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Formal power series
a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are
Generating_function
Concept in probability theory and statistics
probability theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Power series derived from a discrete probability distribution
probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the
Probability generating function
Probability_generating_function
Set of quantities in probability theory
the cumulant generating function (CGF) K(t), which is a generating function that is the natural logarithm of the moment generating function: K ( t ) = log
Cumulant
Operation on formal power series
of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another
Generating function transformation
Generating_function_transformation
Function used to generate other functions
specifically in Hamiltonian mechanics, a generating function is, loosely, a function whose partial derivatives generate the differential equations that determine
Generating_function_(physics)
Uniform distribution on an interval
would be 1 15 . {\displaystyle {\tfrac {1}{15}}.} The moment-generating function of the continuous uniform distribution is: M X = E [ e t X ] =
Continuous uniform distribution
Continuous_uniform_distribution
Formula whose values are the prime numbers
floor function. Then for n≥1, pn=⌊fn⌋. The initial constant f1=2.920050977316 given in the article is precise enough for equation (1) to generate the primes
Formula_for_primes
Fourier transform of the probability density function
moment-generating function, and call the logarithm of the characteristic function the second cumulant generating function. Characteristic functions can be
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Coordinate transformation that preserves the form of Hamilton's equations
canonical. The various generating functions and its properties tabulated below is discussed in detail: The type 1 generating function G1 depends only on the
Canonical_transformation
Probability distribution
generating function exists and is equal to M(t)=E[etX]=f̂(it)=eμteσ2t2/2. For any k, the coefficient of tk/k! in the moment generating function (expressed
Normal_distribution
Topics referred to by the same term
error) Generating function (math) Generating function (physics) Generating set Generating set of a group Generating trigonometric tables Generating a curve
Generate
Number of subsets of a given size
binomial coefficients are to exponential generating series what falling factorials are to ordinary generating series. The product of all binomial coefficients
Binomial_coefficient
Area of combinatorics that deals with the number of ways certain patterns can be formed
enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form. Often, a complicated
Enumerative_combinatorics
Array of nonnegative integers in combinatorics
MacMahon. MacMahon also mentions the generating functions of plane partitions. The formula for the generating function can be written in an alternative way
Plane_partition
Number of ways to pair up n objects
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
Telephone number (mathematics)
Telephone_number_(mathematics)
Sequence of numbers ((2n) choose (n))
}}=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
Central_binomial_coefficient
Number of partitions of an integer
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
Partition function (number theory)
Partition_function_(number_theory)
analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck. The generating function for the general
Difference_polynomials
Formula for the Legendre polynomials
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
Rodrigues'_formula
Graphical aid for deriving some concepts in combinatorics
(because the objects are not distinguished). This is represented by the generating function 1+1x+1x2+1x3+…=1+x+x2+x3+…=11−x. The series is a geometric series
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Mathematical operation
series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A
Dirichlet_series_inversion
Discrete-variable probability distribution
a probability mass function (PMF), discrete probability density function, probability function, or frequency function is a function that gives the probability
Probability_mass_function
Compound probability distribution
characteristic function has the form φX(s)=Mπ(eis−1). Where Mπ is the moment generating function of the density. For the probability generating function, one obtains
Mixed_Poisson_distribution
Continuous probability distribution
{\displaystyle {\text{MTBF}}(k,\lambda )=\lambda \Gamma (1+1/k).} The moment generating function of the logarithm of a Weibull distributed random variable is given
Weibull_distribution
Statistical probability Distribution for discrete event counts
