No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Orthogonal Polynomials and Special Functions
... It might be stated that the importance of linearization formulas in the study of orthogonal polynomials was superbly highlighted in Richard Askey's famous National Science Foundation Regional Conference lecture series at Virginia Polytechnic Institute in June 1974. These lectures resulted in a beautiful set of lecture notes that Askey lovingly assembled in [1], and in particular in his beautiful chapter, Lecture 5: linearization of products where he discusses the importance and history of linearization formulas for Chebyshev, Gegenbauer (ultraspherical), Jacobi, Krawtchouk, Meixner, Laguerre and Hermite polynomials. One of the beautiful things about linearization coefficients for orthogonal polynomials is that they are surprisingly connected with some beautiful combinatorial problems. ...
... |n−m+1|−p+ 0 1 2 1 + |n − m + 1|, 1 2 − m∧(n + 1) ...
... 1) where n ∈ N 0 , x ∈ C. Hermite polynomials are orthogonal on (−∞, ∞), with orthogonality relation[12, (9.15.2)] ∞ −∞ H m (x)H n (x) e −x 2 dx = √ π 2 n n!δ m,n =: h n δ m,n . (5.2) ...
By using the three-term recurrence relation for orthogonal polynomials, we produce a collection of two-dimensional contiguous relations for certain generalized hypergeometric functions. These generalized hypergeometric functions arise through linearization coefficients for some classical orthogonal polynomials in the Askey-scheme, namely Gegenbauer (ultraspherical), Hermite, Jacobi and Laguerre polynomials.
... (1.40) Lemma 1. 5 The elements a,k;b, , S a,k have the following properties: ...
... where in the step " " we used the Chu-Vandermonde identity [5]. The lemma is proved. ...
In this paper, we extend the matrix-resolvent method to the study of the Dubrovin–Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of (M, 1)-type. We show that the Dubrovin–Zhang type tau-function of an arbitrary solution to the bigraded Toda hierarchy of (M, 1)-type is a Dubrovin–Zhang type tau-function for the constrained KP hierarchy, which generalizes the result in Carlet et al. (Mosc Math J 4:313–332, 2004) and Fu and Yang (J Geom Phys 179:104592, 2022) for the Toda lattice hierarchy and the NLS hierarchy corresponding to the \(M=1\) case.
... It is known that 0 ≤ Γ(L; , ) ≤ 1 and + L=0 Γ(L; , ) = 1, see [1,Chapter 5]. Obviously, the following formal equation holds: ...
... In this section, we discuss how to construct a numerical solution to (1). For this purpose, we first establish some notation. ...
In this work, we construct numerical solutions to an inverse problem of a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centered at the origin.Assuming that the values of the forward problem are known at sufficiently many points, we would like to determine the form of the non-linear term on the right-hand side of the equation via its Chebyshev coefficients.
... Here we have used the Chu-Vandermonde's identity [2]: for any integers 0 ≤ r ≤ k ≤ n, ...
We extend several celebrated methods in classical analysis for summing series of complex numbers to series of complex matrices. These include the summation methods of Abel, Borel, Ces\'aro, Euler, Lambert, N\"orlund, and Mittag-Leffler, which are frequently used to sum scalar series that are divergent in the conventional sense. One feature of our matrix extensions is that they are fully noncommutative generalizations of their scalar counterparts -- not only is the scalar series replaced by a matrix series, positive weights are replaced by positive definite matrix weights, order on $\mathbb{R}$ replaced by Loewner order, exponential function replaced by matrix exponential function, etc. We will establish the regularity of our matrix summation methods, i.e., when applied to a matrix series convergent in the conventional sense, we obtain the same value for the sum. Our second goal is to provide numerical algorithms that work in conjunction with these summation methods. We discuss how the block and mixed-block summation algorithms, the Kahan compensated summation algorithm, may be applied to matrix sums with similar roundoff error bounds. These summation methods and algorithms apply not only to power or Taylor series of matrices but to any general matrix series including matrix Fourier and Dirichlet series. We will demonstrate the utility of these summation methods: establishing a Fej\'{e}r's theorem and alleviating the Gibbs phenomenon for matrix Fourier series; extending the domains of matrix functions and accurately evaluating them; enhancing the matrix Pad\'e approximation and Schur--Parlett algorithms; and more.
