Special functions and rod nonlinear theory

There are many physical models or theories in which some functions appear in natural and regular fashion, starting with polynomials and algebraic functions and progressing through the “elementary” transcendental ones, almost all already known to the ancients. Most of special functions of Mathematical Physics are generated from the study of those ordinary differential equations (ODEs) in which the separation of variables breaks many categories of linear partial differential equations, mainly that of Laplace. Among the many special functions that satisfy these second order ODEs the people find the polynomials which bear the names of Tchebychev, Hermite, Jacobi, Laguerre, and the Whittaker functions, and the parabolic cylinder functions. However, there are further special functions not stemming from second order linear ODEs. The problem of “rectification of quadratures” was generally encountered in the setting of ODEs that cannot be solved in closed algebraic form or “by quadratures”.