arXiv subjects
Publications and source records attributed to Christophe Smet.
In this paper we consider a semi-classical variation of the weight related to the little $q$-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence relation.
We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.
We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-β. We combine both lattices to obtain the bi-lattice N \cup (N+1-β) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.
We show how one can use Hermite-Padé approximation and little $q$-Jacobi polynomials to construct rational approximants for $ζ_q(2)$. These numbers are $q$-analogues of the well known $ζ(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $ζ_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $μ(ζ_q(2))\leq 10π^2/(5π^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $ζ(2)$}, Mat. Sb. \textbf{193} (2002), no. 8, 49--70).
We present an asymmetric $q$-Painlevé equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution which gives the recurrence coefficients. We establish a connection with $α-q-P_V$.
We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated to the special case $q_i=q^{r_i}, r_i\in\mathbb{N}, q=1/p, p\in\mathbb{N}\setminus\{1\}$. In this construction we make use of little $q$-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of $h^{\pm}(q_1,q_2)$ and give an upper bound for the irrationality measure.