Related papers: $q$-rious and $q$-riouser
Some achievements of the late Andrzej Kossakowski in the field of statistical physics and quantum theory are presented. We recall his attempt to find an analytical solution of the 3-dimensional Ising model and present a problem concerning…
This is a survey of the diversity of problems in additive number theory. Equity requires the consideration of less currently popular problems, and suggests their inclusion in the additive canon. Of particular interest are problems about the…
This is a tutorial on duality properties of special functions, mainly of orthogonal polynomials in the ($q$-)Askey scheme. It is based on the first part of the 2017 R.P. Agarwal Memorial Lecture delivered by the author.
We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional `discrete' quantum mechanics, in which the Schr\"{o}dinger equation is a difference equation. It reproduces all the known ones whose…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra…
Racah and Wilson polynomials with dilated and translated argument are reparametrized such that the polynomials are continuous in the parameters as long as these are nonnegative, and such that restriction of one or more of the new parameters…
Zagier introduced the term "strange identity" to describe an asymptotic relation between a certain $q$-hypergeometric series and a partial theta function at roots of unity. We show that behind Zagier's strange identity lies a statement…
Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper…
The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the…
We deal with the classification problem of finite-dimensional representations of so called Askey--Wilson algebra in the case when $q$ is not a root of unity. We classify all representations satisfying certain property, which ensures…
The Askey-Wilson algebra and its relatives such as the Racah and Bannai-Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since proved ubiquitous. In particular they admit presentations…
We show that intertwining operators for the discrete Fourier transform form a cubic algebra $\mathcal{C}_q$ with $q$ a root of unity. This algebra is intimately related to the two other well-known realizations of the cubic algebra: the…
An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it…
A riddle is a question or statement with double or veiled meanings, followed by an unexpected answer. Solving riddle is a challenging task for both machine and human, testing the capability of understanding figurative, creative natural…
Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to…
In their recent preprint arXiv:2101.08308, Robert Dougherty-Bliss, Christoph Koutschan and Doron Zeilberger come up with a powerful strategy to prove the irrationality, in a quantitative form, of some numbers that are given as multiple…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
Recently, Andrews and EI Bachraoui discovered several companions for some famous $q$-series formulas, and derived some new identities involving partitions and overpartitions with distinct parts. In this paper, we shall refine their results…
In 1955 George Mackey suggested that there is a fundamental dichotomy in the unitary representation theory of locally compact second countable groups. He felt that there cannnot be a reasonable classification theory for the unitary…