Searcharxiv⌕ Search

arXiv subjects

Wojciech Młotkowski

Publications and source records attributed to Wojciech Młotkowski.

16 recordsLinked to original sources
arXiv2025

Partial Chebyshev Polynomials and Fan Graphs

Motivated by the product formula of the Chebyshev polynomials of the second kind $U_n(x)$, we newly introduce the partial Chebyshev polynomials $U^{\mathrm{e}}_n(x)$ and $U^{\mathrm{o}}_n(x)$ and derive their basic properties, relations to the classical Chebyshev polynomials, and new factorization formulas for $U_n(x)$. In order to calculate the quadratic embedding constant (QEC) of a fan graph $K_1+P_n$, we derive a new polynomial $ϕ_n(x)$ which is factorized by partial Chebyshev polynomial $U^{\mathrm{e}}_n(x)$. We prove that $\mathrm{QEC}(K_1+P_n)$ is given in terms of the minimal zero of $ϕ_n(x)$, and obtain the explicit value of $\mathrm{QEC}(K_1+P_n)$ for an even $n$ and its reasonable exstimate for an odd $n$.

math.CO↗
arXiv2024

Quadratic embedding constants of fan graphs and graph joins

We derive a general formula for the quadratic embedding constant of a graph join $\bar{K}_m+G$, where $\bar{K}_m$ is the empty graph on $m\ge1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where $K_1=\bar{K}_1$ is the singleton graph and $P_n$ is the path on $n\ge1$ vertices, we show that $\mathrm{QEC}(K_1+P_n)=-\tildeα_n-2$, where $\tildeα_n$ is the minimal zero of a new polynomial $Φ_n(x)$ related to Chebyshev polynomials of the second kind. Moreover, for an even $n$ we have $\tildeα_n=\min\mathrm{ev}(A_n)$, where the right-hand side is the An minimal eigenvalue of the adjacency matrix $A_n$ of $P_n$. For an odd $n$ we show that $\min\mathrm{ev}(A_{n+1})\le\tildeα_n<\min\mathrm{ev}(A_n)$.

math.CO↗
arXiv2024

On quadratic embeddability of bipartite graphs and theta graphs

We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points and common endpoints. As a result, we provide an infinite family of primary graphs which are not quadratically embeddable.

math.CO↗
arXiv2022

On freely quasi-infinitely divisible distributions

Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study the class of freely quasi-infinitely divisible (FQID) distributions on $\mathbb{R}$, i.e. distributions which admit the free Lévy-Khintchine-type representation with signed Lévy measure. We prove several properties of the FQID class, some of them in contrast to those of the QID class. For example, a FQID distribution may have negative Gaussian part, and the total mass of its signed Lévy measure may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet, with the Lévy measure having nonzero negative part, which is at the same time classical and free characteristic triplet.

math.PR↗
arXiv2021

Probability distributions with rational free $R$-transform

We study the class $\mathcal{M}_{\mathrm{ratio}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the monotone convolution. We prove a sufficient condition that a rational function is the free $R$-transform of a probability distribution. Several examples are provided, including that of free deconvolution.

math.PR↗
arXiv2019

Some relatives of the Catalan sequence

We study a family of sequences $c_n(a_2,\ldots,a_r)$, where $r\ge2$ and $a_2,\ldots,a_r$ are real parameters. We find a sufficient condition for positive definiteness of the sequence $c_n(a_2,\ldots,a_r)$ and check several examples from OEIS. We also study relations of these sequences with the free and monotonic convolution.

math.CO↗
arXiv2019

Permutations of type $B$ with fixed number of descents and minus signs

We study three dimensional array of numbers $B(n,k,j)$, $0\le j,k\le n$, where $B(n,k,j)$ is the number of type $B$ permutations of order $n$ with $k$ descents and $j$ minus signs. We prove in particular, that $b(n,k,j):=B(n,k,j)/\binom{n}{j}$ is an integer and provide two combinatorial interpretations for these numbers.

math.CO↗
arXiv2018

On Quadratic Embedding Constants of Star Product Graphs

A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is non-positive. For a finite star product of (finite or infinite) graphs $G=G_1\star\dotsb \star G_r$ an estimate of $\mathrm{QEC}(G)$ is obtained after a detailed analysis of the minimal solution of a certain algebraic equation. For the path graph $P_n$ an implicit formula for $\mathrm{QEC}(P_n)$ is derived, and by limit argument $\mathrm{QEC}(\mathbb{Z})=\mathrm{QEC}(\mathbb{Z}_+)=-1/2$ is shown. During the discussion a new integer sequence is found.

math.CO↗
arXiv2017

Positive definite functions on Coxeter groups with applications to operator spaces and noncommutative probability

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.

math.OA↗
arXiv2015

New Eulerian numbers of type D

We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type $A$, $B$ and $D$.

math.CO↗
arXiv2015

A family of sequences of binomial type

For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $ν_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.

math.PR↗
arXiv2015

A Fuss-type family of positive definite sequences

We study a two-parameter family $a_{n}(p,t)$ of deformations of the Fuss numbers. We show a sufficient condition for positive definiteness of $a_n(p,t)$ and prove that some of the corresponding probability measures are infinitely divisible with respect to the additive free convolution.

math.PR↗
arXiv2011

The free Meixner class for pairs of measures

We investigate in more detail the two-state free convolution semigroups of pairs of measures whose Jacobi parameters are linear in the convolution parameter $t$. These semigroups were constructed in arXiv:1001.1540, where we also showed that measures with the analogous property for the usual and free convolution are exactly the classical, resp. free Meixner classes. The class of measures in this paper has not been considered explicitly before, but we show that it also has Meixner-type properties. Specifically, it appears in limit theorems, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.

math.OA↗
arXiv2011

Semigroups of distributions with linear Jacobi parameters

We show that a convolution semigroup of measures has Jacobi parameters polynomial in the convolution parameter $t$ if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha-Lukacs-type characterization, and is related to the $q=0$ case of quadratic harnesses.

math.CO↗