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Mathematical series
In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where α is any
Binomial_series
Algebraic expansion of powers of a binomial
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem
Binomial_theorem
Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is
Binomial_coefficient
Mathematical approximation of a function
binomial series given in the next section. The binomial series is the power series (1+x)α=∑n=0∞(αn)xn whose coefficients are the generalized binomial
Taylor_series
Probability distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes
Binomial_distribution
Probability distribution
In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that
Negative binomial distribution
Negative_binomial_distribution
Topics referred to by the same term
of polynomials Binomial series, a mathematical series Binomial distribution, a type of probability distribution Binomial process Binomial test, a test of
Binomial
Transformation of a mathematical sequence
where δnm is the Kronecker delta. The original series can be regained by an=∑k=0n(−1)k(nk)sk. The binomial transform of a sequence is just the n-th forward
Binomial_transform
Type of polynomial sequence
which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities pn(x+y)=∑k=0n(nk)pk(x)pn−k(y)
Binomial_type
Sequence of numbers ((2n) choose (n))
In mathematics the nth central binomial coefficient is the particular binomial coefficient (2nn)=(2n)!(n!)2 for all n≥0. They are called central since
Central_binomial_coefficient
Approximation of powers of some binomials
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that (1+x)α≈1+αx. It is valid when
Binomial_approximation
In mathematics, a polynomial with two terms
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after
Binomial_(polynomial)
Family of polynomials
mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
Statistical confidence interval for success counts
statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure
Binomial proportion confidence interval
Binomial_proportion_confidence_interval
1/e. That is, limn→∞(1−1n)n=1e Consider the sequence: en=(1+1n)n By the binomial theorem: en=∑k=0n(nk)1nk=∑k=0nnk_k!1nk which converges to e as n increases
List_of_representations_of_e
Graphical aid for deriving some concepts in combinatorics
xk(1−x)k, and the coefficient of xn is found using binomial series to be (n−1k−1). Gaussian binomial coefficient Partition (number theory) Twelvefold way
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Regression analysis technique
a series of n {\displaystyle n} independent Bernoulli trials, where each trial has probability of success p {\displaystyle p} . In binomial regression
Binomial_regression
A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an orthogonal wavelet developed in 1990. The binomial QMF bank with perfect
Binomial_QMF
filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Mathematical set with repetitions allowed
"n choose k" for (nk). Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset
Multiset
Triangular array of the binomial coefficients
mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics
Pascal's_triangle
Topics referred to by the same term
Newton's series may refer to: The Newton series for finite differences, used in interpolation theory. The binomial series, first proved by Isaac Newton
Newton's_series
Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Recursive integer sequence
C0=limx→0c(x)=1. The square root term can be expanded as a power series using the binomial series 1−1−4x=−∑n=1∞(12n)(−4x)n=−∑n=1∞(−1)n−1(2n−3)!!2nn!(−4x)n=−∑n=0∞(−1)n(2n−1)
Catalan_number
Arrangement of trinomial coefficients
triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients
Pascal's_pyramid
Mathematical theorem on convolved binomial coefficients
identity (or Vandermonde's convolution) is the following identity for binomial coefficients: (m+nr)=∑k=0r(mk)(nr−k) for any nonnegative integers r, m
Vandermonde's_identity
Quantity in relativistic physics
1+12β2+38β4+516β6+35128β8+63256β10+⋯, which is a special case of a binomial series. The approximation γ≈1+12β2 may be used to calculate relativistic effects
Lorentz_factor
Near-field diffraction
r=z1+ρ2z2=z[1+ρ22z2−18(ρ2z2)2+⋯]=z+ρ22z−ρ48z3+⋯ If we consider all the terms of binomial series, then there is no approximation. Let us substitute this expression
Fresnel_diffraction
Mathematical concept
for the natural logarithm, the arcsin function, and the generalized binomial series. Jones & Thron (1980) p. 5 C. F. Gauss (1813), Werke, vol. 3 pp. 134–38
Gauss's_continued_fraction
In mathematics, series built from equally spaced terms of another series
series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series ∑
Series_multisection
