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An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the Catalan numbers, C n. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯ ...
For example, with the Hamiltonian = +, where p is the generalized momentum and q is the ... The generating function F for this transformation is of the third kind,
The probability generating function is an example of a generating function of a sequence: see also formal power series. It is equivalent to, and sometimes called, the z-transform of the probability mass function.
The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and ...
Consider the problem of distributing objects given by a generating function into a set of n slots, where a permutation group G of degree n acts on the slots to create an equivalence relation of filled slot configurations, and asking about the generating function of the configurations by weight of the configurations with respect to this equivalence relation, where the weight of a configuration ...
For example, every Dyck word w of length ≥ 2 can be written in a unique way in the form w = Xw 1 Yw 2. with (possibly empty) Dyck words w 1 and w 2. The generating function for the Catalan numbers is defined by = =.
Here are some examples of the moment-generating function and the characteristic function for comparison. It can be seen that the characteristic function is a Wick rotation of the moment-generating function M X ( t ) {\displaystyle M_{X}(t)} when the latter exists.
Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, ... Generating functions. The ordinary generating functions are