recurrence and orthogonality relations
Once you have a whole family of special polynomials, two structural laws make them genuinely usable rather than just a list. A recurrence relation links three consecutive members so you can build the next from the previous two; an orthogonality relation says that two different members are 'perpendicular' when integrated against a suitable weight. Together they are the toolkit that turns a polynomial family into a working basis.
Every classical orthogonal family obeys a three-term recurrence of the form p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x); for Legendre, (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x). This is the stable, efficient way a computer evaluates P_50 — never by expanding a degree-50 polynomial, always by climbing the recurrence from P_0 and P_1. The orthogonality relation states that the integral over the family's natural interval of p_m(x) p_n(x) w(x) dx is zero whenever m is not equal to n, where w(x) is the weight (1 for Legendre on [-1, 1], e^{-x^2} for Hermite on the whole line, e^{-x} for Laguerre on [0, infinity)). When m equals n the integral gives a known normalisation constant.
Orthogonality is the property that makes these families the analogue of the sine-cosine system in Fourier analysis: any reasonable function can be expanded as a series sum c_n p_n(x), and each coefficient c_n is found by a single integral, c_n = (integral of f p_n w) / (integral of p_n^2 w), exactly because the cross terms vanish. This is generalised Fourier expansion, and it is the link from this field into Sturm-Liouville theory, where orthogonality is shown to be a structural consequence of the self-adjoint form of the underlying equation rather than a lucky accident.
Use Legendre's recurrence to climb up. From P_0 = 1 and P_1 = x, set n = 1: 2 P_2 = 3 x P_1 - P_0 = 3x^2 - 1, so P_2 = (3x^2 - 1)/2. Orthogonality then says the integral from -1 to 1 of P_1(x) P_2(x) dx = 0 — and indeed integral of x times (3x^2 - 1)/2 over a symmetric interval is zero because the integrand is odd.
Recurrence builds the polynomials; orthogonality lets you expand functions in them.
Orthogonality always carries a specific weight and interval — Legendre polynomials are orthogonal with weight 1 on [-1, 1], but Hermite need weight e^{-x^2} on the whole line. They are not orthogonal in the plain unweighted sense, so quoting 'the polynomials are orthogonal' without naming the weight is a common and consequential slip.