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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)
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
Uniform distribution on an interval
}}-b\leq x\leq b,\\[8pt]0&{\text{otherwise}};\end{cases}}} the moment-generating function reduces to the simple form: M X = sinh b t b t . {\displaystyle
Continuous uniform distribution
Continuous_uniform_distribution
Probability distribution
\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
Normal_distribution
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)
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
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
Formula whose values are the prime numbers
\lfloor \ \rfloor } is the floor function, which rounds down to the nearest integer. The first few values of the function are 2, 2, 3, 2, 5, 2, 7, 2, 2,
Formula_for_primes
Discrete-variable probability distribution
and statistics, a probability mass function (sometimes called probability function or frequency function) is a function that gives the probability that a
Probability_mass_function
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
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
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 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
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
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
Compound probability distribution
{\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
Mixed_Poisson_distribution
Special mathematical functions defined on the surface of a sphere
can be seen in many ways (perhaps most simply from the Herglotz generating function), with r {\displaystyle \mathbf {r} } being a unit vector, Y ℓ m
Spherical_harmonics
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)
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
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
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
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
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
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
Graphical aid for deriving some concepts in combinatorics
(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
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
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
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
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
Formulation of classical mechanics
{\displaystyle Q_{m}=\beta _{m}} . Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant A {\displaystyle A}
Hamilton–Jacobi_equation
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
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
defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, σ n ( x ) {\displaystyle
Stirling_polynomials
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
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
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
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
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
In mathematics, a quantitative measure of the shape of a set of points
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)
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
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
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
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 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
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
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
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
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)
French polymath (1749–1827)
probability-generating function of the former. Laplace then shows how, by means of interpolation, these coefficients may be determined from the generating function
Pierre-Simon_Laplace
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
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
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
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
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
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
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
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
Stochastic process for effort or wear
where Γ ( z ) {\displaystyle \Gamma (z)} is the Gamma function. The moment generating function is the expected value of exp ( t X ) {\displaystyle \exp(tX)}
Gamma_process
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
Probability distribution
characteristic function of the beta distribution is displayed for symmetric (α = β) and skewed (α ≠ β) cases. It also follows that the moment generating function is
Beta_distribution
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
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
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
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
Statistical function that defines the quantiles of a probability distribution
probability distribution's quantile function is the inverse of its cumulative distribution function. That is, the quantile function of a distribution D {\displaystyle
Quantile_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
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
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)
Recursive integer sequence
binomial coefficients, by Stirling's approximation for n!, or via generating functions. The only Catalan numbers Cn that are odd are those for which n =
Catalan_number
Symmetric function invariant of graphs
function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function
Chromatic_symmetric_function
Discrete probability distribution
applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution: lim n → ∞ P ( n ) ( x
Poisson_distribution
Pair of polynomial sequences
{1-tx}{1-2tx+t^{2}}}.} There are several other generating functions for the Chebyshev polynomials; the exponential generating function is ∑ n = 0 ∞ T n ( x ) t n n !
Chebyshev_polynomials
of F generated by the coordinates of P. The logarithmic derivative of the infinite product Z(X, t) is easily seen to be the generating function discussed
