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Paweł J. Szabłowski

Publications and source records attributed to Paweł J. Szabłowski.

At least 19 recordsLinked to original sources
arXiv2026

Moment sequences and difference equations

We recall the definition and properties of a moment sequence and show that all real sequences whose Hankel matrices have finite rank (see definition in the sequel) satisfy a homogeneous linear equation with constant coefficients. Then we analyze the cases in which a difference equation with constant coefficients and suitably chosen initial conditions and having as an input a positive moment sequence has a solution that is a positive moment sequence. We give one general simple result and give many examples illustrating the theory. The main result states that the roots of the odd multiplicity of the characteristic equation must lie outside the support of the measure that produces the moment sequence that is in the input and the initial conditions suitably chosen.

math.CA↗
arXiv2026

The conditional expectation of the product of the first $n-1$ Hermite polynomials in a multivariate normal distribution with respect to the $n$-th variable. A fresh perspective on the Kibble-Slepian formula

We calculate the conditional expectation of $\prod_{j=1}^{n-1}H_{k_{j}}% (X_{j})$ given $X_{n}=z,$ if random vector $(X_{1},\ldots,X_{n})^{T}$ has multivariate normal distribution and $H_{n}(x) $ denotes $n$-th Hermite polynomial. This expectation is a polynomial in $z$ of order $\sum_{j=1}% ^{n-1}k_{j}$. Our formula has an iterative form with respect to $n$. We also present some auxiliary observations concerning the expansion of the density of the $n$-dimensional normal distribution in the series of the Hermite polynomials. Mostly concerning the properties of the coefficients of this expansion. To perform these calculations, we give a few auxiliary formulas concerning Hermite polynomials and multivariate normal distributions. We apply this result to obtain exact, simple forms of these expansions for $n=2$ and $3$, thus looking at known results from a different perspective.

math.PR↗
arXiv2026

A few identities and integrals involving Pochhammer symbols, Jacobi polynomials, and the generalized hypergeometric function

We first present some identities involving the Pochhammer symbol (rising factorial). We also recall and present some new properties of the Jacobi polynomials. We use them to expand a general hypergeometric function in an orthogonal series of Jacobi polynomials. Then we use these expansions to discover closed forms for certain integrals of Jacobi polynomials that are multiplied by a generalized hypergeometric function and a Beta density. We can also obtain closed forms for some series involving rising factorials that generalise binomial series by using well-known properties of the hypergeometric function. In particular, we get a few new, nontrivial identities involving the Pochhammer symbol. We can also derive some simplifying identities for generalized hypergeometric functions.

math.CA↗
arXiv2026

Applications of infinite lower triangular matrices and their group structure in combinatorics and the theory of orthogonal polynomials

Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect to matrix multiplication, the invertivle elements of the set form a group. The set becomes an algebra (non-commutative in fact) with unity when all three operations are considered together. We indicate important properties of the algebraic structures obtained in this way. In particular, we indicate several sub-groups or sub-rings. Among sub-groups, we consider the group of Riordan matrices and indicate its several sub-groups. We show a variety of examples (approximately 20) of matrices that are composed of the sequences of important polynomial or number families as entries of certain lower-triangular infinite matrices. New, significant relationships between these families can be discovered by applying well-known matrix operations like multiplication and inverse calculation to this representation. The paper intends to compile numerous simple facts about the lower-triangular matrices, specifically the family of Rionian matrices, and briefly review their properties.

math.CO↗
arXiv2025

A few finite and infinite identities involving Pochhammer and $q$- Pochhammer symbols obtained via analytical methods

We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on infinite expansions of the ratio of densities in a Fourier series of polynomials orthogonal with respect to the density in the denominator. We want a unified approach to justify many known and unknown identities. The purpose of studying these identities is to simplify calculations occurring while dealing with Pochhammer and q-Pochhammer symbols. Additional possible applications of the results presented in the paper are applications within the Combinatorics and the transformation formulae of hypergeometric and basic hypergeometric functions.

math.AP↗
arXiv2024

On a class of bivariate distributions built of q-ultraspherical polynomials

Our primary result concerns the positivity of specific kernels constructed using the $q$-ultraspherical polynomials. In other words, it concerns a two-parameter family of bivariate, compactly supported distributions. Moreover, this family has a property that all its conditional moments are polynomials in the conditioning random variable. The significance of this result is evident for individuals working on distribution theory, orthogonal polynomials, $q$-series theory, and the so-called quantum polynomials. Therefore, it may have a limited number of interested researchers. That is why, we put our results into a broader context. We recall the theory of Hilbert-Schmidt operators and the idea of Lancaster expansions of the bivariate distributions absolutely continuous with respect to the product of their marginal distributions. Applications of Lancaster expansion can be found in Mathematical Statistics or the creation of Markov processes with polynomial conditional moments (the most well-known of these processes is the famous Wiener process).

math.FA↗
arXiv2023

Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths

We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are important because they are relatively easy to simulate. One does not have to care about the distribution of their jumps which is always difficult to find. Among them are the Ornstein-Uhlenbeck process, the Gamma process, the process with Arcsin margins and the Theta function transition densities and others. We give a simple criterion for the stationary process to have a continuous path modification expressed in terms of skewness and excess kurtosis of the marginal distribution.

