Related papers: Sequences of symmetric functions of binomial type
We prove an inequality for polynomials applied in a symmetric way to non-commuting operators.
We develop the method of similar operators to study the spectral properties of unbounded perturbed linear operators that can be represented by matrices of various kinds. The class of operators under consideration includes various…
Recently it has been shown that any regular simple symmetric operator with deficiency indices (1,1) is unitarily equivalent to the operator of multiplication in a reproducing kernel Hilbert space of functions on the real line with the…
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta…
We study linear ordinary differential equations which are analytically parametrized on Hermitian symmetric spaces and invariant under the action of symplectic groups. They are generalizations of the classical Lam\'e equation. Our main…
We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate…
We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…
It is well known that there are entire functions whose orbit approximates any other entire function under the action of a sequence of translation operators . This result also holds for an uncountable family of sequences of translation…
We obtain the specialization of monomial symmetric functions on the alphabet (a-b)/(1-q). This gives a remarkable algebraic identity, and four new developments for the Macdonald polynomial associated with a row. The proofs are given in the…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
In combinatorics, P\'{o}lya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of P\'{o}lya's…
The theory of $q$-analogs frequently occurs in a number of areas, including the fractals and dynamical systems. The $q$-derivatives and $q$-integrals play a prominent role in the study of $q$-deformed quantum mechanical simple harmonic…
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions…
We construct a generalization of the theory of symmetric functions involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
This paper develops three related combinatorial results for Dyck-type sequences. First, it constructs a row-insertion algorithm for dual Dyck sequences and extends it to Dyck tableaux. This construction gives a weight-preserving bijection…
Polynomial functors are sums of covariant representable functors from the category of sets to itself. They have a robust theory with many applications -- from operads and opetopes to combinatorial species. In this paper, we define a…
If $T$ is a (densely defined) self-adjoint operator acting on a complex Hilbert space $\mathcal{H}$ and $I$ stands for the identity operator, we introduce the delta function operator $\lambda \mapsto \delta \left(\lambda I-T\right) $ at…
We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The…