The Generating Function

Before we analyze in more detail the expression for the transition probability (3.16), let us consider the generating function of Equation 3.12 for the case N—1 = 2, 

In our mathematically lax maimer, we ignore in Equation 3.20 possible difficulties in interchanging orders of integration and summation. To calculate this fourfold integral, we need the transformation (1.44) of Chapter 1 between the coordinates q and q, which in the two-dimensional case is written as 

In dealing with several modes, which we shall study in Chapter 4, it is customary to use this notation for the displacement vector ki2 = col(k j k 2 ). To facilitate integration, we introduce new coordinates 

With this transformation, the integral appearing in (3.20) is easily performed, 

We see that the fourfold integral (3.24) is written as a product of two Gaussian integrals of the form 

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Table 2.5-2 provides a convenient summary of distributions, means and variances used in reliability analysis. This table also introduces a new property called the generating function (M,0). 

A generating function is defined by equation 2.5-47. To illustrate it use. Table 2.> 2 gives the generating function for an exponential distribution as -A/(0-X). Each moment i.s obtained by successive differentiations. Equation 2.5-48 shows how to obtain the first moment. By taking the limit of higher derivatives higher moments are found. 

In our algebraic formalism, the time evolution of this system is represented by multiplying the generating function by the dipolynomial... 

But since x — 1) contains a constant term, if A x 1) is a conventional polynomial, then A x-, )/ x — 1) must also be a conventional polynomial. Thus, all reachable configurations represented by the generating function 1) have the form... 

Let t x) be the generating function of the topologically different planted trees. 

The definitions of and are, from a purely geometric-combinatorial point of view, somewhat artificial. However, p is related to R like to T. p will be derived from R, and R is the coefficient of x in the power expansion of the generating function... 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

It is easy to see that a combination with no repetitions gives rise to exactly two transitivity systems with respect to Ag. Summarizing the results, we have the rule the number of different transitivity systems of configurations with respect to Ag is the sum of the respective numbers of combinations with and without repetitions. Therefore, the generating function of the permutations which are nonequivalent with respect to Ag is... 

We note that (1.12) relates to the cycle index (1.16) like (1.15) (taking (1.12) and (1.14) into account) to (1.17). The following definitions allow us to state the rules on the construction of the generating functions in a unified way. To introduce the functions /(x), f(x,y) into the cycle index means putting... 

Theorem. The generating function for the configurations [] which are nonequivalent with respect to H is obtained by substituting the generating function of [4>] in the cycle index of U. 

Let jp(5 ) be the number of those configurations with content k,tl,m) which remain invariant under the permutation S in (1.6). Thus, the generating function of interest is... 

According to the nature of the congruence, topological, spatial, or planar, the generating functions of the planted C-H trees are given by... 

Making use of the relationship discussed above, "the number of noncongruent planted trees equals the number of nonequivalent configurations of three planted trees", of the generating function and the main theorem of Chapter 1 (Sec. 16) and taking the special case n 0 into account, we establish for each of the three situations an equation ... 

The terms on the right-hand side correspond to the different possible cases of 1, 2, 3,. .. principal branches. Introducing the generating function... 

We note that chemical substitution of a radical into a basic compound corresponds (in the sense of the main theorem of Chapter 1) to the algebraic substitution of the generating function into the cycle index of the group of the basic compound. 

The generating function 0(x,y) of and its functional equation (2.22) have been established in Sec. 42. Now we derive some properties of the numbers from the functional equation (2.22). 

Replacing the H-atoms by alkyl radicals, that is, replacing / by the generating series r(x) of the alkyl radicals (as in Sec. 58) and representing the m C-atoms of the initial compound by the factor X , we get the generating function of the special disubstituted paraffins discussed here, namely... 

It is the generating function of all structurally isomeric disubstituted paraffins C Hj XY. 

The tree can be constructed in five steps and, correspondingly, the generating function of the structurally isomeric molecules can be built up from five factors (multiplication of the generating function in the case of independence cf. Sec. 17). 

Multiply substituted paraffins. If there are more than three substitutions in a paraffin the generating function of the structural isomers becomes more involved. I will describe the construction but will forego the details of the proof. 

Hence, 5, Pj, can assume only finitely many values For given i there exist only finitely many topologically different reduced trees. The following rule holds The generating function of the structurally... 

In the preceding section we looked only at those multiply substituted paraffins in which all the substituents are distinct. The case where two or more substituents are equal can be treated too however, the description and justification of the formulas become so awkward that I refer the reader to the generating function established elsewhere for two and three substituents. Cf. P6lya 4, p. 440. 

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