Related papers: A Classical Limit of Noumi's $q$-Integral Operator

We study a family of "classical" orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl-type. These polynomials can be obtained from the little $q$-Jacobi…

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…

We prove several results on monodromies associated to Macdonald integrals, that were used in our previous work on the finite field analogue of a conjecture of Macdonald. We also give a new proof of our formula expressing recursively the…

By applying the derivative operator to the corresponding hypergeometric form of a $q$-series transformation due to Andrews [1,Theorem 4], we establish a general harmonic number identity. As the special cases of it, several interesting…

We prove that the radial part of the class one Whittaker function on a split semisimple Lie group can be obtained as an appropriate limit of the Heckman-Opdam hypergeometric function.

We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of…

In this paper, we investigate on a bounded open set of $\mathbb{R}^N$ with smooth boundary, an eigenvalue problem involving the sum of nonlocal operators $(-\Delta)_p^{s_1}+ (-\Delta)_q^{s_2}$ with $s_1,s_2\in (0,1)$, $p,q\in (1,\infty)$…

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set…

The integral of the Wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval…

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric $q$-Whittaker functions. In particular, we show that the series part of the nonsymmetric $q$-Whittaker…

We introduce difference operators on the space of symmetric functions which are a natural generalization of the $(q,t)$-Macdonald operators. In the $t\to\infty$ limit, they satisfy the $A_{N-1}$ quantum $Q$-system. We identify the elements…

Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.

We give generalizations of a finite version of Euler's pentagonal number theorem and of a q-identity of Gauss.

The q-Bessel-Macdonald functions of kinds 1, 2 and 3 are considered. Their representations by classical integral are constructed.

In this paper, we have introduced the Prabhakar fractional $q$-integral and $q$-differential operators. We first study the semi-group property of the Prabhakar fractional $q$-integral operator, which allowed us to introduce the…

In this paper we give a convolution identity for the complete and elementary symmetric functions. This result can be used to proving and discovering some combinatorial identities involving $r$-Stirling numbers, $r$-Whitney numbers and…

Let $q=e^{2\pi i\tau}$, $\Im\tau>0$, $x=e^{2\pi i\xi}\in\CC$ and $(x;q)_\infty=\prod_{n\ge 0}(1-xq^n)$. Let $(q,x)\mapsto(q^*,\iota_q x)$ be the classical modular substitution given by $q^*=e^{-2\pi i/\tau}$ and $\iota_q x=e^{2\pi…

Given constants $x, \nu \in \mathbb{C}$ and the space $\mathscr{H}_0$ of entire functions in $\mathbb{C}$ vanishing at $0$, we consider the integro-differential operator $$ \mathfrak{L} = \left ( \frac{x \, \nu(1-\nu)}{1-x} \right ) \;…

We investigate the homogeneous symmetric Macdonald polynomials $P_\lambda(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\lambda(\X;q,q^k)$ and $P_\lambda(\frac{1-q}{1-q^k}\X;q,q^k)$. As a…

In 1994, Kac and Wakimoto found the denominator identity for classical affine Lie superalgebras, generalizing that for affine Lie algebras. As an application, they obtained power series identities for some powers of $\triangle(q)$, where…