Junk solutions

When you’re interested in studying a family of functions, it can be useful to look at a differential equation that the functions solve. This is a theme I’ve written about several times, most recently here and here, but also three years ago here. Orthogonal polynomials are mathematically elegant as well as very useful in applications [1]. Various families of orthogonal polynomials satisfy various differential equations. These equations have a polynomial and non-polynomial solutions. What use are the latter? If the differential equation modeled something physical, then the second solution would be necessary to have a complete basis of solutions. But if the differential equation is only instrumental in studying the orthogonal polynomials, what use is a non-polynomial solution? These non-polynomial solutions turn out to be useful. Just as “junk” DNA turned out not to be junk, these “junk” solutions are important. Junk DNA doesn’t directly code for proteins, but it regulates DNA that does…