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In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the ...
For a degenerate point mass at c, the cumulant generating function is the straight line () =, and more generally, + = + if and only if X and Y are independent and their cumulant generating functions exist; (subindependence and the existence of second moments sufficing to imply independence.
Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
The left-hand side of the equation is the generating function for the Legendre polynomials. As an example, the electric potential Φ( r , θ ) (in spherical coordinates ) due to a point charge located on the z -axis at z = a (see diagram right) varies as Φ ( r , θ ) ∝ 1 R = 1 r 2 + a 2 − 2 a r cos θ . {\displaystyle \Phi (r,\theta ...
can be proved by the techniques at Stirling numbers and exponential generating functions#Stirling numbers of the first kind and Binomial coefficient#Ordinary generating functions. The table in section 6.1 of Concrete Mathematics provides a plethora of generalized forms of finite sums involving the Stirling numbers. Several particular finite ...
The moment generating function of a real random variable is the expected value of , as a function of the real parameter . For a normal distribution with density f {\displaystyle f} , mean μ {\displaystyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} , the moment generating function exists and is equal to
The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, Analytic Combinatorics, while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions. During two centuries, generating functions ...