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...

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

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

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

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...

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

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

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...

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

\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...

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 (...

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...

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 ∞...

}}=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;...

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

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...

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

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 −...

{\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...

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

natural number. In that case x is the nth pentatope number. The generating function for pentatope numbers is x ( 1 − x ) 5 = x + 5 x 2 + 15 x 3 + 35...

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...

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

{\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...