q−q-Wiener (α,q)−\alpha,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 ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters ({\alpha},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(-{\alpha}|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.Comment: 25 page

On the generalized Kesten--McKay distributions

We examine the properties of distributions with the density of the form: $\frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{\pi \prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ 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 $n=2,4,6.$Comment: 14 page

Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions

We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q-Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R^{3} of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions

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 $$ 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 $\mathbf{\Pi }_{n}.$Comment: 18 page

q-Gaussian distributions. Simplifications and simulations

We present some properties of measures (q-Gaussian) that orthogonalize the set of q-Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having q-Gaussian distribution.Comment: 13 page