Second solutions

This article illustrates the concept of second solutions to differential equations associated with families of polynomials, specifically Legendre and Chebyshev polynomials. For each n, the differential equation has two solutions: the polynomial (Pn or Tn) and a non-polynomial second solution (Qn or Vn) with singularities at ±1. While the polynomial solutions are bounded on [-1,1], the second solutions blow up there. However, for fixed |x|>1, the polynomial solutions grow exponentially with n, whereas the second solutions decay exponentially. This mirrors earlier examples with Bessel functions. The post provides concrete examples to clarify the contrasting behavior with respect to x versus n.