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Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green…

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl…

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric…

In this paper, we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators in two-dimensional setting in the following form: \begin{equation*} L_{\lambda }\left( f;x,y\right)…

A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…

In this work, we consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost…

Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown…

In this note we consider infinite series similar to the "strange" function $F(q)$ of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-Rhoades, Rolen-Schneider, and others in connection to quantum modular forms. We…

We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group $G$ by a family of groups $\mathcal{F}=\{(G_{\alpha},…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$, where…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the "overlap set" $\Ok$ is finite, and which are "invertible" on the attractor $A$, the sense that there is a continuous surjection…

The paper studies logarithmic convexity and concavity of power series with coefficients involving q-gamma functions or q-shifted factorials with respect to a parameter contained in their arguments. The principal motivating examples of such…

In the previous author's paper the Macdonald norm conjecture (including the famous constant term conjecture) was proved. This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation theorem is…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

In the conventional Schr\"{o}dinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent,…

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on…

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincar\'e series, we…

Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function…

Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and…