... The classical Chu-Vandermonde identity [3] Since the norms of these Hilbert spaces coincide, then the corresponding inner products also coincide, and therefore the two-variable extremal problem is the same as (2.5). ...
We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable.
... Lemma 2 [40] Let γ > 0, α 1 , α 2 > −1 and x ∈ (−1, 1). Then, we have ...
In this paper, a high-order and fast numerical method based on the space-time spectral scheme is obtained for solving the space-time fractional telegraph equation. In the proposed method, for discretization of temporal and spatial variables, we consider two cases. We use the Legendre functions for discretization in time. To obtain the full discrete numerical approach, we use a Fourier-like orthogonal function. The convergence and stability analysis for the presented numerical approach is studied and analyzed. Some numerical examples are given for the effectiveness of the numerical approach.
... For n ∈ N 0 and κ, β > −1, the nth Jacobi polynomial P (κ,β) n (x) may be defined by means of Rodrigues' formula (see [4,22,27]): ...
In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their num- bers, and their relationship with the Riemann zeta function. We also derive the Apostol-type gen- eralizations to obtain some of their algebraic and differential properties. We introduce general- ized U-Bernoulli, U-Euler and U-Genocchi polynomial Pascal-type matrix. We deduce some prod- uct formulas related to this matrix. Furthermore, we establish some explicit expressions for the U-Bernoulli, U-Euler, and U-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
... For n ∈ N 0 and κ, β > −1, the nth Jacobi polynomial P (κ,β) n (x) may be defined by means of Rodrigues' formula (see [4,22,27]): ...
In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
... By this and the Chu-Vandermonde identity [8], we see that ...
We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.
... Generating functions as a powerful tool, have many applications in the diverse fields of mathematics such as statistics, analysis of algorithms, combinatorics, probability theory and special functions. Generating function of special functions and polynomials are represented as infinite series over these polynomials in various types [2], [3], [5], [6] and [8] . For some polynomials this summation does not always contain the terms with integer factorial 1 n! such as Laguerre and Legendre polynomials. ...
This paper presents a new generating function for Hermite polynomials of one variable in the form of $g(x,t)=\sum_{n=0}^{\infty }t^{n}H^{e}_{n}(x)$ and reveals its connection with incomplete gamma function.
... Generating functions as a powerful tool, have many applications in the diverse fields of mathematics such as statistics, analysis of algorithms, combinatorics, probability theory and special functions. Generating function of special functions and polynomials are represented as infinite series over these polynomials in various types [1], [2], [4], [5] and [7] . For some polynomials this summation does not always contain the terms with integer factorial 1 n! such as Laguerre and Legendre polynomials. ...
This paper presents a new generating function for Hermite polynomials of one variable in the form of g(x, t) = ∞ n=0 t n H e n (x) and reveals its connection with incomplete gamma function.
... where in the step " " we used the Chu-Vandermonde identity [3]. The lemma is proved. ...
In this paper, we extend the matrix-resolvent method to the study of the Dubrovin--Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type. We show that the Dubrovin--Zhang type tau-function of an arbitrary solution to the bigraded Toda hierarchy of $(M,1)$-type is a Dubrovin--Zhang type tau-function for the constrained KP hierarchy, which generalizes the result in [10, 35] for the Toda lattice hierarchy and the NLS hierarchy corresponding to the $M=1$ case.
... The equation (4.9) is commonly referred to as the Vandermonde's identity or the Vandermonde's convolution. Interested readers can find more information on this topic in the relevant literature, such as [2]. ...
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method.
... More generally, we may consider a mixing base (Q n,η ) n∈N where the density of Q n,η is a polynomial of degree n in η t x. This leads to the consideration of general linear combinations of ultraspherical polynomials; indeed, finding conditions for such polynomials to be nonnegative (on a given interval) is an ongoing research topic, see Askey (1975). ...
We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises--Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.
... The theory of orthogonal polynomials is vast and rich, extending all the way back to the groundbreaking work of Legendre [42], where he introduced the family of polynomials that now bears his name. We direct the interested reader to (some of!) the fundamental treatises on the field [8], [10], [29], [31], [33], [40], [70]. ...
... These functions are widely discussed in every book on special functions. Among many others we cite Abramowitz and Stegun (1972), Askey (1975), Temme (1996) and Mathai and Haubold (2008). Here, we briefly introduce the hypergeometric function with its most relevant properties. ...