Conjecture in combinatorial number theory
appear exactly five or seven times. Binomial coefficient De Weger, Benjamin M.M. (August 1995). "Equal binomial coefficients: some elementary considerations"
Singmaster's_conjecture
Mathematical function
+(s−1)(s−2)(s−3)3⋅3!⋯,ℜ(s)>0,=−γ−∑k=1∞(−1)kk(s−1k)⋯,ℜ(s)>0. where (s k) is the binomial coefficient. It may also be generalized to ψ(s+1)=−γ−1m∑k=1m−1m−ks+k−1
Digamma_function
Data structure that acts as a priority queue
In computer science, a binomial heap is a data structure that acts as a priority queue. It is an example of a mergeable heap (also called meldable heap)
Binomial_heap
Numerical method for calculating the gamma function
entire complex plane by the reflection formula, Γ(1−z)Γ(z)=πsinπz. The series A is convergent, and may be truncated to obtain an approximation with the
Lanczos_approximation
Infinite sum that is considered independently from any notion of convergence
either by composition with the binomial series (1 + x)α, or by composition with the exponential and the logarithmic series, fα=exp(αlog(f)), or as the
Formal_power_series
Generalization of the binomial theorem to other polynomials
of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. For any positive integer m and any non-negative
Multinomial_theorem
Discrete probability distribution
Log(p) distribution, then ∑i=1NXi has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson
Logarithmic_distribution
Recurrence relations of binomial coefficients in Pascal's triangle
partial sum formula for geometric series, we find that Xr+Xr+1+⋯+Xn=Xr−Xn+11−X=Xn+1−Xrx. Further, by the binomial theorem, we also find that
Hockey-stick_identity
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist
Combinatorics
Solution of a confluent hypergeometric equation
integral followed by expanding the binomial series and integrating it formally term by term gives rise to an asymptotic series expansion, valid as x → ∞: U(a
Confluent hypergeometric function
Confluent_hypergeometric_function
Fictional book mentioned in stories of Sherlock Holmes
A Treatise on the Binomial Theorem is a fictional work of mathematics by the young Professor James Moriarty, the criminal mastermind and archenemy of the
A Treatise on the Binomial Theorem
A_Treatise_on_the_Binomial_Theorem
series – see Taylor series Binomial series – the Maclaurin series of the function f given by f(x) = (1 + x) α Telescoping series Alternating series Geometric
List_of_real_analysis_topics
Type of function in mathematics
if α>0 is not an integer, then the binomial series (1+z)α=∑n=0∞(αn)zn has radius of convergence R=1. The series converges everywhere on the closed unit
Analytic_function
Rational numbers in a reciprocal logarithm
and 1 with respect to x, once directly and the second time using the binomial series expansion first. It implies the finite summation formula n!Gn=∑ℓ=0ns(n
Gregory_coefficients
Expectation or average of the falling factorial of a random variable
involve Stirling numbers of the second kind. If a random variable X has a binomial distribution with success probability p ∈ [0,1] and number of trials n
Factorial_moment
Generalization of the binomial distribution
probability theory, the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for each
Multinomial_distribution
Series related to Ramanujan's pi formulas
certain recurrence relation, sequences which may be expressed in terms of binomial coefficients (nk), and A,B,C employing modular forms of higher levels.
Ramanujan–Sato_series
Distance measured along the surface of the Earth
expressions in the FCC formula are derived from the truncation of the binomial series expansion form of M {\displaystyle M\,\!} and N {\displaystyle N\,\
Geographical_distance
∑k=1∞1nkk=ln(nn−1), that is ∀n>1 Series (mathematics) List of integrals Summation § Identities Taylor series Binomial theorem Gregory's series On-Line Encyclopedia
List_of_mathematical_series
Infinite matrices with Pascal's triangle as elements
combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle
Pascal_matrix
Characteristic of electromagnetic radiation
com/~u85920178/data/pathlos.htm#bulges Archived 2009-10-14 at the Wayback Machine Approximating 2-Ray Model by using Binomial series by Matthew Bazajian
Line-of-sight_propagation
Result from multiplying no factors
found in the binomial theorem (which assumes and implies that x0 = 1 for all x), Stirling number, König's theorem, binomial type, binomial series, difference
Empty_product
Geographic coordinate specifying north-south position
map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see Meridian arc for details. The length
Latitude
Numeral system in combinatorics
terms cancel each other, leaving the first and last term (see Telescoping series). However, when using Arabic numerals to write the digits (and not including
