Related papers: Generalized Macdonald functions and quantum toroid…
The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial $P_\kappa(z)$ are known explicitly. These formulas generalise the known $r=1$ case of the Pieri-type formulas…
We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric…
We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct…
In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(\alpha,q)$-Fock space, which satisfy the commutation relation $$…
We introduce and study the quantum toroidal algebra $\mathcal{E}_{m|n}(q_1,q_2,q_3)$ associated with the superalgebra $\mathfrak{gl}_{m|n}$ with $m\neq n$, where the parameters satisfy $q_1q_2q_3=1$. We give an evaluation map. The…
We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop…
This paper is devoted to studying the boundedness of multilinear operartors and their commutators on generalized weighted Morrey spaces, which includes multilinear fractional maximal operator and multilinear fractional integral operator.…
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized…
We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with…
We define cylindric generalisations of skew Macdonald functions when one of their parameters is set to zero. We define these functions as weighted sums over cylindric skew tableaux: fixing two integers n>2 and k>0 we shift an ordinary skew…
In this paper, we establish a generalized Taylor expansion of a given function $f$ in the form $\displaystyle{f(x) = \sum_{j=0}^m c_j^{\alpha,\rho}\left(x^\rho-a^\rho\right)^{j\alpha} + e_m(x)}$ \noindent with $m\in \mathbb{N}$,…
We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians…
A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of…
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal…
We introduce a family of algebras $\mathcal{A}_{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$, wherein the parameters are tuned in a…
The Fuss-Catalan numbers are a generalization of the Catalan numbers. They enumerate a large class of objects and in particular m-Dyck paths and m+1-ary trees. Recently, F. Bergeron defined an analogue for generic m of the Tamari order on…
For the rational Baker-Akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the self-duality property of the usual Gaussian…
The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators…
We propose integral representations of the Whittaker functions for the classical Lie algebras sp(2l), so(2l) and so(2l+1). These integral representations generalize the integral representation of gl(l+1)-Whittaker functions first introduced…