arXiv subjects

Paweł J. Szabłowski

Publications and source records attributed to Paweł J. Szabłowski.

At least 37 records · Page 2Linked to original sources

arXiv2017

Markov processes, polynomial martingales and orthogonal polynomials

We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales $\left\{ M_{n}(X_{t},t)\right\}_{n\geq1}$ that contains complete information on the distribution (both marginal and transitional) of the process. We specify conditions expressed in terms of $M_{n}^{\prime}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic harness. We also give conditions under which some of these martingales are also reversed martingales.

math.PR↗

arXiv2015

Moments of q-Normal and conditional q-Normal distributions

We calculate moments and moment generating functions of two distributions: the so called $q-$Normal and the so called conditional $q-$Normal distributions. These distributions generalize both Normal ($q=1),$ Wigner ($% q=0,$ $q-$Normal) and Kesten-McKay ($q=0,$ conditional $q-$Normal) distributions. As a by product we get asymptotic properties of some expansions in modified Bessel functions.

math.PR↗

arXiv2015

On connection between values of Riemann zeta function at integers and generalized harmonic numbers

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized harmonic numbers carries information about the values of the arguments of Hurwitz function. In particular we prove: $\forall k\in \mathbb{N}:$ $ζ(k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}% \sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},$ where $H_{n}^{(k)}$ are defined below generalized harmonic numbers. Further we find generating function of the numbers $\hatζ(k)=\sum_{j=1}^{\infty }(-1)^{j-1}/j^{k}. $

math.NT↗

arXiv2014

A few remarks on values of Hurwitz Zeta function at natural and rational arguments

We exploit some properties of the Hurwitz zeta function $ζ(n,x)$ in order to study sums of the form $\frac{1}{π^{n}}\sum_{j=-\infty}^{\infty}1/(jk+l)^{n}$ and $\frac{1}{π^{n}}\sum_{j=-\infty}^{\infty}(-1)^{j}/(jk+l)^{n}$ for $% 2\leq n,k\in \mathbb{N},$ and integer $l\leq k/2$. We show that these sums are algebraic numbers. We also show that $1<n\in \mathbb{N}$ and $p\in \mathbb{Q\cap (}0,1\mathbb{)}$ $:$ the numbers $(ζ(n,p)+(-1)^{n}ζ(n,1-p))/π^{n}$ are algebraic. On the way we find polynomials $s_{m}$ and $c_{m}$ of order respectively $2m+1$ and $2m+2$ such that their $n-$th coefficients of sine and cosine Fourier transforms are equal to $% (-1)^{n}/n^{2m+1}$ and $(-1)^{n}/n^{2m+2}$ respectively.

math.NT↗

arXiv2014

Askey--Wilson Integral and its Generalizations

We expand the Askey--Wilson (AW) density in a series of products of continuous $q-$Hermite polynomials times the density that makes these polynomials orthogonal. As a by-product we obtain the value of the AW integral as well as the values of integrals of $q-$Hermite polynomial times the AW density ($q-$Hermite moments of AW density). Our approach uses nice, old formulae of Carlitz and is general enough to venture a generalization. We prove that it is possible and pave the way how to do it. As a result we obtain system of recurrences that if solved successfully gives a sequence of generalized AW densities with more and more parameters.

math.CA↗

arXiv2014

A few remarks on orthogonal polynomials

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $% p_{n}p_{m}$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{Π}_{n}.$

math.AP↗

arXiv2014

Lévy processes, martingales, reversed martingales and orthogonal polynomials

We study class of Lévy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left\{ M_{n}(X_{t},t),\mathcal{F}_{\leq t}\right\} _{n\geq 1},$ where $% \mathcal{F}_{\leq t}$ is a suitable filtration defined below. We present several properties of these martingales. Among others we show that $% M_{1}(X_{t},t)/t$ is a reversed martingale as well as a harness. Main results of the paper concern the question if martingale say $M_{i}$ multiplied by suitable determinstic function $μ_{i}(t)$ is a reversed martingale. We show that for $n\geq 3$ $M_{n}(X_{t},t)$ is a reversed martingale (or orthogonal polynomial) only when the Lévy process in question is Gaussian (i.e. is a Wiener process). We study also a more general question if there are chances for a linear combination (with coefficients depending on $t)$ of martingales $M_{i},$ $i\allowbreak =\allowbreak 1,\ldots ,n$ to be reversed martingales. We analyze case $% n\allowbreak =\allowbreak 2$ in detail listing all possible cases.

math.PR↗

arXiv2014

On moments of Cantor and related distributions

We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having different but related parameters. We give examples relating Euler numbers to the moments of infinite Bernoulli convolutions. One of the examples provides moment interpretation of Pell numbers as well as new identities satisfied by Pell and Lucas numbers.

math.PR↗

arXiv2013

A few remarks on quadratic harnesses

We analyze and partially solve system of recurrences that can be derived from the properties of martingale orthogonal polynomials that characterize quadratic harnesses (QH). We also specify conditions for the existence of moments of one dimensional distribution for large classes of quadratic harnesses that are also Markov processes complementing earlier results.

