Difference Equations with Public Health Applications by Lemuel A. Moyé, Asha Seth Kapadia

By Lemuel A. Moyé, Asha Seth Kapadia

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These examples will be provided in increasing order of complexity. It is important that the reader understand each step of the process for every example. 55) If each of GI(S) and G2(s) can be inverted, the convolution principle will permit us to construct the sequence {yk} associated with the inversion of G(s). 60) first proceed by recognizing that the numerator will utilize the sliding tool. 64) It can be seen from the denominator of G(s) that the convolution principle will be required not once, but twice.

For this range of s, as k gets larger sk+1 approaches zero. Now, if —— is to be a generating function, then there must be an infinite series 1-s oo 1 fo 1-S such that V y k s k = —— . Is there such a series? Yes! Let yk=l for all k 28 Chapter 2 Thus, —— is a generating function, and its inversion is y k = 1 for all integers k 1-s > 0. Using a straightforward summation of the simple geometric series, it was possible to find a one-to-one correspondence between the sum of the series and the coefficients of each terms in that series.

97) 3 •2 and ———- > {(k + 3) • (k + 2) • (k +1)}. In fact, it is easy to see a pattern here. (l-s) In general r! 98) k! (k + r)! r! v r / , we can write Finally, we can change the base series on which we apply the derivative. For example, it has been demonstrated that if G(s) = ———- , then G(s) > {k} . (l-s) Following the pattern of taking a derivative on each side, G'(s) = ———j > |(k + 1)2 1 . 100) Generating Functions I: Inversion Principles 47 This process involves taking a derivative followed by the use of the scaling tool.

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