Related papers: Continuous Analogues for the Pochhammer Symbol

In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.

The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.

Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…

A systematic study of the contributions at infinity for the cohomology of variations of polarized Hodge structures over quasicompact K\"ahler manifolds. Several isomorphisms between different cohomologies given.

Using local cohomology and algebraic D-Modules, we generalize a comparison theorem between relative de Rham cohomology and Dwork cohomology due to N. Katz, A. Adolphson and S. Sperber.

One of the most famous results of Dedekind is the transformation law of $\log \Delta(z)$. After a half-century, Rademacher modified Dedekind's result and introduced an $\mathrm{SL}_2(\mathbb{Z})$-conjugacy class invariant (integer-valued)…

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked…

These notes follow my articles [1, 6], and give some new important details. We propose here a new combinatorial method of encoding of measure spaces with measure preserving transformations, (or groups of transformations) in order to give…

Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's…

We give a combinatorial extension of the classical inequalities of Maclaurin about symmetric functions of several variables. We discuss two problems - one analytical and another combinatorial - and show that they are in some sense…

We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how…

Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs. Our main result is an infinite family of counterexamples to…

We introduce and study analogues of expander and hyperfinite graph sequences in the context of directed acyclic graphs, which we call "extender" and "hypershallow" graph sequences, respectively. Our main result is a probabilistic…

The paper presents a proof of the Hodge Riemann relations for the combinatorial intersection cohomology of a polytope, as fist given by K.Karu, in terms of geometric operations on polytopes.

DHOL is an extensional, classical logic that equips the well-known higher-order logic (HOL) with dependent types. This allows for concise encodings of important domains like size-bounded data structures, category theory, or proof theory.…

We make a progress towards describing the commutants of Toeplitz operators with harmonic symbols on the Bergman space over the unit disk. Our work greatly generalizes several partial results in the field.

In the paper a theorem of Piccard's type is proved and, consequently, the continuity of $\mathcal{D}$-measurable polynomial functions of $n$-th order as well as $\mathcal{D}$-measurable $n$-convex functions is shown. The paper refers to the…

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…