a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are...

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability...

are defined using the cumulant-generating function K(t), which is the natural logarithm of the moment-generating function: K ( t ) = log ⁡ E ⁡ [ e t X ]...

probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the...

specifically in Hamiltonian mechanics, a generating function is, loosely, a function whose partial derivatives generate the differential equations that determine...

of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another...

an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal...

would be ⁠ 1 15 . {\displaystyle {\tfrac {1}{15}}.} ⁠ The moment-generating function of the continuous uniform distribution is: M X = E ⁡ [ e t X ] =...

canonical. The various generating functions and its properties tabulated below is discussed in detail: The type 1 generating function G1 depends only on the...

binomial coefficients are to exponential generating series what falling factorials are to ordinary generating series. The product of all binomial coefficients...

moment-generating function, and call the logarithm of the characteristic function the second cumulant generating function. Characteristic functions can be...

\operatorname {E} [X^{k}]} ⁠. The cumulant generating function is the logarithm of the moment generating function, namely g ( t ) = ln ⁡ M ( t ) = μ t + 1...

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist;...

orthogonal polynomials obtained from the Rodrigues formula have a generating function of the form G ( x , u ) = ∑ n = 0 ∞ u n P n ( x ) G(x,u)=\sum _{n=0}^{\infty...

}}=e^{2x}I_{0}(2x),} where I0 is a modified Bessel function of the first kind. The generating function of the squares of the central binomial coefficients...

expansion at x of the entire function z → e−z2 (in the physicist's case). One can also derive the (physicist's) generating function by using Cauchy's integral...

functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior. A rational...

and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a...

Meijer G-function. The characteristic function has also been obtained by Muraleedharan et al. (2007). The characteristic function and moment generating function...

enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form. Often, a complicated...

the constant of proportionality is 1. This function is closely related to the exponential generating function of the Hermite polynomials, which are the...

calculate the generating function F ( x ) = ∑ n ≥ 0 H ( n ) x n {\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}} . The generating function satisfies F (...

binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence...

roots of the first few spherical Bessel functions are: The spherical Bessel functions have the generating functions 1 z cos ⁡ ( z 2 − 2 z t ) = ∑ n = 0 ∞...

{\displaystyle M_{\pi }} is the moment generating function of the density. For the probability generating function, one obtains m X ( s ) = M π ( s − 1...

its conjugate. The related function z ↦ − s ( − 1 / z ) {\textstyle z\mapsto -s\left(-1/z\right)} is the generating function for the negafibonacci numbers...

and λ {\displaystyle \lambda } as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit...

{\displaystyle n\geq 0} these weighted harmonic number expansions are generated by the generating function 1 n ! [ n + 1 k ] = [ x k ] exp ⁡ ( ∑ m ≥ 1 ( − 1 ) m − 1...

confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated as M ( t...

of F generated by the coordinates of P. The logarithmic derivative of the infinite product Z(X, t) is easily seen to be the generating function discussed...