hypergeometric equation
Legendre, Chebyshev and many other equations look like separate creatures, but most of them are secretly the same animal in different clothing. The hypergeometric equation is that master equation — a single second-order template with three adjustable parameters that contains an astonishing number of the classical equations of physics as special cases. Learn it once and you have learned a whole zoo.
It is x(1 - x) y'' + [c - (a + b + 1) x] y' - a b y = 0, with three parameters a, b, c. It has exactly three regular singular points, conventionally at x = 0, x = 1, and x = infinity — and remarkably, any linear second-order equation with precisely three regular singular points and no others can be transformed into this one. Its Frobenius solution about x = 0 is the hypergeometric series 2F1(a, b; c; x) = sum over k of [(a)_k (b)_k / (c)_k] x^k / k!, where (a)_k means the rising product a(a+1)...(a+k-1). The series is a vast generalisation: choosing the parameters cleverly reduces 2F1 to the geometric series, the binomial series, logarithms, Legendre and Chebyshev polynomials, and the complete elliptic integrals.
Its value is unification and leverage. Once you know the convergence, the recurrences, the integral representation and the transformation formulas of 2F1, every special case inherits them for free, which is why computer algebra systems internally route so many functions through the hypergeometric form. The honest caveat is convergence: the defining series converges only for |x| < 1, and to use the function outside the unit disc — or when c is a non-positive integer, where the series breaks — you need analytic continuation and the connection formulas linking the three singular points.
The plain geometric series is the simplest special case: 2F1(1, b; b; x) = sum x^k = 1/(1 - x). And 2F1(-n, b; b; -x) = (1 + x)^n reproduces the binomial theorem. So the everyday series you already know are tiny corners of the hypergeometric family.
The geometric and binomial series are just two of the countless faces of 2F1.
If c is zero or a negative integer the standard series is undefined (a zero appears in a denominator), and one of the two Frobenius exponents difference becomes an integer — a logarithmic second solution can intrude. The 'three regular singular points' characterisation is what really pins down the equation, not the series formula.