Graphs as an Aid to Understanding Special Functions
Richard Askey · 2020
Graphs of special functions can be very useful, both in teaching and in discovering problems to work on. The use of graphs to help understand properties of functions goes back a long way. Legendre polynomials arise in many places, so many people have to try to find the right formulas to help solve their problems. There are many known formulas about Legendre polynomials, and it is not easy to separate them into formulas which are specific to Legendre polynomials, those which have extensions to some other sets of orthogonal polynomials, and those that hold for general orthogonal polynomials. There are two ways to consider conjugate functions. One is to consider two appropriate solutions of a second order differential equation. However Cooper’s claims for Jacobi polynomials are clearly incorrect, since they disagree with the case α = β, where there are theorems.