Families Of Special Functions Research Articles

A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions

AbstractThe Fox–Wright function is a very general form of function, covering many families of special functions as particular cases. Any special function can be used as a kernel for a fractional integral operator, but which of these operators will satisfy desiderata such as a semigroup property for composition? This paper provides a rigorous categorisation of all such fractional integral operators which have a semigroup property in any of their parameters. We discover that nearly all possible semigroup properties arise from the Chu–Vandermonde identity, with the Prabhakar fractional calculus emerging as one special case. For any integral operator with such a semigroup property, it is possible to construct a complete model of fractional calculus, including both integral and derivative operators which interact with each other in a natural way.

An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

The Harish-Chandra integral: An introduction with examples

This expository paper introduces the theory of Harish-Chandra integrals, a family of special functions that express the integral of an exponential function over the adjoint orbits of a compact Lie group. Originally studied in the context of harmonic analysis on Lie algebras, Harish-Chandra integrals now have diverse applications in many areas of mathematics and physics. We review a number of these applications, present several different proofs of Harish- Chandra’s celebrated exact formula for the integrals, and give detailed derivations of the specific integral formulae for all compact classical groups. These notes are intended for mathematicians and physicists who are familiar with the basics of Lie groups and Lie algebras but who may not be specialists in representation theory or harmonic analysis.

Quasinormal modes of Kerr–de Sitter black holes via the Heun function

This note addresses quasinormal mode (QNM) frequencies of four-dimensional asymptotically de Sitter rotating black holes. The main motivation is that Mathematica 12.1 has implemented a new family of special functions: Heun functions. Using the fact that Teukolsky’s equations for Kerr–de Sitter black holes are mapped to Heun’s equations, we are able to compute their QNM frequencies by the Heun function. In this approach, Mathematica normally evaluates these frequencies to arbitrary numerical precision in a few seconds. We further discuss an application to asymptotically flat rotating black holes.

Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals/derivatives, which is naturally arisen from exact representations of Chebyshev expansion coefficients, for optimal error estimates of Chebyshev approximations to functions with limited regularity. The essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterize in existing literature. We can then derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve the available results in usual Sobolev spaces of integer regularity exponentials in several senses. As a by-product, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi spectral approximations.

ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

Boundary value problems are considered on a simplex <em>F</em> in the real Euclidean space R<sup>2</sup>. The recent discovery of new families of special functions, orthogonal on <em>F</em>, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on <em>F</em>, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of <em>F</em> a Dirichlet condition is fulfilled and on the other Neumann’s works.

Fractional Edgeworth expansion: Corrections to the Gaussian-Lévy central-limit theorem.

In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to PDFs with a diverging variance, which converge to a Lévy α-stable density function. Our correction may be written by means of a series of fractional derivatives of the Lévy and the conjugate Lévy PDFs. This series expansion is general and applies also to the Gaussian regime. To describe the terms in the series expansion, we introduce a new family of special functions and briefly discuss their properties. We implement our generalization to the distribution of the momentum for atoms undergoing Sisyphus cooling, and show the improvement of our leading order approximation compared to previous approximations. In vicinity of the transition between Lévy and Gauss behaviors, convergence to asymptotic results slows down.

On the q-Charlier Multiple Orthogonal Polynomials

We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural properties. Raising and lowering operators as well as Rodrigues-type formulas are obtained. An explicit representation in terms of a $q$-analogue of the second of Appell's hypergeometric functions is given. A high-order linear $q$-difference equation with polynomial coefficients is deduced. Moreover, we show how to obtain the nearest neighbor recurrence relation from some difference operators involved in the Rodrigues-type formula.

Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

Suppose that d ≥ 1 is an integer, \({\alpha \in (0,d)}\) is a fixed parameter and let Iα be the fractional integral operator associated with d-dimensional Walsh–Fourier series on [0, 1)d. The paper contains the proof of the sharp weak-type estimate $$||I_\alpha(f)||_{L^{d/(d-\alpha),\infty}([0,1)^d)}\leq\frac{2^d-1}{(2^{d-\alpha}-1)(2^\alpha-1)}||f||_{L^1([0,1)^d)}.$$ The proof rests on Bellman-function-type method: the above estimate is deduced from the existence of a certain family of special functions.

Discretization of four types of Weyl group orbit functions

The discrete Fourier calculus of the four families of special functions, called C–, S–, Ss– and Sl -functions, is summarized. Functions from each of the four families of special functions are discretely orthogonal over a certain finite set of points. The generalizations of discrete cosine and sine transforms of one variable — the discrete Ss– and Sl-transforms of the group F4 — are considered in detail required for their exploitation in discrete Fourier spectral methods. The continuous interpolations, induced by the discrete expansions, are presented.

Analytical two-center integrals over Slater geminal functions

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.

