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...

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 ]...

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

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

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

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...

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...

incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. A locally finite poset is...

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...

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

{\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...

}}=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...

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

{\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...

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 (...

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

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

is the value at zero of the n-th derivative of this function. The exponential generating function can be derived in a number of ways; for example, taking...

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 −...

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

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

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

exponential function and the nonemptiness constraint ≥1 into subtraction by one. An alternative method for deriving the same generating function uses the...

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

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 ∞...

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...

function, that function converges for all | x | < 1 {\displaystyle |x|<1} , in which case it can be expressed as the meromorphic generating function 1...