Factorial_number_system
Model describing the adsorption of a mono-layer of gas molecules on an ideal flat surface
chemical potential of an adsorbed molecule. As it has the form of binomial series, the summation is reduced to 𝒵(μA)=(1+x)NS, where x=ζLexp(μAkBT)
Langmuir_adsorption_model
Theorem in p-adic analysis
Newton series converges everywhere to f, so that for all x∈ℤp we have f(x)=∑n=0∞(Δnf)(0)(xn), where (xn)=x(x−1)(x−2)⋯(x−n+1)n! is the nth binomial coefficient
Mahler's_theorem
Numbering of combinations of items
argument proves that every N can be written in exactly one way as a sum of k binomial coefficients of the given form. The given formula allows finding the place
Combinatorial_number_system
Formula in mathematics
second application of the binomial expansion, introducing another summation over the index s. The product of the two binomial coefficients is simplified
Trinomial_expansion
Arithmetic operation, inverse of nth power
1665, Isaac Newton discovered the general binomial theorem, which can convert an nth root into an infinite series. Based on approach developed by François
Nth_root
Mathematical sequences in combinatorics
of Stirling number. The notation of brackets and braces, in analogy to binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later
Stirling_number
Motion extremely close to the speed of light
than c. Its kinetic energy can be approximated by first term of the γ binomial series: Ek=(γ−1)mc2=12mv2+[38mv4c2+...+mc2(2n)!22n(n!)2v2nc2n+...] Relativistic
Ultrarelativistic_limit
Type of prime number
Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial type ·Twelvefold way · Abel's binomial theorem · Trinomial expansion
Wilson_prime
Triangular array of natural numbers
Narayana (1930–1987). The Narayana numbers can be expressed in terms of binomial coefficients: N(n,k)=1k(n−1k−1)(nk−1)=1n(nk)(nk−1) The first eight rows
Narayana_number
{\displaystyle p} pairs of identical cards from the two sets, which is the binomial coefficient ( n p ) {\displaystyle {n \choose p}} . The remaining k − 2
Trinomial_triangle
Difference between total and static pressure
{\displaystyle \;{\tfrac {1}{2}}\gamma PM^{2}} and expanding by the binomial series gives: q c = q ( 1 + M 2 4 + M 4 40 + M 6 1600 . . . ) {\displaystyle
Impact_pressure
Solved prime-number problem
mathematical publications. The basic idea is to show that the central binomial coefficients must have a prime factor within the interval (n,2n) in order
Proof_of_Bertrand's_postulate
Upper bound on intersecting set families
Then the theorem states that the number of sets in 𝒜 is at most the binomial coefficient (n−1r−1). The requirement that n≥2r is necessary for the problem
Erdős–Ko–Rado_theorem
n-1} \choose {n-1}}} where ( z n ) {\displaystyle {z \choose n}} is the binomial coefficient. For β = 0 {\displaystyle \beta =0} , the generated polynomials
Difference_polynomials
American statistician (1913–2000)
Grubbs's test for outliers, and the Mann-Grubbs method for calculating a binomial series lower confidence bound, are named after him. He worked at the Ballistic
Frank_E._Grubbs
Product of the first "n" prime numbers
function exceed e and oscillate infinitely around e later on. Since the binomial coefficient (2nn) is divisible by every prime between n+1 and 2n, and
Primorial
Time series models
with the meaning of the term identified using the following formal binomial series expansion ( 1 − B ) d = ∑ k = 0 ∞ ( d k ) ( − B ) k = ∑ k = 0 ∞ ∏ a
Autoregressive fractionally integrated moving average
Autoregressive_fractionally_integrated_moving_average
Mathematical result on arithmetic properties of binomial coefficients
arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. The greatest common divisors of the binomial coefficients forming
Star_of_David_theorem
Generalized chain rule in calculus
the formal power series ∑n≥0anxn in the case a0=1. Stanley gives a version for exponential power series. In the formal power series f(x)=∑nann!xn, we
Faà_di_Bruno's_formula
Spanish mathematician (1955–2026)
on 9 February 2026, at the age of 70. Guillera, Jesús (2002). "Some binomial series obtained by the WZ-method". Advances in Applied Mathematics. 29 (4):
Jesús_Guillera
Divergent sum of positive unit fractions
harmonic series is the infinite series formed by summing all positive unit fractions: ∑i=1∞1i=11+12+13+14+15+⋯. The first n terms of the series sum to approximately
Harmonic_series_(mathematics)
Mathematical integral
difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n. Table of Newtonian series List
Nørlund–Rice_integral
Sequence of data points over time
mathematics, a time series is a sequence of data points indexed, listed, or graphed in chronological order. Most commonly, a time series consists of observations
Time_series
