On the maximal difference between an element and its inverse in residue rings

We investigate the distribution of where

Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on . We also present some heuristic arguments showing that these lower bounds are probably tight, and thus our technique can be a more appropriate tool to study than a more traditional way using exponential sums.

     

Comparison of rigid and flexible simple point charge water models at supercritical conditions

This study investigates the differences between the predictions of various properties of rigid and flexible simple point charge water models at supercritical conditions. Molecular dynamics simulations were conducted for supercritical water in a temperature range of 773–1073 K and densities in the range 115–659 kg/m³. We present thermodynamic data, pair correlation functions, self-diffusivity, power spectra, dielectric constants, and variaous measures of hydrogen bonding at different state conditions. The flexible water model performs better in predicting the pressures along the supercritical isotherms simulated. Agreement between experimental and calculated dielectric constants is superior for the flexible water model, particularly at high densities. The flexible model exhibits a greater degree of hydrogen bonding and more persistent hydrogen bonds than does the rigid model. The structural features of supercritical water at high densities are identical for the two water models. At low densities, however, the flexible potential exhibits pair correlation functions with enhanced peaks. Inclusion of flexibility in the potential model does not result in a significant shift in the position of the rotational/librational peak in the power spectrum. The self-diffusivities obtained from the simulations are within the accuracy of the experimental values for both the rigid and flexible models. On balance the inclusion of flexibility improves agreement with the properties of real supercritical water while incurring little or no additional computational burden. © 1996 by John Wiley & Sons, Inc.     

Ladder operators for Szego polynomials and related biorthogonal rational functions

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

     

A -analogue of the 9- symbols and their orthogonality

We introduce a q-analogue of Wigner’s 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.     

Inverse Operators, q-Fractional Integrals, and q-Bernoulli Polynomials

We introduce operators of q-fractional integration through inverses of the Askey–Wilson operator and use them to introduce a q-fractional calculus. We establish the semigroup property for fractional integrals and fractional derivatives. We study properties of the kernel of q-fractional integral and show how they give rise to a q-analogue of Bernoulli polynomials, which are now polynomials of two variables, x and y. As q→1 the polynomials become polynomials in x−y, a convolution kernel in one variable. We also evaluate explicitly a related kernel of a right inverse of the Askey–Wilson operator on an L² space weighted by the weight function of the Askey–Wilson polynomials.     

Some systems of multivariable orthogonal q-Racah polynomials

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D50; Secondary—33C50 Supported in part by NSERC grant #A6197.     

Linearization formula for the product of associated q-ultraspherical polynomials

We obtain an explicit formula for the linearization coefficient of the product of two associated q-ultraspherical polynomials in terms of a multiple of a balanced, terminating very-well-poised ₁₀φ₉ series. We also discuss the nonnegativity properties of the coefficients as well as some special cases. 2000 Mathematics Subject Classification Primary—33D45; Secondary—33D8 This work was supported in part by an NSERC grant A6197.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

Some Cubic Summation and Transformation Formulas

q-Analogues of two cubic summation formulas that have recently caught the attention of Bill Gosper are found by first showing their connection with the q-binomial formula and then using some known transformation formulas. We also find a q-extension of a cubic transformation formula involving Gauss' hypergeometric function, which turns out to be a relation between balanced and very-well-poised 109 series.     

A q-integral representation of Rogers' q-ultraspherical polynomials and some applications

Aq-integral representation of Rogers'q-ultraspherical polynomialsC _n (x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating₃ φ ₂ series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel ofC _n (x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of theq-ultraspherical polynomials.