Examples of second solutions to differential equations for Legendre and Chebyshev polynomials, contrasting behavior in x and n.
Explores numerical instability in recurrence relations for special functions like Bessel functions, explaining why forward/reverse computations can be unstable and how to handle it.
Explores three-term recurrence formulas in mathematics, with examples from Bessel functions, Chebyshev, Hermite, and Legendre polynomials.
Explains why modified Bessel functions are called 'modified', linking complex analysis and applied contexts like heat and wave equations.
Explores why special function terminology is arcane, linking independent discoveries to varied definitions and names.
Explores how inverse differential equations reveal properties of special functions like Bessel functions, using the indicial equation to understand behavior without solving.
Explores holonomic functions, which solve linear differential equations with polynomial coefficients, and their prevalence among special functions in Abramowitz and Stegun.
Explains why second order linear differential equations with polynomial coefficients are important in physics and applications, linking to separable PDEs.
