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Alphonse P. Magnus

Publications and source records attributed to Alphonse P. Magnus.

7 recordsLinked to original sources
arXiv2025

Difference operators and difference equations on lattices, or grids, up to the elliptic hypergeometric case

It is shown how to define difference operators and equations on particular lattices $\{x_n\}$, $2n\in\mathbb{Z}$, such that the divided difference operator $(\mathcal{D}f)(x_{n+1/2})= (f(x_{n+1})-f(x_n))/(x_{n+1}-x_n)$ has the property that $\mathcal{D}f$ is a rational function of degree $2d$ when $f$ is a rational function of degree $d$. It is then shown that the $x_n$s are in the most general case values of an elliptic function at a sequence of arguments in arithmetic progression (\emph{elliptic lattice}). Many special and limit cases, down to the most elementary ones, are considered too. First and second order difference operators and equations are constructed, up to the simplest elliptic hypergeometric ones. One also shows orthogonality and biorthogonality properties of rational solutions to some of these difference equations.

math.NT↗
arXiv2009

Elliptic Hypergeometric Solutions to Elliptic Difference Equations

It is shown how to define difference equations on particular lattices $\{x_n\}$, $n\in\mathbb{Z}$, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations have remarkable simple interpolatory expansions. Only linear difference equations of first order are considered here.

math.CA↗
arXiv1996

Preud's equations for orthogonal polynomials as discrete Painlevé equations

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2tw_n + w_n(w_n+1 + w_n + w_n-1) for the recurrence coefficients p_n+1 = xp_n - w_np_n-1 of the orthogonal polynomials related to the weight w(x) = exp(-4(tx^3 + x^4)) (notation of [26, pp. 34-36]). This example appears in practically all the references below. The connection with discrete Painlevé equations is described here.

math.CA↗
arXiv1995

Special non uniform lattice ($snul$) orthogonal polynomials on discrete dense sets of points.

Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered.

math.CA↗
arXiv1993

Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ satisfies a Painlevé ${\rm P}_{\rm IV}$ equation.

math.CA↗