Quantum Algebras and Quantum Groups in q-Special Function Theory

1993 ◽  
pp. 83-94
Author(s):  
Roberto Floreanini ◽  
Luc Vinet
Keyword(s):  
Quantum Groups ◽  
Special Function ◽  
Function Theory ◽  
Quantum Algebras

Using quantum algebras in q-special function theory

1992 ◽  
Vol 170 (1) ◽  
pp. 21-28 ◽  
Author(s):  
Roberto Floreanini ◽  
Luc Vinet
Keyword(s):  
Special Function ◽  
Function Theory ◽  
Quantum Algebras

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

scholarly journals Deformed traces and covariant quantum algebras for quantum groups GLqp(2) and

1992 ◽  
Vol 280 (3-4) ◽  
pp. 219-226 ◽  
Author(s):  
A.P. Isaev ◽  
R.P. Malik
Keyword(s):  
Quantum Groups ◽  
Quantum Algebras ◽  
Covariant Quantum

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

2016 ◽  
Vol 15 (10) ◽  
pp. 1650179 ◽  
Author(s):  
Yongjun Xu ◽  
Dingguo Wang ◽  
Jialei Chen
Keyword(s):  
Weyl Group ◽  
Quantum Groups ◽  
Braid Group ◽  
Group Actions ◽  
Necessary Condition ◽  
Quantum Algebras ◽  
Ore Extension ◽  
Schubert Cell ◽  
Sufficient And Necessary Condition

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

APPLICATIONS OF COMPUTER ALGEBRA IN QUANTUM GROUPS

1994 ◽  
Vol 05 (04) ◽  
pp. 701-706
Author(s):  
W.-H. STEEB
Keyword(s):  
Quantum Groups ◽  
Computer Algebra ◽  
Kronecker Product ◽  
Quantum Algebra ◽  
Theoretical Physics ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Helpful Tool

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

How joe gillis discovered combinatorial special function theory

1995 ◽  
Vol 17 (2) ◽  
pp. 65-66 ◽  
Author(s):  
Doron Zeilberger
Keyword(s):  
Special Function ◽  
Function Theory

scholarly journals THE SHARP BOUND FOR THE DEFORMATION OF A DISC UNDER A HYPERBOLICALLY CONVEX MAP

2006 ◽  
Vol 93 (2) ◽  
pp. 395-417 ◽  
Author(s):  
ROGER W. BARNARD ◽  
LEAH COLE ◽  
KENT PEARCE ◽  
G. BROCK WILLIAMS
Keyword(s):  
Extremal Function ◽  
Schwarzian Derivative ◽  
Special Function ◽  
Function Theory ◽  
Sharp Bound ◽  
Spherical Case ◽  
Euclidean Case ◽  
Step Down ◽  
The Extremal Function

We complete the determination of how far convex maps can deform discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mejía and Pommerenke in 2000. We find the sharp bound on the Schwarzian derivative of a hyperbolically convex function and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers published in the 1980s. Mejía and Pommerenke then produced partial results and a conjecture as to the extremal function in 2000. Their function maps onto a domain bounded by two proper geodesic sides, a ‘hyperbolic strip’. Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that, among the remaining candidates, the two-sided domain of Mejía and Pommerenke is in fact extremal. This correlates nicely with the euclidean and spherically convex cases in which the extremal is known to be a map onto a two-sided ‘strip’.

scholarly journals CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS

2012 ◽  
Vol 53 (3) ◽  
pp. 171-189 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Special Function ◽  
Function Theory ◽  
Applied Mathematics ◽  
Building Blocks ◽  
Review Article ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Mathematical Problems ◽  
Explicit Formulae ◽  
Prime Function

AbstractConformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

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