the recurrence relation
/ ri-KUR-ents /
When you feed a power series into a differential equation and collect terms, you do not get the coefficients handed to you all at once. Instead you get a rule that links them — a formula that tells you each new coefficient in terms of earlier ones. That rule is the recurrence relation, and it is the engine that actually generates a series solution, coefficient after coefficient, like a chain of dominoes once you tip the first one or two.
Concretely, after substituting y = sum a_n x^n and matching the coefficient of each power of x to zero, you typically arrive at something like a_(n+2) = -a_n / ((n+2)(n+1)) for all n >= 0. Read it as a recipe: choose the starting values a_0 and a_1 freely (they are the two arbitrary constants of a second-order equation), then crank — a_2 comes from a_0, a_3 from a_1, a_4 from a_2, and so on indefinitely. A two-term recurrence (relating a_(n+2) to a_n) splits the series neatly into an even-power solution built from a_0 and an odd-power solution built from a_1; longer recurrences mix more terms but work the same way.
The recurrence is where the equation's personality lives: its exact form decides the pattern of the coefficients, whether the series terminates into a polynomial (as for special values that give Legendre or Hermite polynomials), and ultimately what function the series sums to. In practice solving the differential equation IS solving its recurrence — once you have a closed form or a clear pattern for a_n, you have the solution.
Substituting into y'' + y = 0 gives a_(n+2) = -a_n/((n+2)(n+1)). Starting from a_0: a_2 = -a_0/2!, a_4 = a_0/4!, ... reproducing cos(x). Starting from a_1: a_3 = -a_1/3!, a_5 = a_1/5!, ... reproducing sin(x). The two free constants a_0, a_1 give the two independent solutions.
A two-term recurrence couples coefficients two apart, splitting the solution into even and odd series — here exactly cos and sin.
A recurrence that relates only a_(n+2) to a_n is unusually clean; many equations give three-term recurrences that mix a_(n+2), a_(n+1), a_n, which still work but resist a simple closed form.