Hypergeometric functions!

Any natural scientist or engineer would know at least one of the followings: Hermite polynomials, Legendre polynomials, Airy function, Bessel function, Chebyshev polynomials, Spherical harmonics, Error function, Gamma function, Gegenbauer polynomials, and Wigner D-matrix.

For example, a biomedial engineer would know Chebyshev polynomials from the Chebyshev filter, an electrical engineer would know Error function from stochastic process analysis, an astronomer would know Airy function from a diffraction analysis, a chemist would know spherical harmonics from molecular shells, a physicists would know Legendre polynomials from quantum mechanics; honestly, you should hesitate to call yourself a natural scientist or an engineer if you haven’t heard any of them!  (Gamma and Error functions are the ones most likely to be known!)

Well, if you know more than one of them, then you may be surprised to learn that all these functions, and many many more, are actually different special cases of one big fat function: Hypergeometric function!

For a mathematician this is not that surprising; after all, all these functions are solutions to some differential equation and hypergeometric function is simply the solution of the more general differential equation. Still, I cherish this knowledge, and so shall you!