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Howard S. Cohl

Publications and source records attributed to Howard S. Cohl.

At least 19 recordsLinked to original sources
arXiv2026

Connection formulas for Askey--Wilson polynomials and related expansions

We derive and study expansions of and over the Askey--Wilson polynomials. We study these expansions and examine some limits to the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials and their $q^{-1}$-analogues. The Poisson kernel for the infinite discrete orthogonality relation for the $q^{-1}$-Al-Salam--Chihara polynomials is derived which in a special case reduces to the Gupta--Masson biorthogonal rational ${}_4\phi_3$-functions. This Poisson kernel implies new infinite series connection relations for the Askey--Wilson polynomials involving these rational ${}_4\phi_3$-functions. We also consider various interesting limits.

math.CA↗
arXiv2025

Integral representation for a product of two Jacobi functions of the second kind

By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation $\expe^{ix}\expe^{-ix}=1$ to those functions.

math.CA↗
arXiv2025

Terminating representations, transformations and summations for the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials

In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly setting one of the free parameters (not $q$) equal to zero until no parameters are left. These subfamilies (and their $q^{-1}$ counterparts) are the continuous dual $q$-Hahn, Al-Salam--Chihara, continuous big $q$-Hermite, and the continuous $q$-Hermite polynomials. 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 formulas which these polynomials satisfy. We then study the terminating transformation structure which are implied by the terminating representations of these polynomials. We conclude by describing the symmetry group structure of the $q$-Askey scheme.

math.CA↗
arXiv2025

Transformations and summations for bilateral basic hypergeometric series

We review and derive transformation and summation formulas for bilateral basic hypergeometric series. Our study focuses on consequences of certain bilateral extensions of two important results by Bailey, namely a transformation for very-well-poised $_8W_7$ series in terms of two balanced $_4\phi_3$ series, and a transformation connecting three $_8W_7$ series. Two rather recently discovered transformations of bilateral basic very-well-poised ${}_8\Psi_8$, one by Zhang and Zhang, the other by Wei and Yu, serve as the starting point of our investigations. From these transformations we work out interesting special cases that were not considered before, including explicit bilateral quadratic and cubic summations. We further explicitly record noteworthy lower-level transformations derived by taking suitable limits and deduce more transformations by exploiting the symmetry of the parameters in the series.

math.CA↗
arXiv2024

Asymptotics, orthogonality relations and duality for the $q$ and $q^{-1}$-symmetric polynomials in the $q$-Askey scheme

In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials and their dual counterparts which are connected with the big $q$-Jacobi polynomials, the little $q$-Jacobi polynomials and the $q$ and $q^{-1}$-Bessel polynomials. The $q^{-1}$-symmetric polynomials in the $q$-Askey scheme satisfy an indeterminate moment problem, satisfying an infinite number of orthogonality relations for these polynomials. Among the infinite number of orthogonality relations for the $q^{-1}$-symmetric families, we attempt to summarize those currently known. These fall into several classes, including continuous orthogonality relations and infinite discrete (including bilateral) orthogonality relations. Using symmetric limits, we derive a new infinite discrete orthogonality relation for the continuous big $q^{-1}$-Hermite polynomials. Using duality relations, we explore orthogonality relations for and from the dual families associated with the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. In order to give a complete description of the convergence properties for these polynomials, we provide the large degree asymptotics using the Darboux method for these polynomials. In order to apply the Darboux method, we derive a generating function with two free parameters for the $q^{-1}$-Al-Salam--Chihara polynomials which has natural limits to the lower $q^{-1}$-symmetric families.

math.CA↗
arXiv2024

Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme

In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a $q$-beta integral which comes from the continuous orthogonality relation for the $q^{-1}$-Askey--Wilson polynomials. In the $q\to 1^{-}$ limit, this $q$-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.

