confluent hypergeometric equation
The hypergeometric equation has three regular singular points; many equations of physics have only two interesting ones, because the third has run off to infinity and merged with another. The confluent hypergeometric equation is what you get when you let two of the hypergeometric equation's singular points collide and merge — 'confluence' literally means flowing together. It is the streamlined cousin that governs bound-state problems in quantum mechanics.
It is x y'' + (c - x) y' - a y = 0, also called Kummer's equation, with just two parameters a and c. It has a regular singular point at x = 0 and now an irregular singular point at infinity (the residue of the merger). Its Frobenius solution about the origin is the confluent hypergeometric function 1F1(a; c; x) = sum over k of [(a)_k / (c)_k] x^k / k!, often written M(a, c, x). Unlike 2F1 this series converges for every x, because the bad singularity has been pushed off to infinity. When the parameter a is a non-positive integer the series terminates into a polynomial — and that is exactly how Hermite and Laguerre polynomials reappear as special cases.
This single equation is the spine of quantum bound states. The radial hydrogen equation and the harmonic oscillator both reduce to Kummer's equation after factoring out their asymptotic exponential behaviour; the requirement that 1F1 terminate (so the wavefunction stays normalizable) is what quantizes the energy. The error function, the incomplete gamma function, and the Coulomb wavefunctions are all 1F1 in disguise. So confluent hypergeometric is the common engine room beneath several apparently different special functions.
With a = 1, c = 2 the series 1F1(1; 2; x) = sum x^k/(k+1)! = (e^x - 1)/x. With a = -n (a negative integer) it truncates: 1F1(-n; 1/2; x^2) reproduces a Hermite polynomial up to scaling. The same template gives a smooth transcendental function or a polynomial depending only on the parameter a.
The parameter a decides whether 1F1 is an infinite transcendental or a terminating polynomial.
There are two standard solutions: Kummer's M(a, c, x), regular at the origin, and Tricomi's U(a, c, x), which captures the correct behaviour at infinity. Physics problems usually need the linear combination that is finite at one end and decaying at the other — using M alone can give a non-normalizable answer.