Yesterday I wrote that a lot of the special functions that pop up in mathematical physics are solutions to second order linear differential equations with polynomial coefficients. More generally, holonomic functions are defined to be those functions that are the solutions to linear differential equations, of any order, with polynomial coefficients.
Most special functions are holonomic. To quantify that statement, I went through the special functions covered in Abramowitz and Stegun. The large majority are holonomic, thought some common functions like the gamma function are not holonomic.
This report goes through the functions in A&S. For those that are holonomic, it gives the differential equation that the function solves. The large majority of these equations are second order, but not all. And the coefficients are nearly always first or second order polynomials, rarely higher order.