relations involving Apostol type numbers and parametric type polynomials. By using generating functions and their functional equations, we give many relations among the certain family of combinatorial numbers, the Vieta polynomials, the two-parametric types of the Apostol-Euler polynomials, the Apostol-Bernoulli polynomials, the Apostol-Genocchi polynomials, the Fibonacci and Lucas numbers, the Chebyshev polynomials, and other special numbers and polynomials. Moreover, we give some formulas related to trigonometric functions, special numbers and special polynomials. Finally, some remarks and observations on the results of this paper are given.

Abstract: The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.

Abstract: The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms.

Abstract: The article focuses on introducing some properties of representative special functions with basic steps as the clue to follow the logic of each property. It is a review paper, so no results come out from it, but it will be a good paper for beginners to take a closer look at special functions. It also includes some extra stuff like Gamma Function and Applications of Orthogonal Polynomials.

Abstract: The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown.

Abstract: A remarkably large number of polynomials and their extensions have been presented and studied. In this paper, we consider a new type of degenerate Changhee–Genocchi numbers and polynomials which are different from those previously introduced by Kim et al. (J. Ineq. Appl. 294, 2017). We investigate some properties of these numbers and polynomials. We also introduce a higher-order new type of degenerate Changhee–Genocchi numbers and polynomials which can be represented in terms of the degenerate logarithm function. Finally, we derive their summation formulae.

Abstract: . Making use of Gegenbauer polynomials, we initiate and explore two sets of normalized regular and bi-univalent (or bi-Schlicht) functions in D = { z ∈ C : | z | < 1 } linked with Gegenbauer polynomials. We investi-gate certain coefficients bounds and the Fekete-Szeg¨o functional for functions in these families. We also present few interesting observations and provide relevant connections of the results investigated.