The Art of Probability by Richard W. Hamming

Author:Richard W. Hamming
Language: eng
Format: epub
Publisher: CRC Press

We easily see that the initial conditions for state S3(3) are

Multiply the difference equation through by p2 to get it in the standard form

(4.4–2)

To check this equation we try the two extreme cases, p = 0 which gives S3(n) = 0 for all n, and the case p = 1 which gives q = 0 and S3(n) = 1 for n = 2 and 0 otherwise.

Equation (4.4–2) is a homogeneous second order linear difference equation with constant coefficients, and according to Appendix 4.A it generally has eigenfunctions of the form rn. Substituting rn into the difference equation (4.4–2) and dividing out the rn we get the characteristic equation

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