Related papers: Well--Poised Macdonald Functions W_lambda and Jack…
We give an alternative proof of an elliptic summation formula of type $BC_n$ by applying the fundamental $BC_n$ invariants to the study of Jackson integrals associated with the summation formula.
We propose three kinds of explicit formulas for the elliptic lambda function by the elliptic modular function. Further, we derive incredible cubic identities as a corollary of our explicit formulas and evaluate some singular values of the…
We observe that approximate copies of the function $\Lambda _{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} \Lambda _{n}(x)=\exp \left( -x_{1}-\pi \sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description…
The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $\theta_k$ are introduced, and the Hamiltonian is represented as a…
Elliptic Macdonald polynomials of sl(2)-type and level 2 are introduced. Suitable limits of elliptic Macdonald polynomials are the standard Macdonald polynomials and conformal blocks. Identities for elliptic Macdonald polynomials, in…
The Macdonald polynomials with prescribed symmetry are obtained from the nonsymmetric Macdonald polynomials via the operations of $t$-symmetrisation, $t$-antisymmetrisation and normalisation. Motivated by corresponding results in Jack…
We consider the Jack--Laurent symmetric functions for special values of parameters p_0=n+k^{-1}m, where k is not rational and m and n are natural numbers. In general, the coefficients of such functions may have poles at these values of p_0.…
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and…
Coefficients in the expansions of the form $\partial P_{n}(\lambda;z)}/\partial\lambda=\sum_{k=0}^{n}a_{nk}(\lambda)P_{k}(\lambda;z)$, where $P_{n}(\lambda;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi)…
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic…
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials,…
Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
The two variable Kostka functions are the scalar products of the Macdonald polynomials with the Schur polynomials with respect to the scalar product which makes the Hall-Littlewood polynomials pairwise orthogonal. A conjecture of Macdonald…
Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t in [0,1). We prove several results about these…
We give a domination condition implying good-$\lambda$ and exponential inequalities for couples of measurable functions. Those inequalities recover several classical and new estimations involving some operators in Harminic Analysis. Among…
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…
Let $m_{\lambda }$ be the monomial symmetric functions, $ \lambda $ being an integer partition of $n\in \mathbb{N}^{\ast }$. For the specialization corresponding to the $q$-deformation of the exponential, we prove that each $m_{\lambda }$…