By D. Jungnickel, K. Vedder
Read or Download Combinatorial Theory. Proc. conf. Rauischholzhausen, 1982 PDF
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Additional resources for Combinatorial Theory. Proc. conf. Rauischholzhausen, 1982
To do that, begin with the fact that and notice that if we apply (xD) to both sides of this relation and then set x = 1, the left side will be the sum of squares that we seek, and the right side will be the answer! ), which you no doubt knew already. Do notice, however, that the generating function machine is capable of doing, quite mechanically, many formidablelooking problems involving sums. | Our third rule will be a restatement of the way that two opsgf's are multiplied. Rule 3. 3) Now consider the product of more than two series.
Thus the composition f(g(x)) of two formal power series is defined if and only if go — 0 or f is a polynomial For instance, the series e is a well defined formal series, whereas the series e is not defined, at least from the general definition of composition of functions. 5) = x f(9(x))=9(f(x)) must both make sense and be true. We claim that if / ( 0 ) = 0 the inverse series exists if and only if the coefficient of x is nonzero in the series / . Proposition. 5) and / ( 0 ) = 0. 0). 2 lX 2 2 9 l Proof.
Take the function f(z) = z/(e — 1) ( / ( 0 ) = 1). Estimate the size of the coefficients of its power series about the origin directly from the analyticity properties of the function. This is where things start getting more interesting. , except possibly at the points z — 2km for integer k. The nearest of these to the origin is the origin itself (k = 0). However, / is not singular at z — 0 because even though the denominator of / is 0 there, the numerator is also, and L'Hospital's rule, or whatever, reveals that the value f(0) — 1 removes the singularity.