This paper introduces inter-order formulas for partial and complete moments of a Student $t$ distribution with $n$ degrees of freedom. We show how the partial moment of order $n - j$ about any real value $m$ can be expressed in terms of the partial moment of order $j - 1$ for $j$ in $\{1,\dots, n \}$. Closed form expressions for the complete moments are also established. We then focus on $L_p$-quantiles, which represent a class of generalized quantiles defined through an asymmetric $p$-power loss function. Based on the results obtained, we also show that for a Student $t$ distribution the $L_{n-j+1}$-quantile and the $L_j$-quantile coincide at any confidence level $\tau$ in $(0, 1)$.
... By this and the Chu-Vandermonde identity [7], we see that ...
We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is a homogeneous quasi-arithmetic mean, and under some conditions, several limit theorems hold for the power mean as well as for the arithmetic mean of i.i.d. integrable random variables. We establish integrabilities and a limit theorem for the variances of the power mean of i.i.d. non-integrable random variables. We also consider behaviors of the power mean when the parameter of the power varies. Our feature is that the generator of the power mean is allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. The complex-valued power mean is an unbiased strongly-consistent estimator for the joint of the location and scale parameters of the Cauchy distribution.
... Since inequalities for trigonometric sums have remarkable applications in various fields, like, for instance, geometric function theory and the theory of absolutely and completely monotonic functions, they have attracted the attention of numerous researchers. Detailed information on this subject, including many references and interesting historical comments, can be found in Askey [1], Askey and Gasper [2], Milovanović et al. [5, chapter 4], Raigorodskii and Rassias [6]. ...
The inequalities An(x)=∑k=1nsin((2k-1)x)2k-1>0(n≥1;0<x<π)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}$$\end{document}and Bn(x)=∑k=1ncos((2k-1)x)2k-1>0(n≥1;0≤x<π/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}$$\end{document}are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts: An(x)+Bn(x)>0(n≥1;0≤x<3π/4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}$$\end{document}and Bn(x)≥6475cos3(x)(n≥2;-π/2<x<π/2).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}$$\end{document}The constants 3π/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\pi /4$$\end{document} and 64/75 are best possible.
This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms.
We present several new inequalities for trigonometric sums. Among others, we show that the inequality
holds for all \(n\ge 1\) and \(x\in (0, 2\pi /3)\). The constant factor 2/9 is sharp. This refines the classical Szegö-Schweitzer inequality which states that the sine sum is positive for all \(n\ge 1\) and \(x\in (0,2 \pi /3)\). Moreover, as an application of one of our results we obtain a two-parameter class of absolutely monotonic functions.
The sine polynomials of Fejér and Lukács are defined by
respectively. We prove that for all \(n\ge 2\) and \(x\in (0,\pi )\), we have
with the best possible constants
An application of the first inequality leads to a class of absolutely monotonic functions involving the arctan function.
The main crux of this work is to define the Wigner transform associated with the Laguerre-Bessel operator and to give some results related to this transform. Next we introduce a new class of pseudo-differential operator \(\mathscr {L}_{u, v}(\sigma )\) called localization operator which depend on a symbol \(\sigma\) and two functions u and v. We give a criteria in terms of the symbol \(\sigma\) for its boundedness and compactness, we also show that these operators belongs to the Schatten-Von Neumann class \(S^p\) for all \(p \in [1; +\infty ]\) and we give a trace formula.
It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.
Boundedness conditions are found for the Hilbert transform H in Besov spaces with Muckenhoupt weights. The operator H in this situation acts on subclasses of functions from Hardy spaces. The results are obtained by representing the Hilbert transform H via Riemann–Liouville operators of fractional integration on ℝ, on the norms of images and pre-images of which independent estimates are established in the paper. Separately, a boundedness criterion is given for the transform H in weighted Besov and Triebel–Lizorkin spaces restricted to the subclass of Schwartz functions.
In this work we develop the Gaussian quadrature rule for weight functions involving powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified Chebyshev algorithm, here we present a very stable algorithm based on the preconditioning of the moment matrix. Numerical experiments are provided and a geophysical application is considered.