math.CA↗
arXiv2023

Orthogonality of the big $-1$ Jacobi polynomials for non-standard parameters

The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;α,β,c))_n$ have been classically defined for $α,β\in(-1,\infty)$, $c\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard. Assuming initial conditions $Q^{(0)}_0(x)=1$, $Q^{(0)}_{-1}(x)=0$, we consider the big $-1$ Jacobi polynomials as monic orthogonal polynomials which therefore satisfy the following three-term recurrence relation \[ xQ^{(0)}_n(x)=Q^{(0)}_{n+1}(x)+b_{n} Q^{(0)}_n(x)+ u_{n} Q^{(0)}_{n-1}(x), \quad n=0, 1, 2,\ldots. \] For standard parameters, the coefficients $u_n>0$ for all $n$. We discuss the situation where Favard's theorem cannot be directly applied for some positive integer $n$ such that $u_n=0$. We express the big $-1$ Jacobi polynomials for non-standard parameters as a product of two polynomials. Using this factorization, we obtain a bilinear form with respect to which these polynomials are orthogonal.

math.CA↗
arXiv2023

Multi-integral representations for Jacobi functions of the first and second kind

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to as Jacobi functions. In a similar fashion as associated Legendre functions, these break into two categories, functions which are analytically continued from the real line segment $(-1,1)$ and those continued from the real ray $(1, \infty)$. Using properties of Gauss hypergeometric functions, we derive multi-derivative and multi-integral representations for the Jacobi functions of the first and second kind.

math.CA↗
arXiv2023

The $q$ and $q^{-1}$-symmetric orthogonal polynomials in the $q$-Askey scheme, their dual polynomials and functions, orthogonality, generating functions and relations and nonterminating $q$-Chaundy double product representations

We derive double-product representations of nonterminating basic hypergeometric series using diagonalization, a method introduced by Theo William Chaundy in 1943. We refer to this result as the $q$-Chaundy theorem and several limiting $q\to 1^{-}$ cases are considered. Using the $q$-Chaundy theorem, we explore properties of the symmetric and $q^{-1}$-symmetric basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. These are the continuous dual $q$ and $q^{-1}$-Hahn polynomials, the $q$ and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big $q$ and $q^{-1}$-Hermite polynomials and the continuous $q$ and $q^{-1}$-Hermite polynomials. For instance, we show how many known (and unknown) generating functions can be easily derived for these polynomials. We also explore other methods to find generating functions for these polynomials. By applying the $q$-Chaundy theorem to the Ismail--Masson $q$-exponential generating function for continuous $q$ and $q^{-1}$-Hermite polynomials, we are able to derive alternative expansions of these generating functions, and from these, new terminating basic hypergeometric representations for the continuous $q$ and $q^{-1}$-Hermite polynomials. New quadratic transformations for the terminating basic hypergeometric series involved connect these representations. For the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials and as well their dual polynomials, which include the big and little $q$-Jacobi polynomials and the $q^{-1}$-Bessel polynomials, we discuss, and show how to exploit special orthogonality relations (integral and infinite series), connection formulas, and duality relations for these infinite families to derive new generating relations, and as well summation and integration formulas.

math.CA↗
arXiv2023

Double summation addition theorems for Jacobi functions of the first and second kind

In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation between symmetric, antisymmetric and odd-half-integer parameter values for the Jacobi functions with certain Gauss hypergeometric functions which satisfy a quadratic transformation, including associated Legendre, Gegenbauer and Ferrers functions of the first and second kind. We also introduce Olver normalizations of the Jacobi functions which are particularly useful in the derivation of expansion formulas when the parameters are integers. We introduce an application of the addition theorems for the Jacobi functions of the second kind to separated eigenfunction expansions of a fundamental solution of the Laplace-Beltrami operator on the compact and noncompact rank one symmetric spaces.

math.CA↗
arXiv2023

Special values for continuous $q$-Jacobi polynomials

We study special values for the continuous $q$-Jacobi polynomials and present applications of these special values which arise from bilinear generating functions, and in particular the Poisson kernel for these polynomials.