"Hermite distribution" from the fact its probability function and the moment generating function can be expressed in terms of the coefficients of (modified)
Hermite_distribution
Methods used in combinatorics
double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate
Combinatorial_principles
Mathematical transformation on sequences
numbers—also known as secant or tangent numbers. The exponential generating function of a sequence (an) is defined by E G ( a n ; x ) = ∑ n = 0 ∞ a n
Boustrophedon_transform
Polynomial sequence
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
Hermite_polynomials
Number that represents a hexagon with a dot in the center
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 (
Centered_hexagonal_number
exponential generating function, and the Stirling (convolution) polynomials, σn(x), which also satisfy a characteristic ordinary generating function and that
Stirling_polynomials
Associative algebra used in combinatorics
incidence algebras give a natural construction of various types of generating functions used in combinatorics and number theory. A locally finite poset is
Incidence_algebra
Function of a matrix
{\det {\big (}I-ZS{\big )}}}{\Big .}} , is in fact a multivariate generating function for a series of hafnians, and the right-hand side constitutes its
Hafnian
Family of solutions to related differential equations
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 ∞
Bessel_function
Infinite integer series where the next number is the sum of the two preceding it
322 − 4 ( 18 ) + 6 {\displaystyle 256=322-4(18)+6} The ordinary generating function of the sequence of Lucas numbers is the power series Φ ( x ) = ∑
Lucas_number
In probability theory and statistics, the factorial moment generating function (FMGF) of the probability distribution of a real-valued random variable
Factorial moment generating function
Factorial_moment_generating_function
Formula for number of orbits of a group action
branches of a rooted tree. Thus the generating function f for the colors is derived from the generating function F for arrangements, and the Pólya enumeration
Pólya_enumeration_theorem
Numbers obtained by adding the two previous ones
F_{1}=F^{\prime }(0)=1} , the exponential generating function of the Fibonacci numbers is given by the entire function F ( x ) = e φ x − e ψ x 5 {\displaystyle
Fibonacci_sequence
∑n=1∞Hn(r)nm<+∞ when m>r. The generating function of the hyperharmonic numbers is ∑n=0∞Hn(r)zn=−ln(1−z)(1−z)r. The exponential generating function is much harder to
Hyperharmonic_number
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Set of probability distributions
same dimension as X {\displaystyle \mathbf {X} } . The cumulant-generating function of Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma
Exponential_dispersion_model
Special mathematical functions defined on the surface of a sphere
and λ {\displaystyle \lambda } as real parameters. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, §VII.7, who credit
Spherical_harmonics
Transformation of a mathematical sequence
binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence
Binomial_transform
Statistical approximation method
formula for any PDF or probability mass function of a distribution, based on the moment generating function. There is also a formula for the CDF of the
Saddlepoint approximation method
Saddlepoint_approximation_method
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Type of polynomial sequence
Polynomial sequences of binomial type are precisely those whose generating functions are formal (not necessarily convergent) power series of the form
Binomial_type
Count of permutations by cycles
{\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
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
{d}{dz}}p_{n}(z)=np_{n-1}(z).} The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely A (
Q-difference_polynomial
Integer partition attribute, in number theory
author has no conception." The Durfee square method leads to this generating function for the integer partitions: P ( x ) = ∑ k = 0 ∞ x k 2 ∏ i = 1 k (
Durfee_square
Probability distribution in mathematics
series itself, and are therefore undefined for large n. The moment generating function is defined as M ( t ; s ) = E ( e t X ) = 1 ζ ( s ) ∑ k = 1 ∞ e t
Zeta_distribution
Mathematics concept
form expressions or have a generating function with a simple form. The following rules are notable: The sequence generated is 1, 3, 5, 11, 21, 43, 85
Elementary_cellular_automaton
Noncentral generalization of the chi-squared distribution
the series are (1 + 2i) + (k − 1) = k + 2i as required. The moment-generating function is given by M ( t ; k , λ ) = exp ( λ t 1 − 2 t ) ( 1 − 2 t ) k
Noncentral chi-squared distribution
Noncentral_chi-squared_distribution
Gives a functional equation satisfied by the generating function of any rational cone
equation is satisfied by the integer-point generating function of a rational cone and the generating function of the cone's interior. A rational cone is
