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...

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...

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 ]...

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

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

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

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

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

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

In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable...

}}=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...

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...

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

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...

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

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 (...

{\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...

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 ∞...

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

cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,...

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...

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...

a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given...

ordinary generating function of the Fibonacci sequence, ∑ i = 0 ∞ F i z i {\displaystyle \sum _{i=0}^{\infty }F_{i}z^{i}} , is the rational function z 1 −...

(because the objects are not distinguished). This is represented by the generating function 1 + 1 x + 1 x 2 + 1 x 3 + … = 1 + x + x 2 + x 3 + … = 1 1 − x . {\displaystyle...

gamma function. Using that f ( . ; m, r, ps) for s ∈ (0, 1] is also a probability mass function, it follows that the probability generating function is given...

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...