The heart of 19th century math was the study of special functions arising from mathematical physics.
It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.
The above quote was the judgment of Felix Klein (of Klein bottle fame) in 1893. The differential equations he had in mind were the second order differential equations of mathematical physics.
Special functions were the core of 19th century math, and hypergeometric series were the grand unifying theory of special functions. (Not every special function is hypergeometric, but quite a few are.) And yet they’re hardly taught any more. I never heard of hypergeometric series in college, even though I studied differential equations and applied math. Later I encountered hypergeometric functions first in combinatorics and only later in differential equations.
It’s odd that what was “the central problem of the whole of modern mathematics” could become almost a lost art a century later. How could this be? I believe part of the explanation is that special functions, and hypergeometric function in particular, fall between two stools: too advanced for undergraduate programs but not a hot enough of a research area for graduate programs.
Related: Consulting in differential equations