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pro vyhledávání: '"Szabłowski, Paweł J."'
Autor:
Szabłowski, Paweł J.
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 infin
Externí odkaz:
http://arxiv.org/abs/2411.00647
Autor:
Szabłowski, Paweł J.
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 prope
Externí odkaz:
http://arxiv.org/abs/2309.13652
Autor:
Szabłowski, Paweł J.
We consider the set of lower-triangular, infinite matrices with natural operations such as addition, multiplication by a number or matrix multiplication. With respect to every of these operations separately, the set preserves the group structure. Wit
Externí odkaz:
http://arxiv.org/abs/2303.12373
Autor:
Szabłowski, Paweł J.
Publikováno v:
Stochastics 2023
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 c
Externí odkaz:
http://arxiv.org/abs/2206.11798
Autor:
Szabłowski, Paweł J.
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 a
Externí odkaz:
http://arxiv.org/abs/2206.02201
Autor:
Szabłowski, Paweł J.
We recall the definition and the properties of a moment sequence and recall that all real sequences that have a finite rank of its Hankel matrix (see definition in the sequel) satisfy a homogeneous linear equation with constant coefficients. Then we
Externí odkaz:
http://arxiv.org/abs/2204.04706
Autor:
Szabłowski, Paweł J.
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 identifi
Externí odkaz:
http://arxiv.org/abs/2112.13420
Autor:
Szabłowski, Paweł J.
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
Externí odkaz:
http://arxiv.org/abs/2106.10461
Autor:
Szabłowski, Paweł J.
Publikováno v:
Positivity 26, Aricle number 19(2022)
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 co
Externí odkaz:
http://arxiv.org/abs/2011.02710
Autor:
Szabłowski, Paweł J.
Publikováno v:
Infinite Dimensional Analysis, Quantum Probability and Related Topics, (2022) 2230001
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
Externí odkaz:
http://arxiv.org/abs/2007.03267