the hypergeometric equation
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Many of the famous special equations — Legendre, Chebyshev, and others — look like different problems, but they are secretly the same equation in disguise. The hypergeometric equation is that master equation: a single, flexible second-order equation with three adjustable parameters that, by suitable changes of variable, contains a huge swath of the special functions as special cases. It is the grand unifier of the subject.
It is written x(1 - x) y'' + (c - (a + b + 1) x) y' - a b y = 0, with three constant parameters a, b, c. It has exactly three regular singular points — at x = 0, x = 1, and x = infinity — and that precise structure is what makes it the universal template: any second-order linear equation with three regular singular points can be transformed into this one. Its Frobenius solution about x = 0 is the hypergeometric series 2F1(a, b; c; x) = sum from n=0 of ((a)_n (b)_n / (c)_n) x^n / n!, where (a)_n = a(a+1)...(a+n-1) is the rising factorial. The series's coefficients are ratios that change by a fixed rule from one term to the next, which is exactly the signature of a 'hypergeometric' series and the reason for the name.
Why care about an abstract template? Because recognising that your equation is a hypergeometric equation in disguise instantly hands you its solution, its convergence (the series converges for |x| < 1), its connection formulas, and a century of known identities — no need to redo the Frobenius analysis from scratch. Legendre's P_n, Chebyshev's T_n, and many others are particular hypergeometric functions for special parameter choices, which is how a single theory ties the whole zoo of special functions together.
Chebyshev's T_n is a disguised hypergeometric: T_n(x) = 2F1(-n, n; 1/2; (1 - x)/2). Because the first parameter is the negative integer -n, the rising factorial (-n)_k becomes zero past k = n, so the series terminates into a degree-n polynomial — exactly the termination that produces a special polynomial.
A negative-integer parameter truncates the hypergeometric series into a polynomial — the mechanism behind many classical orthogonal polynomials.
The hypergeometric series converges only for |x| < 1 in general; beyond the unit disk you must use analytic continuation or the connection formulas, because x = 1 is itself one of the three singular points.