COMS W3203 Lecture Notes - Lecture 15: Royal Institute Of Technology, Second Order (Religious), Hermite Polynomials
14 Dec 2017
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Recall a k-th order linear rst order recurrence: an = f (n) + pk f (n) is the inhomogeneous part and pk i=1 hi(n)an 1. i=1 hi(n)an 1 is the homogenous part. Summary for rst-order: an = san 1 + t, a0 = k such that k 2 r. Theorem 12. 16: if an = can 1 + f (n) for n 1, then 9 constant a and polynomial p such that an = acn + p(n). Divide by rn 2 to get: r2 ar b = 0: proceed based on one of three di erent cases: (a) two distinct real roots, i. e. (r x1)(r x2) = 0, x1, x2 2 r. Solution: an = (x1)n + (x2)n, use initial conditions to nd , . (b) one real root, i. e. (r x)n = 0. Solution: an = (x)n + n(x)n, use intial conditions to nd , . (c) complex roots, x1, x2 2 c, r < .