math.PR↗

arXiv2013

Befriending Askey-Wilson polynomials

We recall five families of polynomials constituting a part of the so-called Askey-Wilson scheme. We do this to expose properties of the Askey-Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In doing so we express AW density as a product of the density that makes $q-$Hermite polynomials orthogonal times a product of four characteristic function of $q-$Hermite polynomials (\ref{fAW}) just pawing the way to a generalization of AW integral. Our main results concentrate mostly on the complex parameters case forming conjugate pairs. We present new fascinating symmetries between the variables and some newly defined (by the appropriate conjugate pair) parameters. In particular in (\ref% {rozwiniecie1}) we generalize substantially famous Poisson-Mehler expansion formula (\ref{PM}) in which $q-$Hermite polynomials are replaced by Al-Salam-Chihara polynomials. Further we express Askey-Wilson polynomials as linear combinations of Al-Salam-Chihara (ASC) polynomials. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we reach AW polynomials and AW densities that have clear probabilistic interpretation.

math.FA↗

arXiv2013

On Markov processes with polynomials conditional moments

We study properties of a subclass of Markov processes that have all moments that are continuous functions of the time parameter and more importantly are characterized by the property that say their $n-$th conditional moment given the past is also a polynomial of degree not exceeding $n.$ Of course all processes with independent increments with all moments belong to this class. We give characterization of them within the studied class. We indicate other examples of such process. Besides we indicate families of polynomials that have the property of constituting martingales. We also study conditions under which processes from the analysed class have orthogonal polynomial martingales and further are harnesses or quadratic harnesses. We provide examples illustrating developed theory and also provide some interesting open questions. To make paper interesting for a wider range of readers we provide short introduction formulated in the language of measures on the plane.

math.PR↗

arXiv2013

A few remarks on Euler and Bernoulli polynomials and their connections with binomial coefficients and modified Pascal matrices

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain lower triangular built of binomial coefficients. Another words we interpret Euler and Bernoulli numbers in terms of modified Pascal matrices.

math.RA↗

arXiv2013

On q-Hermite polynomials and their relationship with some other families of orthogonal polynomials

We review properties of the $q-$Hermite polynomials and indicate their links with the Chebyshev, Rogers--Szegö, Al-Salam--Chihara, continuous $q-$% utraspherical polynomials. In particular we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.

math.CO↗

arXiv2013

Around Poisson--Mehler summation formula

We study polynomials in $x$ and $y$ of degree $n+m:\allowbreak \{Q_{m,n}(x,y|t,q)\}_{n,m\geq 0}$ that appeared recently in the following identity: $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak γ_{0,0}(x,y|t,q) \allowbreak Q_{m,n}(x,y|t,q) $ where $γ_{m,n}(x,y|t,q) \allowbreak =\allowbreak \sum_{i\geq 0}\frac{t^{i}}{[i]_{q}}H_{i+n}(x|q) H_{m+i}(y|q)$, $\allowbreak $ $\{H_{n}(x|q)}_{n\geq -1}$ are the so-called $q-$% Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y|t,q) :i=0,...,n\}_{n\geq 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,q)-$Normal distribution) on the square $\{(x,y):|x|,|y|\leq 2/\sqrt{1-q}\} . $ We study structure of these polynomials expressing them with the help of the so-called Al-Salam--Chihara (ASC) polynomials and showing that they are rational functions of parameters $t$ and $q$. We use them in various infinite expansions that can be viewed as simple generalization of the Poisson-Mehler summation formula. Further we use them in the expansion of the reciprocal of the right hand side of the Poisson-Mehler formula.

math.CA↗

arXiv2012

On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) is symmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+κ_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + λ_{n}b_{n-2}(x) for some sequences {κ_{n}} and {λ_{n}}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {κ_{n}} that have a form of Fourier series expansion of the Radon--Nikodym derivative of one measure with respect to the other.

math.CA↗

arXiv2012

On peculiar properties of generating functions of some orthogonal polynomials

We prove that for |x|,|t|<1, -1 <q \leq1 and n\geq0: Σ_{i\geq0}((t^{i})/((q)_{i}))h_{n+i}(x|q) = h_{n}(x|t,q) Σ_{i\geq0}((t^{i})/((q)_{i}))h_{i}(x|q), where h_{n}(x|q) and h_{n}(x|t,q) are respectively the so called q-Hermite and the big q-Hermite polynomials and (q)_{n} denotes the so called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam---Chihara (ASC) polynomials and ASC and the so called continuous dual q-Hahn (c2h) polynomials. Moreover we are able to relate in this way some other 'ordinary ' orthogonal polynomials such as e.g. Hermite, Chebyshev or Laguerre. These equalities give new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey--Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case) thus paving the way to generalization of AW polynomials at least in these two cases.

math.AP↗

arXiv2012

On summable, positive Poisson-Mehler kernels built of Al-Salam--Chihara and related polynomials

Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so called Al-Salam-Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting nice identities built of the ASC polynomials involving 6 variables like e.g. formula (nice). These expansions lead to asymmetric, positive and summable kernels. The particular cases (referring to q=1 and q=0) lead to the kernels build of certain linear combinations of the ordinary Hermite and Chebyshev polynomials.

math.CA↗

arXiv2011

$q-$Wiener ($α,q)-$ Ornstein-Uhlenbeck processes. A generalization of known processes

We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called q-Wiener) depends on one parameter q\in(-1,1] and of the second one (say Y=(Y_{t})_{t\inR} called (α,q)- Ornstein--Uhlenbeck) on two parameters (α,q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener process in the sense that for q=1 it is Wiener process but also that for |q|<1 and \foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the so called q-Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU process. For |q|<1 it is also stationary with correlation function equal to exp(-α|t-s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.

math.PR↗