Stanley's_reciprocity_theorem
Probability distribution
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
Extended negative binomial distribution
Extended_negative_binomial_distribution
Theorem In probability theory and statistics
by Harry Bateman. In Campbell's work, he presents the moments and generating functions of the random sum of a Poisson process on the real line, but remarks
Campbell's theorem (probability)
Campbell's_theorem_(probability)
Count of the possible partitions of a set
exponential function and the nonemptiness constraint ≥1 into subtraction by one. An alternative method for deriving the same generating function uses the
Bell_number
Integral approximation method popular in condensed matter physics
obtain higher order terms in the Sommerfeld expansion by use of a generating function for moments of the Fermi distribution. This is given by ∫ − ∞ ∞ d
Sommerfeld_expansion
Decomposition of an integer as a sum of positive integers
3010, 3718, 4565, 5604, ... (sequence A000041 in the OEIS). The generating function of p {\displaystyle p} is ∑ n = 0 ∞ p ( n ) q n = ∏ j = 1 ∞ ∑ i =
Integer_partition
Generating function in integrable systems
Tau functions also appear as matrix model partition functions in the spectral theory of random matrices, and may also serve as generating functions, in
Tau function (integrable systems)
Tau_function_(integrable_systems)
Pair of polynomial sequences
polynomials; the exponential generating function is ∑n=0∞Tn(x)tnn!=12(exp(t(x−x2−1 ))+exp(t(x+x2−1 )))=etxcosh(tx2−1 ). The generating function relevant for 2-dimensional
Chebyshev_polynomials
Centered figurate number that represents a pentagon with a dot in the center
( P 0 = 1 ) {\displaystyle P_{n}=5T_{n}+1,n\geq 0,(P_{0}=1)} The generating function that gives the centered pentagonal numbers is: 1 + 5 x + 13 x 2 +
Centered_pentagonal_number
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
{1}{120n^{4}}}-\cdots ,\end{aligned}}} where Bk are the Bernoulli numbers. A generating function for the harmonic numbers is ∑ n = 1 ∞ z n H n = − ln ( 1 − z )
Harmonic_number
Generalization of polynomials
only its generating function Q ( x ) := ∑ n ≥ 0 q ( n ) x n {\displaystyle Q(x):=\sum _{n\geq 0}q(n)x^{n}} evaluates to a rational function of the form
Quasi-polynomial
System of complete and orthogonal polynomials
two polynomials P0 and P1, allows all the rest to be generated recursively. The generating function approach is directly connected to the multipole expansion
Legendre_polynomials
Continuous probability distribution, named after Benjamin Gompertz
{\displaystyle \eta ,b>0,} and x ≥ 0 . {\displaystyle x\geq 0\,.} The moment generating function is: E ( e − t X ) = η e η E t / b ( η ) {\displaystyle
Gompertz_distribution
Probability distribution
by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value E [ e t X
Log-normal_distribution
Rational number sequence
between the generating functions for B m + {\displaystyle B_{m}^{+}} and B m − {\displaystyle B_{m}^{-}} is t. The (ordinary) generating function z − 1 ψ
Bernoulli_number
Number of integers coprime to and less than n
converges for ℜ ( s ) > 2 {\displaystyle \Re (s)>2} . The Lambert series generating function is ∑ n = 1 ∞ φ ( n ) q n 1 − q n = q ( 1 − q ) 2 {\displaystyle \sum
Euler's_totient_function
Number in the 5th cell of any row of Pascal's triangle
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
Pentatope_number
Exponentially decreasing bounds on tail distributions of random variables
upper bound on the tail of a random variable based on its moment generating function. The minimum of all such exponential bounds forms the Chernoff or
Chernoff_bound
Discrete probability distribution
applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution: limn→∞P(n)(x)=limn→∞
Poisson_distribution
Concept in network science
{\displaystyle G_{1}(x)={\frac {G'_{0}(x)}{G'_{0}(1)}}} If we know the generating function for a probability distribution P ( k ) {\displaystyle P(k)} then
Degree_distribution
Centered figurate number that represents a triangle with a dot in the center
function, that function converges for all | x | < 1 {\displaystyle |x|<1} , in which case it can be expressed as the meromorphic generating function 1
Centered_triangular_number
Conjecture in algebraic geometry
these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential
Witten_conjecture
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Probability distribution
fractional absolute moments exist. The Cauchy distribution has no moment generating function. In mathematics, it is closely related to the Poisson kernel, which
Cauchy_distribution
Pair of non-standard six-sided dice
number is 1/n. Consider the canonical cubical (six-sided) die. The generating function for the throws of such a die is x + x 2 + x 3 + x 4 + x 5 + x 6 {\displaystyle