math.CA↗
arXiv2023

Discovery and Recognition of Formula Concepts using Machine Learning

Citation-based Information Retrieval (IR) methods for scientific documents have proven effective for IR applications, such as Plagiarism Detection or Literature Recommender Systems in academic disciplines that use many references. In science, technology, engineering, and mathematics, researchers often employ mathematical concepts through formula notation to refer to prior knowledge. Our long-term goal is to generalize citation-based IR methods and apply this generalized method to both classical references and mathematical concepts. In this paper, we suggest how mathematical formulas could be cited and define a Formula Concept Retrieval task with two subtasks: Formula Concept Discovery (FCD) and Formula Concept Recognition (FCR). While FCD aims at the definition and exploration of a 'Formula Concept' that names bundled equivalent representations of a formula, FCR is designed to match a given formula to a prior assigned unique mathematical concept identifier. We present machine learning-based approaches to address the FCD and FCR tasks. We then evaluate these approaches on a standardized test collection (NTCIR arXiv dataset). Our FCD approach yields a precision of 68% for retrieving equivalent representations of frequent formulas and a recall of 72% for extracting the formula name from the surrounding text. FCD and FCR enable the citation of formulas within mathematical documents and facilitate semantic search and question answering as well as document similarity assessments for plagiarism detection or recommender systems.

cs.IR↗
arXiv2023

Internal and external harmonics in bi-cyclide coordinates

The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé-Wangerin functions called internal and external bi-cyclide harmonics. An expansion for the fundamental solution of Laplace's equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansion in bi-spherical and prolate spheroidal coordinates.

math.CA↗
arXiv2022

Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey--Wilson polynomials. We express a certain integral of a ratio of infinite $q$-shifted factorials as a symmetric sum of two basic hypergeometric series with argument $q$. The result is then expressed as a $q$-integral. Examples of integral representations applied to the derivation of generating functions for Askey--Wilson are given and as well the computation of a missing generating function for the continuous dual $q$-Hahn polynomials.

math.CA↗
arXiv2022

Nonterminating transformations and summations associated with some q-Mellin--Barnes integrals

In many cases one may encounter an integral which is of $q$-Mellin--Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting $q$-Mellin--Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey--Wilson moments, nonterminating well-poised ${}_3ϕ_2$, nonterminating very-well-poised ${}_5W_4$, ${}_8W_7$, products of two nonterminating ${}_2ϕ_1$'s, square of a nonterminating well-poised ${}_2ϕ_1$, and nonterminating ${}_{12}W_{11}$ and ${}_{10}W_9$.

math.CA↗
arXiv2022

Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Cyclide Coordinates

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring 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 "flat rings". These internal and external flat-ring harmonic functions are expressed in terms of simply-periodic Lamé functions. In a limiting case we obtain the expansion of the fundamental solution in toroidal coordinates.

math.CA↗
arXiv2022

Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

Digital mathematical libraries assemble the knowledge of years of mathematical research. Numerous disciplines (e.g., physics, engineering, pure and applied mathematics) rely heavily on compendia gathered findings. Likewise, modern research applications rely more and more on computational solutions, which are often calculated and verified by computer algebra systems. Hence, the correctness, accuracy, and reliability of both digital mathematical libraries and computer algebra systems is a crucial attribute for modern research. In this paper, we present a novel approach to verify a digital mathematical library and two computer algebra systems with one another by converting mathematical expressions from one system to the other. We use our previously eveloped conversion tool (referred to as LaCASt) to translate formulae from the NIST Digital Library of Mathematical Functions to the computer algebra systems Maple and Mathematica. The contributions of our presented work are as follows: (1) we present the most comprehensive verification of computer algebra systems and digital mathematical libraries with one another; (2) we significantly enhance the performance of the underlying translator in terms of coverage and accuracy; and (3) we provide open access to translations for Maple and Mathematica of the formulae in the NIST Digital Library of Mathematical Functions.

cs.DL↗