Search Results (25)

This paper establishes a new characterization of symmetric q-Dunkl-classical orthogonal polynomials through a second structure relation. These symmetric polynomials generalize the $q^2$-analogues of Hermite and Gegenbauer polynomials. Our main result provides a finite expansion of each polynomial in terms of its q-Dunkl derivatives, offering a new effective classification method. We derive explicit structure relations for the $q^2$-analogue of generalized Hermite and the $q^2$-analogue of generalized Gegenbauer polynomials. Full article

The paper advances a new technique for constructing the exceptional differential polynomial systems (X-DPSs) and their infinite and finite orthogonal subsets. First, using Wronskians of Jacobi polynomials (JPWs) with a common pair of the indexes, we generate the Darboux–Crum nets of the rational canonical Sturm–Liouville equations (RCSLEs). It is shown that each RCSLE in question has four infinite sequences of quasi-rational solutions (q-RSs) such that their polynomial components from each sequence form a X-Jacobi DPS composed of simple pseudo-Wronskian polynomials (p-WPs). For each p-th order rational Darboux Crum transform of the Jacobi-reference (JRef) CSLE, used as the starting point, we formulate two rational Sturm–Liouville problems (RSLPs) by imposing the Dirichlet boundary conditions on the solutions of the so-called ‘prime’ SLE (p-SLE) at the ends of the intervals (−1, +1) or (+1, ∞). Finally, we demonstrate that the polynomial components of the q-RSs representing the eigenfunctions of these two problems have the form of simple p-WPs composed of p Romanovski–Jacobi (R-Jacobi) polynomials with the same pair of indexes and a single classical Jacobi polynomial, or, accordingly, p classical Jacobi polynomials with the same pair of positive indexes and a single R-Jacobi polynomial. The common, fundamentally important feature of all the simple p-WPs involved is that they do not vanish at the finite singular endpoints—the main reason why they were selected for the current analysis in the first place. The discussion is accompanied by a sketch of the one-dimensional quantum-mechanical problems exactly solvable by the aforementioned infinite and finite EOP sequences. Full article

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X₁-Jacobi DPSs composed of Heun polynomials contains both the X₁-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way. Full article

The present article aims to significantly improve geometric function theory by making an important contribution to p-valent meromorphic and analytic functions. It focuses on subordination, which describes the relationships of analytic functions. In order to achieve this, we utilize a technique that is based on the properties of differential subordination. This approach, which is one of the most recent developments in this field, may obtain a number of conclusions about differential subordination for p-valent meromorphic functions described by the new operator ${IH}_{p,\mathcal{q},s }^{\mathcal{j},p}\left({\nu }_1,\mathcal{n},\alpha ,\mathcal{l}\right)\mathcal{J}(\zeta )$  within the porous unit disk $\Delta$. Numerous mathematical and practical issues involving orthogonal polynomials, such as system identification, signal processing, fluid dynamics, antenna technology, and approximation theory, can benefit from the results presented in this article. The knowledge and comprehension of the unit’s analytical functions and its interacting higher relations are also greatly expanded by this text. Full article

We give the Karhunen–Loève expansion of the covariance function of a family of discrete weighted Brownian bridges, appearing as discrete analogues of continuous Gaussian processes related to Cramér –von Mises and Anderson–Darling statistics. This analogy enables us to introduce a discrete Cramér–von Mises statistic and show that this statistic satisfies a property of local asymptotic Bahadur optimality for a statistical test involving the classical hypergeometric distributions. Our statistic and the goodness-of-fit problem we deal with are based on basic properties of Hahn polynomials and are, therefore, subject to some extension to all families of classical orthogonal polynomials, as well as their q-analogues. Due probably to computational difficulties, the family of discrete Cramér–von Mises statistics has received less attention than its continuous counterpart—the aim of this article is to bridge part of this gap. Full article

