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...

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

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

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...

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

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

\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...

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

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

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...

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...

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 (...

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...

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

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

-th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain...

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...

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...

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...

MacMahon. MacMahon also mentions the generating functions of plane partitions. The formula for the generating function can be written in an alternative way...

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...