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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
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)
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
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
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)
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)
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
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
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
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
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
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
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
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
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 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)
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
Let p 3 ( 0 ) ≡ 1 {\displaystyle p_{3}(0)\equiv 1} . Define the generating function of solid partitions, P 3 ( q ) {\displaystyle P_{3}(q)} , by P 3
Solid_partition
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
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
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
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)
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
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
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
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
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
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)
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 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
Class of probability distributions
NEF. This follows from the properties of the cumulant generating function. The variance function for random variables with an NEF distribution can be written
Natural_exponential_family
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}} The moment generating function is given by: E ( e − t x ) = { β s s b t + s b 2 F 1 ( s + 1 ,
Gamma/Gompertz_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
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
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
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
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
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
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
{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
defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, σ n ( x ) {\displaystyle
Stirling_polynomials
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
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
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
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
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
Wigner distribution function in physics as opposed to in signal processing
probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x)
Wigner quasiprobability distribution
Wigner_quasiprobability_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
GENERATING FUNCTION
GENERATING FUNCTION
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi
Generation; Coming Generation of Father; Family
Girl/Female
Biblical
Birth, generation.
Boy/Male
Japanese Welsh
Large; generation.
Girl/Female
Biblical
Generation, habitation.
Boy/Male
Tamil
Forthcoming generation
Boy/Male
Biblical
Nativity, generation.
Boy/Male
Muslim
Old generation
Girl/Female
Indian
Generation
Boy/Male
Tamil
Young generation
Boy/Male
Gujarati, Hindu, Indian, Kannada
Era; Generation
Boy/Male
Indian, Punjabi, Sikh
New Generation
Girl/Female
Biblical
Nativity, generation.
Boy/Male
British, Czech, Hindu, Indian
New Generation
Boy/Male
Indian
Young Generation
Girl/Female
Indian, Tamil
Generation
Girl/Female
Biblical
A generation.
Boy/Male
Biblical, British, English
Nativity; Generation
Boy/Male
Hindu, Indian
Young Generation
Boy/Male
Indian, Modern
Generations
Boy/Male
Biblical
Nativity, generation.
GENERATING FUNCTION
GENERATING FUNCTION
Girl/Female
Arabic, Muslim
Perfect; Complete
Girl/Female
Hindu, Indian
Prayer Ceremony; Great
Girl/Female
Hindu
Boy/Male
Hindu, Indian, Kannada, Sanskrit
Unencumbered; Sky-clad
Boy/Male
Hindu, Indian
Dharm
Girl/Female
Hindu, Indian
Like a Goddess
Boy/Male
Hindu, Indian, Punjabi, Sikh
One who Remains Aware of Guru's Word
Male
Egyptian
, first king of Egypt; the son of Io.
Girl/Female
Greek
Manly. Brave. Feminine form of Andrew.
Boy/Male
English
from Elijah 'My God is Jehovah.
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
GENERATING FUNCTION
p. pr. & vb. n.
of Generate
a.
Producing or generating pus.
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.
v. i.
Generation.
a.
Having the power of entering, piercing, or pervading; sharp; subtile; penetrative; as, a penetrating odor.
n.
Alternate generation. See under Generation.
a.
Acute; discerning; sagacious; quick to discover; as, a penetrating mind.
n.
The power of generating.
a.
Generating or causing phlegm.
a.
Generating or containing pus; purulent.
a.
Windy; generating wind.
n.
Origination by some process, mathematical, chemical, or vital; production; formation; as, the generation of sounds, of gases, of curves, etc.
a.
Pertaining to generation, or to the generative organs.
a.
Generating phosphorescence; as, phosphorogenic rays.
n.
The act of generating or begetting; procreation, as of animals.
a.
Generating mucus.
a.
generating or producing dew.
a.
Generating bile.
a.
Having the power of generating, propagating, originating, or producing.
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.