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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.
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.
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.
For any , the coefficient of /! in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value [] . The cumulant generating function is the logarithm of the moment generating function, namely
Moment-generating function; P. Probability-generating function; R. Rook polynomial; T. Tau function (integrable systems) W. Weisner's method
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 ...
Alternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients: = (+), where δ ij is the Kronecker delta function and the x k are the N Gauss–Chebyshev zeros of T N (x): = ((+)).
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.