Abstract
For a finite (or countably infinite) sequence an, the generating function (or series) is defined as the polynomial (resp. formal series) ∑antn . For a multisequence am,n,…, the generating function is a polynomial or series in several variables ∑a m,n,… t m u n Various problems of combinatorial analysis and probability theory are successfully explored with such powerful tools as generating functions. In this chapter readers will encounter several problems related to the combinatorics of binomial coefficients, theory of partitions, and renewal processes in probability theory that can be explored using generating functions. The problems of the first two groups does not require the ability to deal with power series (likewise, the problems of the second group assume no familiarity with the theory of partitions) and can be solved by readers with limited experience.
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Notes
- 1.
Readers experienced in analysis will probably notice that the generating function resembles a Laplace transform.
- 2.
The algorithm described subsequently is important for the design of recursive digital filters (Williams 1986).
- 3.
How would the equation for F change if sequences were assumed to obey the recursion relation for n ≥ n f , depending on the sequence?
- 4.
It is a k-dimensional space isomorphic to the space of all k-tuples (f 0,…, f k−1).
- 5.
It can also be proved without using generating functions. (How?) A “continuous” analog, related to functions and ordinary linear differential equations instead of discrete sequences and linear recursion relations, was discussed previously in sections P10.20, P10.21 and P10.22 ( “One-Parameter Groups of Linear Transformations” problem group). To learn more about the close ties between linear recursion relations and autonomous ordinary linear differential equations see Arnol’d (1975).
- 6.
m q can also be found by a different method − from a system of linear equations with a Vandermonde determinant as h k j = m 1 h k−1 j + … m k h 0 j: j = 0,…, k − 1, using h l = l, ∀l = 0,…, k; the solution of the system, \( {m_{k-i }}=\prod\limits_{{s<k,s\ne i}} {{{{({h_k}-{h_s})}} \left/ {{({h_i}-{h_s})}} \right.}} ={(-1)^{k-1-i }}\left( {\begin{array}{ll} k \\i \\\end{array}} \right) \), i = 0,…, k − 1, is obtained by Cramer’s rule taking into account the explicit expression for the Vandermonde determinant (see, e.g., section E1.1, the “Jacobi Identities and Related Combinatorial Formulas” problem group above), by means of the Jacobi identities (section P1.1), or using the fact that m k−i , as a function of h k , is a Lagrange interpolating polynomial of degree k − 1 in h k becoming δ ij for h k = h j , j = 0,…, k − 1. See also a proof in Zalenko (2006).
- 7.
For k = j the result is implied by the related Jacobi identity from section P1.1 using x l = n − l, l = 0,…, k. (Similarly, the result can be deduced from the identities of section P1.1 for any k ≥ j; readers can also formulate the results for k < j and for negative j, which can be derived from the identities of sections P1.5 and P1.7, respectively. Fillin the details.) In addition, readers may prove these identities differently using combinatorial arguments.
- 8.
However, readers familiar with these techniques will be able to find shorter solutions – due to more efficient use of generating functions!
- 9.
In particular, this symbol vanishes when α is not greater than or equal to β (i.e., β i > α i for some i).
- 10.
In the terminology of combinatorial analysis, p n,k (m) is the number of partitions of a natural number m into k distinct natural summands, not exceeding n − 1.
- 11.
Here we are interested in a quite elementary combinatorial aspect. The probabilistic meaning and a far-reaching development are exhaustively discussed in Feller (1966) and references therein.
- 12.
In particular, u grows by an exponential law. Note that the derivative A’(t 0) is finite. [Why? Use arguments similar to those in section E11.13, using convergence of the power series A(t) for |t| < 1.]
- 13.
An involution is an automorphism of a set equal to its own inversion.
- 14.
This action is referred to as an induced action.
- 15.
Orbits of an action of a group H on a set S are equivalence classes with respect to the equivalence relation s 1 ≡ s 2 ⇔ ∃ h∈ H: hs 1 = s 2. An equivalent definition is that the orbits are the sets Hs = {hs: h∈ H} (s∈S).
- 16.
Convergence of a series implies convergence of the tails.
- 17.
As \( U(t) = {{\left( {1-F(t)} \right)}^{-1 }} = \sum\limits_{n=0}^{\infty } {{{{\left[ {F(t)} \right]}}^n}} \) for ∣t∣ < 1 and [F(t)]n > ½ for t close to 1 − 0, if s f = 1, then U(t) > 1 + n/2 for t close to 1 − 0 when s f = 1; what does this imply?
- 18.
By induction, we will have infinite differentiability.
- 19.
For a nonnegative-element series ∑c n , if c n > 0, then \( \sum\limits_{k=0}^n {{c_k}{t^k} > \sum\limits_{k=0}^{n-1 } {{c_k}} } \) for t close to 1 − 0. Hence, \( \mathop{\lim}\limits_{{t\to1-0}}\sum {{c_n}{t^{{_n}}}} =\sum {{c_n}} \), including the case where ∑c n = ∞. (Why?) A similar limit relation holds for a convergent series with arbitrary elements ∑c n (which is proved on the basis of Lemma 2′ (see below on this page); prove it!).
- 20.
We will have a one-to-one correspondence of the orbit to the set of right residue classes if we denote the group’s action by h: s \( \mapsto \) sh (h∈H).
- 21.
This does not hold for composite numbers, which may be verified immediately by considering the first one, 4. (Do this!) Accordingly, the claim of section P11.11* does not hold for composite q. (Consider an example of q = 4 and k = 2.)
- 22.
The series ∑c n t n will converge (even absolutely) for |t| < 1 by Abel’s lemma.
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Roytvarf, A.A. (2013). Some Problems in Combinatorics and Analysis That Can Be Explored Using Generating Functions. In: Thinking in Problems. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8406-8_11
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