Using \D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

We introduce the concept of \D-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra \A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, \beta_n\in \RR. Using the concept of \D-operator, we determine the structure of the sequence $(\beta_n)_n$ in order that the polynomials $(q_n)_n$ are common eigenfunctions of a higher order difference operator. In addition, we generate sequences $(\beta_n)_n$ for which the polynomials $(q_n)_n$ are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.Comment: 43 page

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $\mu$. Let $T$ be a linear operator acting in the linear space of polynomials \PP and satisfying that \dgr(T(p))=\dgr(p)-1, for all polynomial $p$. We then construct a sequence of polynomials $(s_n)_n$, depending on $T$ but not on $\mu$, such that the Wronskian type $n\times n$ determinant $\det \left(T^{i-1}(p_{m+j-1}(x))\right)_{i,j=1}^n$ is equal to the $m\times m$ determinant $\det \left(q^{j-1}_{n+i-1}(x)\right)_{i,j=1}^m$, up to multiplicative constants, where the polynomials $q_n^i$, $n,i\ge 0$, are defined by $q_n^i(x)=\sum_{j=0}^n\mu_j^is_{n-j}(x)$, and $\mu_j^i$ are certain generalized moments of the measure $\mu$. For $T=d/dx$ we recover a Theorem by Leclerc which extends the well-known Karlin and Szeg\H o identities for Hankel determinants whose entries are ultraspherical, Laguerre and Hermite polynomials. For $T=\Delta$, the first order difference operator, we get some very elegant symmetries for Casorati determinants of classical discrete orthogonal polynomials. We also show that for certain operators $T$, the second determinant above can be rewritten in terms of Selberg type integrals, and that for certain operators $T$ and certain families of orthogonal polynomials $(p_n)_n$, one (or both) of these determinants can also be rewritten as the constant term of certain multivariate Laurent expansions.Comment: 36 page

The fixed point for a transformation of Hausdorff moment sequences and iteration of a rational function

We study the fixed point for a non-linear transformation in the set of Hausdorff moment sequences, defined by the formula: $T((a_n))_n=1/(a_0+... +a_n)$. We determine the corresponding measure $\mu$, which has an increasing and convex density on $]0,1[$, and we study some analytic functions related to it. The Mellin transform $F$ of $\mu$ extends to a meromorphic function in the whole complex plane. It can be characterized in analogy with the Gamma function as the unique log-convex function on $]-1,\infty[$ satisfying $F(0)=1$ and the functional equation $1/F(s)=1/F(s+1)-F(s+1), s>-1$.Comment: 29 pages,1 figur

Admissibility condition for exceptional Laguerre polynomials

We prove a necessary and sufficient condition for the integrability of the weight associated to the exceptional Laguerre polynomials. This condition is very much related to the fact that the associated second order differential operator has no singularities in $(0,+\infty)$.Comment: 12 page