math.PR↗
arXiv2023

Moment sequences of Beta distribution

We recall some basic properties of the Beta distribution and some of its modifications. We identified around $20$ of the moment sequences of Beta distributions as important integer sequences in the OEIS base of integer sequences. Among those identified are Catalan, Riordan, Motzkin, or 'super ballot numbers'. By applying a method of expansion of the ratio of densities of involved distributions we are able to obtain some known and many unknown relationships between e.g. Catalan numbers and other moment sequences of the Beta distributions.

math.PR↗
arXiv2023

Polynomial identities that involve binomial coefficients, double and rising factorials and their probabilistic interpretations and proofs

We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The origins and the proofs of these identities are probabilistic. However, their form suggests universal applications in simplifying expressions. Many useful simplifying formulae are presented in the sequel.

math.PR↗
arXiv2022

On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications

We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey--Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. The paper is based on 65 items in the predominantly recent literature.

math.CA↗
arXiv2022

Yet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangles

We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.

math.CO↗
arXiv2021

On positivity of orthogonal series and its applications in probability

We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion $% \sum_{n\geq 0}c_{n}α_{n}(x)β_{n}(y)$ with two sets of orthogonal polynomials $\left\{ α_{n}\right\} $ and $\left\{ β_{n}\right\} $ to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set $C(α,β)$ of the sequences $\left\{ c_{n}\right\} ,$ for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.

math.PR↗
arXiv2021

Multivariate generating functions built of Chebyshev polynomials and some of its applications and generalizations

We sum multivariate generating functions composed of products of Chebyshev polynomials of the first and the second kind. That is, we find closed forms of expressions of the type $\sum_{j\geq0}ρ^{j}\prod_{m=1}^{k}T_{j+t_{m}}% (x_{m})\prod_{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),$ for different integers $t_{m},$ $m=1,...,n+k.$ We also find a Kibble-Slepian formula of $n$ variables with Hermite polynomials replaced by Chebyshev polynomials of the first or the second kind. In all the considered cases, the obtained closed forms are rational functions with positive denominators. We show how to apply the obtained results to integrate some rational functions or sum some related series of Chebyshev polynomials. We hope that the obtained formulae will be useful in the so-called free probability. We expect also that the obtained results should inspire further research and generalizations. In particular, that, following methods presented in this paper, one would be able to obtain similar formulae for the so-called $q-$Hermite polynomials. Since the Chebyshev polynomials of the second kind considered here are the $q$-Hermite polynomials for $q=0$. We have applied these methods in the one- and two-dimensional cases and were able to obtain nontrivial identities concerning $q-$Hermite polynomials.

math.CA↗
arXiv2019

On the generalized Kesten--McKay distributions

We examine the properties of distributions with the density of the form: $% \frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{π\prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ where $c,a_{1},\ldots ,a_{n}$ are some parameters and $A_{n}$ a suitable constant. We find general forms of $% A_{n}$, of $k-$th moment and of $k-$th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so-called Askey--Wilson scheme. On the way, we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters $a_{1},\ldots ,a_{n}$ forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases $n=2,4,6.$

math.PR↗
arXiv2019

On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials

We study properties of compactly supported, 4 parameter \newline $(ρ_{12},ρ_{23},ρ_{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q\rightarrow 1^{-}$ this family tends to some 3 dimensional Normal distribution. For $q=0$ we deal with 3 dimensional generalization of Kesten--McKay distribution. In a very special case when $ρ_{12}ρ_{13}ρ_{23}=q$ all one dimensional marginals are identical, semicircle distributions. We find both all marginal as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order $n$ and are polynomials of the same order $n$ in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey-Wilson, Al-Salam--Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials.

math.PR↗
arXiv2018

On probabilistic aspects of Chebyshev polynomials

} The main goal of this note is to provide new, mostly multidimensional densities, compactly supported and list many of its properties that enable effective calculations. The idea of obtaining such densities is firstly to build some one-dimensional densities depending on many parameters and then treat the constructed in this way distributions as conditional ones. Then of course by imposing certain distribution on the parameters and multiplying the two distributions we arrive at new multivariate distribution. To enable effective calculations, we utilize nice, simple and widely known properties of Chebyshev polynomials. Thus, in particular, the one-dimensional distribution mentioned above will have a form of arcsine distribution multiplied by some rational function. The fact that we use Chebyshev polynomials allows us to calculate all moments of this one-dimensional distribution as well as to find a family of polynomials that are orthogonal with respect to this distribution.

math.CA↗
arXiv2017

On stationary Markov processes with polynomial conditional moments

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say $m$ is a polynomial of degree at most $m\;\text{.}\;$ We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence $\{(α_{n} ,p_{n})\}_{n \geq 0}$ where $α_{n}^{ \prime } s$ are some positive reals while $p_{n}^{ \prime } s$ are some monic orthogonal polynomials. Paper of Bakry\&Mazet(2003) assures that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression. We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein-Uhlenbeck (OU) processes or can also be understood as time scale transformations of L{é}vy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of L{é}vy processes then they are not stationary unless we deal with classical OU- process. Conversely, time scale transformations of stationary processes with independent regression property are not L{é}vy unless we deal with classical OU process.

math.PR↗