In this paper, the analytical solution of a class of Lane–Emden equation is considered using the Adomian decomposition method. The nonlinear term of the proposed equation is given by normalised Jacobi functions \({\mathscr {P}}_{\gamma }^{(\alpha , \beta )}(y(x))\) (\(\gamma \in {\mathbb {C}};\alpha ,\beta >-1\)). The Adomian polynomials for the Jacobi functions \({\mathscr {P}}_{\gamma }^{(\alpha , \beta )}(y(x))\) are constructed and the power series solutions are presented. For the special cases \(\gamma =0,1\); closed form solutions are obtained. Interestingly, the functions \({\mathscr {P}}_{\gamma }^{(\alpha , \beta )}\) are the spherical functions (normalised eigenfunctions) of the Laplacian on rank one symmetric spaces. In order to present several examples of Lane-Emden type equations and their solutions, we specialise to the spherical functions on the real hyperbolic space and sphere (\(\alpha =\beta =(n-2)/2\)), the complex hyperbolic space (\(\alpha =n-1,\beta =0\)), the quaternionic hyperbolic space (\(\alpha =2n-1,\beta =1\)), and the Cayley hyperbolic plane (\(\alpha =7,\beta =3\)), as well as their corresponding projective spaces. Comparisons of the results from the present method with other published results show that the Adomian decomposition method gives accurate and reliable approximate solutions of Lane–Emden equations involving Jacobi functions.
In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli-Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli-Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.
In this paper, we provide a novel generalization of quantum orthogonal polynomials from [Formula: see text]-deformed quantum algebras introduced in earlier works. We construct related quantum Jacobi polynomials and their probability distribution, factorial moments, recurrence relation, and governing difference equation. Surprisingly, these polynomials obey non-conventional recurrence relations. Particular cases of generalized quantum little Legendre, little Laguerre, Laguerre, Bessel, Rogers–Szegö, Stieltjes–Wigert and Kemp binomial polynomials are derived and discussed.
Orthogonality property in some kinds of polynomials has led to significant attention on them. Classical orthogonal polynomials have been under investigation for many years, with Jacobi polynomials being the most common. In particular, these polynomials are used to tackle multiple mathematical, physics, and engineering problems as well as their usage as the kernels has been investigated in the SVM algorithm classification problem. Here, by introducing the novel fractional form, a corresponding kernel is going to be proposed to extend the SVM’s application. Through transforming the input dataset to a fractional space, the fractional Jacobi kernel-based SVM shows the excellent capability of solving nonlinear problems in the classification task. In this chapter, the literature, basics, and properties of this family of polynomials will be reviewed and the corresponding kernel will be explained, the fractional form of Jacobi polynomials will be introduced, and the validation according to Mercer conditions will be proved. Finally, a comparison of the obtained results over a well-known dataset will be provided, using the mentioned kernels with some other orthogonal kernels as well as RBF and polynomial kernels.KeywordsJacobi polynomialFractional jacobi functionsKernel trickOrthogonal functionsMercer’s theorem
The FLIP cipher was proposed at Eurocrypt 2016 for the purpose of meliorating the efficiency of fully homomorphic cryptosystems. Weightwise perfectly balanced Boolean functions meet the balancedness requirement of the filter function in FLIP ciphers, and the construction of them has attracted serious attention from researchers. Nevertheless, the literature is still thin. Modifying the supports of functions with a low degree is a general construction technique whose key problem is to find a class of available low-degree functions. We first seek out a class of quadratic functions and then, based on these functions, present the new construction of weightwise perfectly balanced Boolean functions by adopting an iterative approach. It is worth mentioning that the functions we construct have good performance in weightwise nonlinearity. In particular, some p-weight nonlinearities achieve the highest values in the literature for a small number of variables.
We obtain discrete mixture representations for parametric families of probability distributions on Euclidean spheres, such as the von Mises–Fisher, the Watson and the angular Gaussian families. In addition to several special results we present a general approach to isotropic distribution families that is based on density expansions in terms of special surface harmonics. We discuss the connections to stochastic processes on spheres, in particular random walks, discrete mixture representations derived from spherical diffusions, and the use of Markov representations for the mixing base to obtain representations for families of spherical distributions.