Given a sequence of orthogonal polynomials ${\left\{L_n\right\}}_{n=0}^{\infty }$, orthogonal with respect to a positive Borel $\nu$ measure supported on ${\mathbb{R}}_{+}$, let ${\left\{Q_n\right\}}_{n=0}^{\infty }$ be the the sequence of orthogonal polynomials with respect to the modified measure $r\left(x\right)d\nu \left(x\right)$, where r is certain rational function. This work is devoted to the proof of the relative asymptotic formula $\frac{Q_n^{\left(d\right)}\left(z\right)}{L_n^{\left(d\right)}\left(z\right)}{\rightrightarrows }_n{\prod }_{k=1}^{N_1}{\left(\frac{\sqrt{a_k}+i}{\sqrt{z}+\sqrt{a_k}}\right)}^{A_k}{\prod }_{j=1}^{N_2}{\left(\frac{\sqrt{z}+\sqrt{b_j}}{\sqrt{b_j}+i}\right)}^{B_j}$, on compact subsets of $\mathbb{C}\setminus {\mathbb{R}}_{+}$, where $a_k$ and $b_j$ are the zeros and poles of r, and the $A_k$, $B_j$ are their respective multiplicities. Full article

In the survey Kiryakova: “A Guide to Special Functions in Fractional Calculus” (published in this same journal in 2021) we proposed an overview of this huge class of special functions, including the Fox H-functions, the Fox–Wright generalized hypergeometric functions ${}_p{\Psi }_q$ and a large number of their representatives. Among these, the Mittag-Leffler-type functions are the most popular and frequently used in fractional calculus. Naturally, these also include all “Classical Special Functions” of the class of the Meijer’s G- and ${}_p{F}_q$-functions, orthogonal polynomials and many elementary functions. However, it so happened that almost simultaneously with the appearance of the Mittag-Leffler function, another “fractionalized” variant of the exponential function was introduced by Le Roy, and in recent years, several authors have extended this special function and mentioned its applications. Then, we introduced a general class of so-called (multi-index) Le Roy-type functions, and observed that they fall in an “Extended Class of SF of FC”. This includes the I-functions of Rathie and, in particular, the $\overline{H}$-functions of Inayat-Hussain, studied also by Buschman and Srivastava and by other authors. These functions initially arose in the theory of the Feynman integrals in statistical physics, but also include some important special functions that are well known in math, like the polylogarithms, Riemann Zeta functions, some famous polynomials and number sequences, etc. The I- and $\overline{H}$-functions are introduced by Mellin–Barnes-type integral representations involving multi-valued fractional order powers of $\Gamma$-functions with a lot of singularities that are branch points. Here, we present briefly some preliminaries on the theory of these functions, and then our ideas and results as to how the considered Le Roy-type functions can be presented in their terms. Next, we also introduce Gelfond–Leontiev generalized operators of differentiation and integration for which the Le Roy-type functions are eigenfunctions. As shown, these “generalized integrations” can be extended as kinds of generalized operators of fractional integration, and are also compositions of “Le Roy type” Erdélyi–Kober integrals. A close analogy appears with the Generalized Fractional Calculus with H- and G-kernel functions, thus leading the way to its further development. Since the theory of the I- and $\overline{H}$-functions still needs clarification of some details, we consider this work as a “Discussion Survey” and also provide a list of open problems. Full article

In this paper, we consider the orthogonal polynomials with respect to the weight $w\left(x\right)=w(x;s):=x^{\lambda }{\mathrm{e}}^{-N[x+s(x^3-x)]},x\in {\mathbb{R}}^{+},$ where $\lambda >0$, $N>0$ and $0\le s\le 1$. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients ${\alpha }_n\left(s\right)$ and ${\beta }_n\left(s\right)$. We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when $q:=n/N$ is fixed as $n\to \infty$. Full article

Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath ($hb$), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution $W_{eq}$ generates orthogonal (Hermite) polynomials $H_n$ in momenta. Suitable moments $W_n$ of W (using the $H_n$’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the $W_n$’s yield irreversibly approach to equilibrium. The approach is extended (without $hb$) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ${\varphi }^4$ field theory (without $hb$). The extension to one non-relativistic quantum particle (with $hb$) employs the non-equilibrium Wigner function ($W_Q$): difficulties related to non-positivity of $W_Q$ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ${\varphi }^4$ field theory (a meson gas off-equilibrium, without $hb$), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ${\varphi }^4$ theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ${\varphi }^4$ one, yielding an approach to equilibrium. Full article

Named essentially after their close relationship with the modified Bessel function $K_{\nu }\left(z\right)$ of the second kind, which is known also as the Macdonald function (or, with a slightly different definition, the Basset function), the so-called Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ stemmed naturally in some systematic investigations of the classical wave equation in spherical polar coordinates. Our main purpose in this invited survey-cum-expository review article is to present an introductory overview of the Bessel polynomials $y_n\left(x\right)$ and the generalized Bessel polynomials $y_n(x;\alpha ,\beta )$ involving the asymmetric parameters $\alpha$ and $\beta$. Each of these polynomial systems, as well as their reversed forms ${\theta }_n\left(x\right)$ and ${\theta }_n(x;\alpha ,\beta )$, has been widely and extensively investigated and applied in the existing literature on the subject. We also briefly consider some recent developments based upon the basic (or quantum or q-) extensions of the Bessel polynomials. Several general families of hypergeometric polynomials, which are actually the truncated or terminating forms of the series representing the generalized hypergeometric function ${}_r{F}_s$ with r symmetric numerator parameters and s symmetric denominator parameters, are also investigated, together with the corresponding basic (or quantum or q-) hypergeometric functions and the basic (or quantum or q-) hypergeometric polynomials associated with ${}_r{\Phi }_s$ which also involves r symmetric numerator parameters and s symmetric denominator parameters. Full article

Extended Dynamic Mode Decomposition (EDMD) allows an approximation of the Koopman operator to be derived in the form of a truncated (finite dimensional) linear operator in a lifted space of (nonlinear) observable functions. EDMD can operate in a purely data-driven way using either data generated by a numerical simulator of arbitrary complexity or actual experimental data. An important question at this stage is the selection of basis functions to construct the observable functions, which in turn is determinant of the sparsity and efficiency of the approximation. In this study, attention is focused on orthogonal polynomial expansions and an order-reduction procedure called p-q quasi-norm reduction. The objective of this article is to present a Matlab library to automate the computation of the EDMD based on the above-mentioned tools and to illustrate the performance of this library with a few representative examples. Full article

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of $|a_2|$, $|a_3|$, and $|a_4|$, results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class. Full article

In this contribution, we consider the sequence ${\{{\mathbb{H}}_n(x;q)\}}_{n\ge 0}$ of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of ${\{{\mathbb{H}}_n(x;q)\}}_{n\ge 0}$ in a watermarking problem. Several differences are found in this watermarking application for the non-standard cases (when $j>0$) with respect to the standard classical Krawtchouk case $\lambda =\mu =0$. Full article

In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the ${\mathfrak{L}}_q$-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant. Full article

Since Gottlieb introduced and investigated the so-called Gottlieb polynomials in 1938, which are discrete orthogonal polynomials, many researchers have investigated these polynomials from diverse angles. In this paper, we aimed to investigate the q-extensions of these polynomials to provide certain q-generating functions for three sequences associated with a finite power series whose coefficients are products of the known q-extended multivariable and multiparameter Gottlieb polynomials and another non-vanishing multivariable function. Furthermore, numerous possible particular cases of our main identities are considered. Finally, we return to Khan and Asif’s q-Gottlieb polynomials to highlight certain connections with several other known q-polynomials, and provide its q-integral representation. Furthermore, we conclude this paper by disclosing our future investigation plan. Full article