Differential equation for Jacobi-Pineiro polynomials

Abstract

For $r\in {\mathbb{Z}}_{\ge 0}$, we present a linear differential operator %$(\di)^{r+1}+ a_1(x)(\di)^{r}+...+a_{r+1}(x)$ of order $r+1$ with rational coefficients and depending on parameters. This operator annihilates the $r$-multiple Jacobi-Piñeiro polynomial. For integer values of parameters satisfying suitable inequalities, it is the unique Fuchsian operator with kernel consisting of polynomials only and having three singular points at $x=0,1,\infty$ with arbitrary non-negative integer exponents $0,m_1+1,>...,m_1+...+m_r+r$ at $x=0$, special exponents $0,k+1,k+2,...,k+r$ at $x=1$ and arbitrary exponents at $x=\infty$.

Latex, 9 pages, revised version, misprints are corrected