Any experiment with two possible random outcomes
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success"
Bernoulli_trial
Method for evaluating stock options that divides time into discrete intervals
Edgeworth binomial trees may be employed, as these allow for an analyst-specified skew and kurtosis in spot-price returns (see Edgeworth series). Here,
Lattice_model_(finance)
Discrete probability distribution
Poisson distribution. The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial is p=λn
Poisson_distribution
Number that is the product of factorials
"Comparability graphs and a new matroid", Journal of Combinatorial Theory, Series B, 22 (1): 68–90, doi:10.1016/0095-8956(77)90049-1, MR 0439689 Sloane, N
Jordan–Pólya_number
Mathematical operation
{F}}f(e^{-x})\right\}(-is).} The Mellin transform also connects the Newton series or binomial transform to the Poisson generating function by means of the Poisson–Mellin–Newton
Mellin_transform
Mathematical work by Isaac Newton
contains also the sine series and cosine series and arc series, the logarithmic series and the binomial series. Newton's method The Mathematical Association
De analysi per aequationes numero terminorum infinitas
De_analysi_per_aequationes_numero_terminorum_infinitas
Compound probability distribution
In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X equal to the number of failures
Beta negative binomial distribution
Beta_negative_binomial_distribution
Inequality about exponentiations of ''1+x''
get again (4). One can prove Bernoulli's inequality for x ≥ 0 using the binomial theorem. It is true trivially for r = 0, so suppose r is a positive integer
Bernoulli's_inequality
Mathematical fallacy
also known as freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or the Frobenius identity is the generally
Freshman's_dream
Mathematical sequence
the unsigned Lah numbers L(n,k) are given by the formula involving the binomial coefficient L(n,k)=(n−1k−1)n!k! for n≥k≥1, and the signed Lah numbers L′(n
Lah_number
Uniqueness theorem in complex analysis
Consider then the Newton seriesg(z)=∑n=0∞(zn)Δnf(0)where (zn) is the binomial coefficient and Δnf(0) is the n-th forward difference. By construction
Carlson's_theorem
Type of number in combinatorial mathematics and statistics
means that they are related to the binomial coefficient. The key difference between Fuss-Catalan and the binomial coefficient is that there are no "illegal"
Fuss–Catalan_number
Infinite sum approximating a probability distribution in terms of its cumulants
Edgeworth binomial tree Stuart, A., & Kendall, M. G. (1968). The advanced theory of statistics. Hafner Publishing Company. Kolassa, John E. (2006). Series approximation
Edgeworth_series
Mathematical functions
himself actually used (x)n with yet another meaning, namely to denote the binomial coefficient (xn). In this article, the symbol (x)n is used to represent
Falling_and_rising_factorials
Number of partitions of an integer
N, then the generating function of p(N,M,n) is the following Gaussian binomial coefficient: ∑n=0∞p(N,M,n)qn=(N+MM)q=(1−qN+M)(1−qN+M−1)⋯(1−qN+1)(1−q)(1−q2)⋯(1−qM)
Partition function (number theory)
Partition_function_(number_theory)
Addition of several numbers or other values
expresses the sum of the binomial distribution ∑i=0ni(ni)=n(2n−1), the value at a = b = 1 of the derivative with respect to a of the binomial theorem ∑i=0n(ni)i+1=2n+1−1n+1
Summation
Domesticated species of canid
WHO expert consultation on rabies: Third report. WHO Technical Report Series, 931. World Health Organization. 2018. hdl:10665/272364. ISBN 978-92-4-121021-8
Dog
Infinite sum
the theory of power series by his expansion of a complex function in such a form. Abel (1826) in his memoir on the binomial series 1+m1!x+m(m−1)2!x2+⋯
Series_(mathematics)
Instantaneous rate of change (mathematics)
under the integral sign Risch algorithm Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit
Derivative
Mathematical expression with disputed status
identities. For example, the binomial theorem (1+x)n=∑k=0n(nk)xk holds for x = 0 only if 00 = 1. Similarly, rings of power series require x0 to be defined
Zero_to_the_power_of_zero
Study of health and disease within a population
without the disease to the cases in the series. A prospective study would involve following the case series over time to evaluate the disease's natural
Epidemiology
Integer sequence
(starting from n = 0) gives the highest power of 2 that divides the central binomial coefficient (2nn), and it gives the numerator of 2n/n! (expressed as a
Gould's_sequence
Generalisation of the generalised hypergeometric function pFq(z)
the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society. Second Series: 71–79. doi:10.1112/JLMS/S1-8
Fox–Wright_function
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