Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of $q$-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

On the Two q-Analogue Logarithmic Functions

There is a simple, multi-sheet Riemann surface associated with e_q(z)'s inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be defined. However, the topology of the Riemann surface for ln_q(w) changes each time "q" increases above the collision point of a pair of the turning points of e_q(x). There is also a power series representation for ln_q(1+w). An infinite-product representation for e_q(z) is used to obtain the ordinary natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power series in terms of values of these sum rules. The values of the sum rules for the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the usual Bernoulli numbers.Comment: This is the final version to appear in J.Phys.A: Math. & General. Some explict formulas added, and to update the reference

Expansion of analytic functions in q-orthogonal polynomials.

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey-Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at +/-1. Some examples of explicit expansions are discussed

Relativistic Coulomb Integrals and Zeilberger’s Holonomic Systems Approach II

Abstract. We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way

Higgs Boson Studies at the Tevatron

We combine searches by the CDF and D0 Collaborations for the standard model Higgs boson with mass in the range 90--200 GeV$/c^2$ produced in the gluon-gluon fusion, $WH$, $ZH$, $t{\bar{t}}H$, and vector boson fusion processes, and decaying in the $H\rightarrow b{\bar{b}}$, $H\rightarrow W^+W^-$, $H\rightarrow ZZ$, $H\rightarrow\tau^+\tau^-$, and $H\rightarrow \gamma\gamma$ modes. The data correspond to integrated luminosities of up to 10 fb$^{-1}$ and were collected at the Fermilab Tevatron in $p{\bar{p}}$ collisions at $\sqrt{s}=1.96$ TeV. The searches are also interpreted in the context of fermiophobic and fourth generation models. We observe a significant excess of events in the mass range between 115 and 140 GeV/$c^2$. The local significance corresponds to 3.0 standard deviations at $m_H=125$ GeV/$c^2$, consistent with the mass of the Higgs boson observed at the LHC, and we expect a local significance of 1.9 standard deviations. We separately combine searches for $H \to b\bar{b}$, $H \to W^+W^-$, $H\rightarrow\tau^+\tau^-$, and $H\rightarrow\gamma\gamma$. The observed signal strengths in all channels are consistent with the presence of a standard model Higgs boson with a mass of 125 GeV/$c^2$