Local_zeta_function
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
Family of probability distributions
central limit-like effect of generating distributions that manifest variance-to-mean power functions, will also generate processes that manifest 1/f noise
Tweedie_distribution
Family of probability distributions related to the normal distribution
for the moment-generating function for the distribution of x. In particular, using the properties of the cumulant generating function, E ( T j ) = ∂
Exponential_family
Numbers parameterizing ways to partition a set
{(-1)^{k-i}i^{n}}{(k-i)!i!}}.} (See also Stirling numbers and exponential generating functions in symbolic combinatorics#Stirling numbers of the second kind for
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
\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
Compound Poisson-family discrete probability distribution
generating function is, G Y ( z ) = exp ( λ ( e ϕ ( z − 1 ) − 1 ) ) {\displaystyle G_{Y}(z)=\exp(\lambda (e^{\phi (z-1)}-1))} From the generating function
Neyman_Type_A_distribution
Average value of a random variable
variables can be used to specify their distributions, via their moment generating functions. To empirically estimate the expected value of a random variable
Expected_value
Probability distribution
moment-generating function is actually undefined. Like all stable distributions except the normal distribution, the wing of the probability density function
Lévy_distribution
Probability distribution
6 1 / 6 = 5 {\displaystyle {\frac {1-1/6}{1/6}}=5} . The moment generating function of the geometric distribution when defined over N {\displaystyle
Geometric_distribution
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
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and
Stirling numbers and exponential generating functions in symbolic combinatorics
Stirling_numbers_and_exponential_generating_functions_in_symbolic_combinatorics
{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
Mathematical series
equation are respectively defined in. The sequence an generated by a Dirichlet series generating function corresponding to: ζ ( s ) m = ∑ n = 1 ∞ a n n s {\displaystyle
Dirichlet_series
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
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
Linear transform from the time domain to the frequency domain
important example of the unilateral Z-transform is the probability-generating function, where the component x [ n ] {\displaystyle x[n]} is the probability
Z-transform
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
Infinite sequence of numbers satisfying a linear equation
constant-recursive. This is because the generating function of the Catalan numbers is not a rational function (see #Equivalent definitions). A sequence
Constant-recursive_sequence
Probability distribution
^{(1)}} is the trigamma function. This can be derived using the exponential family formula for the moment generating function of the sufficient statistic
Gamma_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)
Sequence of differential equation solutions
L_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k}.} The generating function for them likewise follows, ∑ n = 0 ∞ t n L n ( x ) = 1 1 − t e −
Laguerre_polynomials
GENERATING FUNCTION
GENERATING FUNCTION
Boy/Male
Tamil
Young generation
Girl/Female
Biblical
Nativity, generation.
Boy/Male
Indian
Young Generation
Boy/Male
Indian, Punjabi, Sikh
New Generation
Boy/Male
Hindu, Indian
Young Generation
Girl/Female
Biblical
Generation, habitation.
Boy/Male
Tamil
Forthcoming generation
Boy/Male
Biblical, British, English
Nativity; Generation
Boy/Male
Muslim
Old generation
Girl/Female
Indian, Tamil
Generation
Boy/Male
Biblical
Nativity, generation.
Boy/Male
Japanese Welsh
Large; generation.
Girl/Female
Biblical
A generation.
Boy/Male
British, Czech, Hindu, Indian
New Generation
Boy/Male
Gujarati, Hindu, Indian, Kannada
Era; Generation
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi
Generation; Coming Generation of Father; Family
Boy/Male
Biblical
Nativity, generation.
Boy/Male
Indian, Modern
Generations
Girl/Female
Indian
Generation
Girl/Female
Biblical
Birth, generation.
GENERATING FUNCTION
GENERATING FUNCTION
Girl/Female
American, German
Strong as Man; Free Woman
Boy/Male
Hindu, Indian, Marathi
Idol of Lord Shiva
Male
Iranian/Persian
Persian form of Arabic Haidar, HEYDAR means "lion."
Boy/Male
Tamil
Celebrated
Boy/Male
English
Contemporary phonetic'enduring.
Girl/Female
Hindu, Indian, Tamil, Telugu
Green; Lovely; Beautiful Princess; Sweet
Biblical
bitter contrition, without judgment
Boy/Male
Sikh
God of peace
Boy/Male
Hindu
Behavior
Boy/Male
Arabic, Muslim
Friend of Allah; An Epithet of Prophet Ibrahim
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
n.
The power of generating.
n.
The formation or production of any geometrical magnitude, as a line, a surface, a solid, by the motion, in accordance with a mathematical law, of a point or a magnitude; as, the generation of a line or curve by the motion of a point, of a surface by a line, a sphere by a semicircle, etc.
a.
Generating or causing phlegm.
n.
Alternate generation. See under Generation.
a.
Having the power of entering, piercing, or pervading; sharp; subtile; penetrative; as, a penetrating odor.
a.
Pertaining to generation, or to the generative organs.
v. i.
Generation.
a.
Generating mucus.
n.
Origination by some process, mathematical, chemical, or vital; production; formation; as, the generation of sounds, of gases, of curves, etc.
a.
Generating phosphorescence; as, phosphorogenic rays.
a.
Producing or generating pus.
p. pr. & vb. n.
of Generate
a.
Acute; discerning; sagacious; quick to discover; as, a penetrating mind.
a.
Having the power of generating, propagating, originating, or producing.
a.
Generating bile.
n.
That form of alternate generation in which two kinds of sexual generation, or a sexual and a parthenogenetic generation, alternate; -- in distinction from metagenesis, where sexual and asexual generations alternate.
a.
Generating or containing pus; purulent.
a.
generating or producing dew.
a.
Windy; generating wind.
n.
The act of generating or begetting; procreation, as of animals.