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Howard S. Cohl

Publications and source records attributed to Howard S. Cohl.

At least 37 records · Page 2Linked to original sources
arXiv2022

Gegenbauer expansions and addition theorems for a binomial and logarithmic fundamental solution of the even-dimensional Euclidean polyharmonic equation

On even-dimensional Euclidean space for integer powers of the Laplace operator greater than or equal to half the dimension, a fundamental solution of the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer polynomial expansions of these fundamental solutions are obtained through a limit applied to Gegenbauer expansions of a power-law fundamental solution of the polyharmonic equation. This limit is accomplished through parameter differentiation. By combining these results with previously derived azimuthal Fourier series expansions for these binomial and logarithmic fundamental solutions, we are able to obtain addition theorems for the azimuthal Fourier coefficients. These logarithmic and binomial addition theorems are expressed in Vilenkin polyspherical geodesic polar coordinate systems and as well in generalized Hopf coordinates on spheres in arbitrary even dimensions.

math.CA↗
arXiv2022

Symmetry of terminating series representations of the Askey-Wilson polynomials

In this paper, we explore the symmetric nature of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. In particular we identify and classify the set of 4 and 7 equivalence classes of terminating balanced ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ basic hypergeometric series which are connected with the Askey--Wilson polynomials. We study the inversion properties of these equivalence classes and also identify the connection of both sets of equivalence classes with the symmetric group $S_6$, the symmetry group of the terminating balanced ${}_4ϕ_3$. We then use terminating balanced ${}_4ϕ_3$ and terminating very-well poised ${}_8W_7$ transformations to give a broader interpretation of Watson's $q$-analog of Whipple's theorem and its converse. We give a broad description of the symmetry structure of the terminating basic hypergeometric series representations of the Askey--Wilson polynomials.

math.CA↗
arXiv2022

Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lamé-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lamé-Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé-Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.

math.CA↗
arXiv2021

Automated Symbolic and Numerical Testing of DLMF Formulae using Computer Algebra Systems

We have developed an automated procedure for symbolic and numerical testing of formulae extracted from the NIST Digital Library of Mathematical Functions (DLMF). For the NIST Digital Repository of Mathematical Formulae, we have developed conversion tools from semantic LaTeX to the Computer Algebra System (CAS) Maple which relies on Youssef's part-of-math tagger. We convert a test data subset of 4,078 semantic LaTeX DLMF formulae %extracted from the DLMF to the native CAS representation and then apply an automated scheme for symbolic and numerical testing and verification. Our framework is implemented using Java and Maple. We describe in detail the conversion process which is required so that the CAS can correctly interpret the mathematical representation of the formulae. We describe the improvement of the effectiveness of our automated scheme through incremental enhancement (making more precise) of the mathematical semantic markup for the formulae.

cs.DL↗
arXiv2021

Semantic Preserving Bijective Mappings of Mathematical Formulae between Document Preparation Systems and Computer Algebra Systems

Document preparation systems like LaTeX offer the ability to render mathematical expressions as one would write these on paper. Using LaTeX, LaTeXML, and tools generated for use in the National Institute of Standards (NIST) Digital Library of Mathematical Functions, semantically enhanced mathematical LaTeX markup (semantic LaTeX) is achieved by using a semantic macro set. Computer algebra systems (CAS) such as Maple and Mathematica use alternative markup to represent mathematical expressions. By taking advantage of Youssef's Part-of-Math tagger and CAS internal representations, we develop algorithms to translate mathematical expressions represented in semantic LaTeX to corresponding CAS representations and vice versa. We have also developed tools for translating the entire Wolfram Encoding Continued Fraction Knowledge and University of Antwerp Continued Fractions for Special Functions datasets, for use in the NIST Digital Repository of Mathematical Formulae. The overall goal of these efforts is to provide semantically enriched standard conforming MathML representations to the public for formulae in digital mathematics libraries. These representations include presentation MathML, content MathML, generic LaTeX, semantic LaTeX, and now CAS representations as well.

cs.IR↗
arXiv2021

MathTools: An Open API for Convenient MathML Handling

Mathematical formulae carry complex and essential semantic information in a variety of formats. Accessing this information with different systems requires a standardized machine-readable format that is capable of encoding presentational and semantic information. Even though MathML is an official recommendation by W3C and an ISO standard for representing mathematical expressions, we could identify only very few systems which use the full descriptiveness of MathML. MathML's high complexity results in a steep learning curve for novice users. We hypothesize that this complexity is the reason why many community-driven projects refrain from using MathML, and instead develop problem-specific data formats for their purposes. We provide a user-friendly, open-source application programming interface for controlling MathML data. Our API is written in JAVA and allows to create, manipulate, and efficiently access commonly needed information in presentation and content MathML. Our interface also provides tools for calculating differences and similarities between MathML expressions. The API also allows to determine the distance between expressions using different similarity measures. In addition, we provide adapters for numerous conversion tools and the canonicalization project. Our toolkit facilitates processing of mathematics for digital libraries, without the need to obtain XML expertise.

cs.DL↗
arXiv2021

On the relation between Gegenbauer polynomials and the Ferrers function of the first kind

Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then compute Rodrigues-type and orthogonality relations for Ferrers functions of the first and second kinds. In the remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, some asymptotic expansions of Mehler-Heine type, Christoffel-Darboux summation formulas, and infinite series closure relations (Dirac delta distribution).

math.CA↗
arXiv2021

Discovering Mathematical Objects of Interest -- A Study of Mathematical Notations

Mathematical notation, i.e., the writing system used to communicate concepts in mathematics, encodes valuable information for a variety of information search and retrieval systems. Yet, mathematical notations remain mostly unutilized by today's systems. In this paper, we present the first in-depth study on the distributions of mathematical notation in two large scientific corpora: the open access arXiv (2.5B mathematical objects) and the mathematical reviewing service for pure and applied mathematics zbMATH (61M mathematical objects). Our study lays a foundation for future research projects on mathematical information retrieval for large scientific corpora. Further, we demonstrate the relevance of our results to a variety of use-cases. For example, to assist semantic extraction systems, to improve scientific search engines, and to facilitate specialized math recommendation systems. The contributions of our presented research are as follows: (1) we present the first distributional analysis of mathematical formulae on arXiv and zbMATH; (2) we retrieve relevant mathematical objects for given textual search queries (e.g., linking $P_{n}^{(α, β)}\!\left(x\right)$ with `Jacobi polynomial'); (3) we extend zbMATH's search engine by providing relevant mathematical formulae; and (4) we exemplify the applicability of the results by presenting auto-completion for math inputs as the first contribution to math recommendation systems. To expedite future research projects, we have made available our source code and data.

cs.DL↗
arXiv2021

Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at $[1,\infty)$. In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

math.CA↗
arXiv2020

Terminating basic hypergeometric representations and transformations for the Askey-Wilson polynomials

In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. From the terminating basic hypergeometric representations of these polynomials, and due to symmetry in their free parameters, we are able to exhaustively explore the terminating basic hypergeometric transformation formulae which these polynomials satisfy.

math.CA↗
arXiv2020

Multi-integral representations for associated Legendre and Ferrers functions

For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind including parameter values for which this function is identically zero.

math.CA↗
arXiv2019

Semantic Preserving Bijective Mappings for Expressions involving Special Functions in Computer Algebra Systems and Document Preparation Systems

Purpose: Modern mathematicians and scientists of math-related disciplines often use Document Preparation Systems (DPS) to write and Computer Algebra Systems (CAS) to calculate mathematical expressions. Usually, they translate the expressions manually between DPS and CAS. This process is time-consuming and error-prone. Our goal is to automate this translation. This paper uses Maple and Mathematica as the CAS, and LaTeX as our DPS. Design/methodology/approach: Bruce Miller at the National Institute of Standards and Technology (NIST) developed a collection of special LaTeX macros that create links from mathematical symbols to their definitions in the NIST Digital Library of Mathematical Functions (DLMF). We are using these macros to perform rule-based translations between the formulae in the DLMF and CAS. Moreover, we develop software to ease the creation of new rules and to discover inconsistencies. Findings: We created 396 mappings and translated 58.8% of DLMF formulae (2,405 expressions) successfully between Maple and DLMF. For a significant percentage, the special function definitions in Maple and the DLMF were different. Therefore, an atomic symbol in one system maps to a composite expression in the other system. The translator was also successfully used for automatic verification of mathematical online compendia and CAS. Our evaluation techniques discovered two errors in the DLMF and one defect in Maple. Originality: This paper introduces the first translation tool for special functions between LaTeX and CAS. The approach improves error-prone manual translations and can be used to verify mathematical online compendia and CAS.

cs.DL↗
arXiv2018

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators $\big({-}Δ\pmβ^2\big)$ in $d$-dimensional, $R$-radius hyperbolic ${\mathbf H}_R^d$ and hyperspherical ${\mathbf S}_R^d$ geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of $\big({-}Δ\pm β^2\big)$ on ${\mathbf H}_R^d$, $\big({-}Δ+β^2\big)$ on ${\mathbf S}_R^d$, and present two candidate fundamental solutions for $\big({-}Δ-β^2\big)$ on ${\mathbf S}_R^d$. Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.

math.AP↗
arXiv2018

Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf S}_R^d$ with $R>0$, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by $d/2-1$ and $1-d/2$ respectively, with real argument $x\in(-1,1)$.

math-ph↗
arXiv2018

Generalizations of generating functions for basic hypergeometric orthogonal polynomials

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one free parameter to them. In particular, we generalize generating functions for the Askey-Wilson, continuous $q$-ultraspherical/Rogers, little $q$-Laguerre/Wall, and $q$-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and $q$-integrals.

math.CA↗
arXiv2018

On a generalization of the Rogers generating function

We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corresponding generalized expansions for the continuous $q$-Jacobi, and Wilson polynomials with two and four free parameters respectively. Comparing the coefficients of the Askey-Wilson expansion to our continuous $q$-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric series connecting ${}_2ϕ_1$ and ${}_8ϕ_7$.

math.CA↗