In this paper, a spectral Petrov–Galerkin method is developed to solve an optimal control problem governed by a two-sided space-fractional diffusion-reaction equation with additive fractional noise. In order to compensate weak singularities of the solution near boundaries, regularities of both the fractional noise and the optimal control problem are analyzed in weighted Sobolev space. The spectral Petrov–Galerkin method is presented by employing truncated spectral expansion of the fractional Brownian motion (fBm) type noise, and error estimates are given based on the obtained regularity of the optimal control problem. Numerical experiments are carried out to verify the theoretical findings.
We present a highly accurate and efficient spectral Galerkin method for the advection–diffusion–reaction equations with fractional lower-order terms in one dimension. We first prove sharp a priori regularity estimates of solutions in weighted Sobolev spaces. Then we obtain optimal convergence orders of the spectral Galerkin method. We also describe its implementation and present an efficient solver with a quasi-optimal complexity. Finally, we perform the numerical experiments and report numerical observations to support the theoretical predictions.
We study the ratio [Formula: see text] asymptotically as [Formula: see text] where the polynomials [Formula: see text] are orthogonal with respect to a discrete linear functional and [Formula: see text] denote the falling factorial polynomials. We give recurrences that allow the computation of high order asymptotic expansions of [Formula: see text] and give examples for most discrete semiclassical polynomials of class [Formula: see text] We show several plots illustrating the accuracy of our results.
Serving both as an introduction to the subject and as a reference, this book presents the theory in elegant form and with modern concepts and notation. It covers the general theory and emphasizes the classical types of orthogonal polynomials whose weight functions are supported on standard domains. The approach is a blend of classical analysis and symmetry group theoretic methods. Finite reflection groups are used to motivate and classify symmetries of weight functions and the associated polynomials. This revised edition has been updated throughout to reflect recent developments in the field. It contains 25% new material, including two brand new chapters on orthogonal polynomials in two variables, which will be especially useful for applications, and orthogonal polynomials on the unit sphere. The most modern and complete treatment of the subject available, it will be useful to a wide audience of mathematicians and applied scientists, including physicists, chemists and engineers.
We give an explicit expansion series and an integral representation for the heat kernel associated with the magnetic Laplacian on the quantized Riemann sphere. We also derive the asymptotic expansion of the associated heat operator.
Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher proposed by Méaux et al. at the conference Eurocrypt 2016. After providing a security analysis on the FLIP cipher, those functions were nicely-investigated firstly by Carlet et al. in 2017 before taking a high interest by the community. Handling such Boolean functions and designing those with optimal characteristic cryptographic properties is no easy assignment. This article attempts to broaden the range of choices for these functions by offering two new concrete constructions of weightwise perfectly balanced (WPB) functions on 2m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^m$$\end{document} variables (where m is a positive integer) with optimal algebraic immunity. It is worth noting that the second class of WPB functions can be linearly transformed to be 2-rotation symmetric. Simultaneously, the k-weight nonlinearities of these newly constructed WPB functions on 2m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^m$$\end{document} variables are discussed for small values of m. Lastly, comparisons of the k-weight nonlinearities of all the known WPB functions are given, including the known results from computer investigations. The comparison to the current literature shows that despite its simplicity (an advantage from the implementation point of view), the WPB functions presented in this paper are the best in behavior from the algebraic immunity and the k-weight nonlinearities. Specifically, the even-weight nonlinearities of our second class of WPB functions are much higher than all the known WPB functions in the literature.
The generalized order of growth and generalized type of an entire function \(F^{\alpha,\beta}\) (generalized biaxisymmetric potentials) have been obtained in terms of the sequence \(E_n^p(F^{\alpha,\beta},\Sigma_r^{\alpha,\beta})\) of best real biaxially symmetric harmonic polynomial approximation on open hyper sphere \(\Sigma_r^{\alpha,\beta}\). Moreover, the results of McCoy [8] have been extended for the cases of fast growth as well as slow growth.
Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.
