Elementary matrices (problem book, Tutorial Text 3) by Mitrinovic D.S.

By Mitrinovic D.S.

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Thus, we should expect a connection with the Hermite polynomials, as indeed we have seen. 7 DYNAMIC DATA STRUCTURES WITH FINITE POPULATION So far, we have assumed that the universe of keys is infinite. Since in practice there are only a finite number of keys, it is of some interest to see this explicitly taken into account. Flajolet and Fran^on considered the markovian model and gave the time cost generating functions. Here we will follow Frangon, Randrianarimanana&;Schott[34]. We will state the results of their analysis, without presenting details here.

2. 1 P r o p o s i t i o n . 2 Corollary. have the expansion For the polynomials (t>k{x,t) = f^'I'^Ukix 12-^1) of Prop. QT wijere M{s) is the corresponding moment generating ... 5, we need to expand the coefficients V(s)" in powers of s. This is where we use Lagrange inversion. 3 P r o p o s i t i o n . Let V(s) = (1 - ^ 1 -As'^t)/2s. 'f(,„,V)'Proof: Let x = V{s) = (1 - Vl - 'is'^t)/2s. Then one readily finds that X s = X2 +t Applying the Lagrange inversion formula, Ch. 3, we have, with XQ = SQ = 0, writing D for d/dx, jt=i Expanding (x'^ + <)* by the binomial theorem and differentiating accordingly, the result follows.

Thus, e*^ tpb = e'* V'ti and the result follows. )" ^—' n! n=0 Thus Hence the result. For the product, we have Now apply Prop. 5. 2 Corollary. • We have the coherent state {R)ab=at, representations {V)ab = b and {RV)ab = abt, Proof: {{RVf)ab = aHh"" + abt For example, ^ (e'"')ab = {Re'"')ab = ate""* and evaluating at p = 0 yields the result for R. In general, differentiate with respect to the appropriate parameters and evaluate at 0 for R and V, 1 for RV, to get the result. • From the CSR's we see that the adjoint of R is tV, with CSR tb.

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