Four types of special functions of G 2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G 2, are compared and described. Two of the four families (called here C- and S-functions) are well known. New results of the paper are in description of two new families of G 2-functions not found in the literature. They are denoted as S L - and S S -functions. It is shown that all four families have analogous useful properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space. They are also orthogonal as discrete functions when their values are sampled at the lattice points F M ⊂F and added up with appropriate weight function. The weight functions are determined for the new families. Products of 10 types among the four families of functions, namely CC, CS, SS, SS L , CS S , SS L , SS S , S S S S , S L S S and S L S L , are completely decomposable into the finite sum of the functions belonging to just one of the families. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.

Three-dimensional C-, S- and E-transforms

Three-dimensional continuous and discrete Fourier-like transforms, based on the three simple and four semisimple compact Lie groups of rank 3, are presented. For each simple Lie group, there are three families of special functions (C-, S- and E-functions) on which the transforms are built. Pertinent properties of the functions are described in detail, such as their orthogonality within each family, when integrated over a finite region F of the three-dimensional Euclidean space (continuous orthogonality), as well as when summed up over a lattice grid FM ⊂ F (discrete orthogonality). The positive integer M sets up the density of the lattice containing FM. The expansion of functions given either on F or on FM is the paper's main focus.

(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms

Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix elements are sine or cosine functions of one variable each. These functions are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space. Discrete and continuous orthogonality on F of the functions within each family allows one to introduce symmetrized and antisymmetrized multivariate Fourier-like transforms involving the symmetric and antisymmetric multivariate sine and cosine functions.

Global solutions and smoothing effects for semi-linear evolution equations in circular domains

The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the corresponding region. A forced Cahn–Hilliard-type equation in a unit disc Ω is considered as an example. The current work focuses on revealing the mechanism of nonlinear smoothing, i.e., on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem. To this end convolutions of Rayleigh functions with respect to the Bessel index are employed. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424; V. Varlamov, Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains, J. Math. Anal. Appl. 327 (2007) 1461–1478]. In order to reveal the effect of nonlinear smoothing, anisotropic Sobolev spaces H s , α ( Ω ) are introduced. They are based on the Sobolev spaces H s ( Ω ) “weighted” by tangential derivatives, so that the index α is responsible for the smoothness in θ and s is the usual Sobolev index. Global-in-time solutions of the mixed problem in question are constructed, and additional smoothness with respect to the angular coordinate is established in the anisotropic Sobolev spaces. The above mentioned special functions are essentially used for this purpose.

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Rayleigh functions σ l ( ν ) are defined as series in inverse powers of the Bessel function zeros λ ν , n ≠ 0 , σ l ( ν ) = ∑ n = 1 ∞ 1 λ ν , n 2 l , where l = 1 , 2 , … ; ν is the index of the Bessel function J ν ( x ) and n = 1 , 2 , … is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l ( m ) = ∑ k = − ∞ ∞ σ l ( | m − k | ) σ l ( | k | ) for l = 1 , 2 , … ; m = 0 , ± 1 , ± 2 , … , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699–725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424], where the properties of R 1 ( m ) were investigated. In the present work a general representation of R l ( m ) in terms of σ l ( ν ) is deduced. On the basis of this a representation for the function R 2 ( m ) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 ( m ) as | m | → ∞ . Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded domains with periodicity conditions in one coordinate. As an example of application of R l ( m ) a forced Boussinesq equation u t t − 2 b Δ u t = − α Δ 2 u + Δ u + β Δ ( u 2 ) + f with α , b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 ( m ) and R 2 ( m ) which are responsible for the nonlinear smoothing effect.

Special functions arising in the study of semi-linear equations in circular domains

Rayleigh functions are defined by the formula σ l ( ν ) = ∑ n = 1 ∞ 1 λ ν , n 2 l , where l = 1 , 2 , 3 , … ; λ ν , n ≠ 0 are zeros of the Bessel function J ν ( x ) and n = 1 , 2 , 3 , … , is the number of the zero. These functions appear in the classical problems of vibrating circular membranes, heat conduction in cylinders and diffraction through circular apertures. In the present paper it is shown that a new family of special functions, convolutions of Rayleigh functions with respect to the Bessel index, (1) R l ( m ) = ∑ p , k = - ∞ ; p + k = m ∞ ∑ q , s = 1 ∞ 1 λ p , q 2 l 1 λ k , s 2 l for l = 1 , 2 , … ; m = 0 , ± 1 , ± 2 , … , arises in constructing solutions of semi-linear evolution equations in circular domains (see also [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]). As an example of its application a forced Cahn-Hilliard equation is considered in a unit disc with homogeneous boundary and initial conditions. Construction of its global-in-time solutions involves the use of R 1 ( m ) and R 2 ( m ) . A general representation of R l ( m ) is deduced and on the basis of that a particular result for R 2 ( m ) is obtained convenient for computing its asymptotics as | m | → ∞ . The latter issue is important for establishing a function space to which a solution of the corresponding problem belongs.

On new families of integral transforms for the solution of partial differential equations

The integral transform method, associated with operatorial techniques, can be often applied in order to construct solutions of initial value problems for partial differential equations. In this article, we apply these techniques, introducing suitable families of special functions, in order to derive a very efficient procedure for obtaining solutions of particular families of evolution-type problems.