The classical Lane-Emden equation, a nonlinear second-order ordinary differential equation, models several phenomena in mathematical physics, mathematical chemistry and astrophysics. In this paper, the analytical solution of a class of fractional Lane–Emden differential equation is considered using a power series method. The nonlinear term \(y^{m}(x)\) (\(m\ge 0\), \(0<x\le 1\)) appearing in the classical Lane–Emden equation is replaced with the normalised Jacobi polynomials \({\mathscr {P}}_{m}^{(\alpha , \beta )}(y(x))\) (\(m=0,1,2,3,\dots ;\alpha ,\beta >-1\)) in y(x). Useful and important special cases of \({\mathscr {P}}_{m}^{(\alpha , \beta )}\), namely, the Gegenbauer or ultraspherical polynomials (\(\alpha =\beta \)), the Legendre polynomials (\(\alpha =\beta =0\)), and the Chebyshev polynomials of the first kind (\(\alpha =\beta =-1/2\)), the second kind (\(\alpha =\beta =1/2\)), the third kind (\(\alpha =-1/2,\beta =1/2\)) & the fourth kind (\(\alpha =1/2,\beta =-1/2\)) are considered explicitly. Coincidentally, the polynomials \({\mathscr {P}}_{m}^{(\alpha , \beta )}\) are the spherical functions (normalised eigenfunctions) of the Laplacians on rank one symmetric spaces of compact type; in order to present several examples of fractional Lane–Emden equations and their solutions, we specialise to the spherical functions on the spheres (\(\alpha =\beta =(n-2)/2\)), the complex projective spaces (\(\alpha =n-1,\beta =0\)), the quaternionic projective spaces (\(\alpha =2n-1,\beta =1\)) and the Cayley projective plane (\(\alpha =7,\beta =3\)). To demonstrate the accuracy and reliability of the proposed method, some well-known Lane–Emden type equations are solved. The results show that the proposed method gives accurate and reliable approximate solutions of ordinary and fractional Lane–Emden type equations.
The seminal works by George Matheron provide the foundations of walks through dimensions for positive definite functions defined in Euclidean spaces. For a d-dimensional space and a class of positive definite functions therein, Matheron called montée and descente two operators that allow for obtaining new classes of positive definite functions in lower and higher dimensional spaces, respectively. The present work examines three different constructions to dimension walks for continuous positive definite functions on hyperspheres. First, we define montée and descente operators on the basis of the spectral representation of isotropic covariance functions on hyperspheres. The second approach provides walks through dimensions following Yadrenko’s construction of random fields on spheres. Under this approach, walks through unit dimensions are not permissible, while it is possible to walk under ±2 dimensions from a covariance function that is valid on a d-dimensional sphere. The third construction relies on the integration of a given isotropic random field over latitudinal arcs. In each approach, we provide spectral representations of the montée and descente as well as illustrative examples.
In this paper, we seek to present some new identities for the elementary symmetric polynomials and use these identities to construct new explicit formulas for the Legendre polynomials. First, we shed light on the variable nature of elementary symmetric polynomials in terms of repetition and additive inverse by listing the results related to these. Sequentially, we have proven an important formula for the Legendre polynomials, in which the exponent moves one walk instead of twice as known. The importance of this formula appears throughout presenting Vieta’s formula for the Legendre polynomials in terms of their zeros and the results mentioned therein. We provide new identities for the elementary symmetric polynomials of the zeros of the Legendre polynomials. Finally, we propose the relationship between elementary symmetric polynomials for the zeros of P n x and the zeros of P n − 1 x .
In this paper, we express the expansion coefficients of the Jacobi polynomial series expansion in terms of the Hankel transform, and expand functions involving Bessel, Struve and Lommel functions in the series of Jacobi polynomials. Apart from expanding trigonometric functions in the series of Jacobi polynomials, we establish identities relating the classical Taylor coefficients of trigonometric functions to finite sums involving Bessel functions and certain spectral polynomials in the eigenvalues λkα,β:=k(k+α+β+1),k≥0, whose coefficients are the local Jacobi coefficients. By extension, we describe the Riemann-Liouville fractional derivative of the normalised Jacobi polynomials Pk(α,β)(η) by fractional Jacobi coefficients. The fractional Taylor expansion of the Jacobi polynomial series are computed and the associated coefficients - Jacobi–Taylor coefficients are introduced. We describe the fractional Jacobi–Taylor coefficients of trigonometric functions by finite sums involving Bessel functions and the fractional Jacobi coefficients. As special cases, the Chebyshev–Taylor coefficients and Legendre–Taylor coefficients are also presented. Fractional Jacobi coefficients are useful in describing the constants appearing in the fractional Taylor expansion of any eigenfunction involving Jacobi polynomials. The first fractional Jacobi coefficients are explicitly computed.
ResearchGate has not been able to resolve any references for this publication.