Sicherman_dice
Measure of the shape of a function
n} th moment of the function given in the brackets. This identity follows by the convolution theorem for moment generating function and applying the chain
Moment_(mathematics)
Infinite sequence in mathematics
repeated in reverse order, replacing 0 with 1 and vice versa. The generating function of the paperfolding sequence is given by G ( t n ; x ) = ∑ n = 1
Regular_paperfolding_sequence
\right),} where w(x;λ,φ) is the weight function given above. The Meixner–Pollaczek polynomials have the generating function ∑ n = 0 ∞ t n P n ( λ ) ( x ; ϕ )
Meixner–Pollaczek_polynomials
Probability distribution
confluent hypergeometric function and J1 is the Bessel function of the first kind. Likewise the moment generating function can be calculated as M ( t
Wigner semicircle distribution
Wigner_semicircle_distribution
Number of orderings allowing ties
ordered Bell numbers causes their ordinary generating function to diverge; instead the exponential generating function is used. For the ordered Bell numbers
Ordered_Bell_number
Compound Poisson-family discrete probability distribution
generating function exists and is equal to M(t)=GY(et)=exp(λ(eϕ(et−1)−1)) The cumulant generating function is the logarithm of the moment generating
Neyman_Type_A_distribution
Probability distribution
, x ) {\displaystyle P(k,x)} is the regularized gamma function. The moment-generating function is given by: M ( t ) = M ( k 2 , 1 2 , t 2 2 ) + t 2 Γ
Chi_distribution
Fundamental result in the theory of large deviations
Harald Cramér in 1938. The logarithmic moment generating function (which is the cumulant-generating function) of a random variable is defined as: Λ ( t )
Cramér's theorem (large deviations)
Cramér's_theorem_(large_deviations)
Function whose domain is the positive integers
(Because Ω(1) = ω(1) = 0.) τ(n), the Ramanujan tau function, is defined by its generating function identity: ∑n≥1τ(n)qn=q∏n≥1(1−qn)24. Although it is
Arithmetic_function
Mathematics concept
factorial). The Bessel polynomials, with index shifted, have the generating function ∑ n = 0 ∞ 2 π x n + 1 2 e x K n − 1 2 ( x ) t n n ! = 1 + x ∑ n =
Bessel_polynomials
Sequence of differential equation solutions
The generating function for them is ∑n=0∞tnLn(α)(x)=1(1−t)α+1e−tx/(1−t). Laguerre functions are defined by confluent hypergeometric functions and Kummer's
Laguerre_polynomials
Discrete probability distribution
ν)=∑k=0nxk(nk)ν. Then, the probability generating function, moment generating function and characteristic function are given, respectively, by: G(t)=T(tp/(1−p)
Conway–Maxwell–binomial distribution
Conway–Maxwell–binomial_distribution
Infinite binary sequence generated by repeated complementation and concatenation
string as follows: n = 7 print(f"{thue_morse_bits(n):0{1<<n}b}") A generating function for the sequence can be defined by: ∏ i = 0 ∞ ( 1 − x 2 i ) = ∑ j
Thue–Morse_sequence
Type of generating function in mathematics
In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on
Igusa_zeta_function
Numbers parameterizing ways to partition a set
representation has no two consecutive 1s. For a fixed integer n, the ordinary generating function for Stirling numbers of the second kind { n 0 } , { n 1 } , … {\displaystyle
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Integral transform useful in probability theory, physics, and engineering
of generating functions (1814), and the integral form of the Laplace transform evolved naturally as a result. Laplace's use of generating functions was
Laplace_transform
Figurate number representing a dodecahedron
the dodecahedral graph fall along (n+1) unit balls. The ordinary generating function of the Dodecahedral numbers is x ( 1 + 16 x + 10 x 2 ) ( − 1 + x
Dodecahedral_number
Number-theoretical function
} The generating function of the sequence r k ( n ) {\displaystyle r_{k}(n)} for fixed k can be expressed in terms of the Jacobi theta function: ϑ ( 0
Sum_of_squares_function
Measure of the deviation of position over time
moment-generating function, an extremely useful, and general function when dealing with probability densities. The moment-generating function describes
Mean_squared_displacement
Number of dots in a centred dot square
a consequence of (2n − 1)2 + (2n2 − 2n)2 = (2n2 − 2n + 1)2. The generating function that gives the centered square numbers is: ( x + 1 ) 2 ( 1 − x )
Centered_square_number
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Inequality in probability theory
theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable, implying that such variables are
Hoeffding's_lemma
Centered figurate number that represents a decagon with a dot in the center
a Centered decagonal number iff 20N + 5 is a Square number. The generating function of the centered decagonal number is x ∗ ( 1 + 8 x + x 2 ) ( 1 − x
Centered_decagonal_number
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