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Topics referred to by the same term
Binomial identity may refer to: Binomial theorem Binomial type Binomial (disambiguation) This disambiguation page lists articles associated with the title
Binomial_identity
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
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
Algebraic expansion of powers of a binomial
2\cdot 1}}.} This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely
Binomial_theorem
Mathematical theorem on convolved binomial coefficients
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: ( m + n r ) = ∑ k = 0 r (
Vandermonde's_identity
mathematical identities, that is, identically true relations holding in mathematics. Binet-cauchy identity Binomial inverse theorem Binomial identity Brahmagupta–Fibonacci
List of mathematical identities
List_of_mathematical_identities
Identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002
In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002: ( x + m
Sun's_curious_identity
Theorem of matrix ranks
In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction
Woodbury_matrix_identity
Type of polynomial sequence
polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n
Binomial_type
On finite sums of products of three binomial coefficients, and a hypergeometric sum
finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon
Dixon's_identity
same as λ-rings for which all Adams operations are the identity. Elliott, Jesse (2006), "Binomial rings, integer-valued polynomials, and λ-rings", Journal
Binomial_ring
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
Mathematical functions
Sheffer sequence of binomial type, the Mittag-Leffler polynomials M n ( x ) {\displaystyle M_{n}(x)} also satisfy the binomial identity M n ( x + y ) = ∑
Mittag-Leffler_polynomials
Transformation of a mathematical sequence
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely
Binomial_transform
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 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 {\displaystyle
Binomial_series
Combinatorial identity about binomial coefficients
Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. The binomial coefficients are the numbers that appear in Pascal's
Pascal's_rule
Result in enumerative combinatorics and linear algebra
Combinatory analysis (1916). It is often used to derive binomial identities, most notably Dixon's identity. In the monograph, MacMahon found so many applications
MacMahon's_master_theorem
Identity in mathematical combinatorics
q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that (
Q-Vandermonde_identity
Fictional book mentioned in stories of Sherlock Holmes
strange binomial identities of Professor Moriarty" (PDF). Fibonacci Quarterly. 10 (4): 381–392, 402. Anderson, Poul. A Treatise on the Binomial Theorem
A Treatise on the Binomial Theorem
A_Treatise_on_the_Binomial_Theorem
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
Stochastic process in probability theory
_{n}(t)=E(X_{t}^{n})} , is a polynomial function of t; these functions satisfy a binomial identity: μ n ( t + s ) = ∑ k = 0 n ( n k ) μ k ( t ) μ n − k ( s ) . {\displaystyle
Lévy_process
Class of statistical models
).} The identity link g(p) = p is also sometimes used for binomial data to yield a linear probability model. However, the identity link can predict
Generalized_linear_model
Recurrence relations of binomial coefficients in Pascal's triangle
In combinatorics, the hockey-stick identity, Christmas stocking identity, boomerang identity, Fermat's identity or Chu's Theorem, states that if n ≥ r
Hockey-stick_identity
using De Moivre's formula, Euler's formula and the binomial theorem. The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding
List of trigonometric identities
List_of_trigonometric_identities
Polynomials in combinatorial mathematics
a_{n-k+1})x^{k}.} Then this polynomial sequence is of binomial type, i.e. it satisfies the binomial identity p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n
Bell_polynomials
case that P {\displaystyle P} is a chain, this recovers the negative binomial identity. There are similar results for the chromatic polynomial and Ehrhart
Order_polynomial
Triangular array of the binomial coefficients
Bernoulli's triangle Binomial expansion Cellular automata Euler triangle Floyd's triangle Gaussian binomial coefficient Hockey-stick identity Leibniz harmonic
Pascal's_triangle
Equalities involving sums over the coefficients occurring in hypergeometric series
hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur
Hypergeometric_identity
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
Vector calculus formulas relating the bulk with the boundary of a region
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential
Green's_identities
Mathematical fallacy
freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or the Frobenius identity is the generally-false equation (x + y)n = xn + yn
Freshman's_dream
Addition of several numbers or other values
arithmetico–geometric sequence) There exist very many summation identities involving binomial coefficients (a whole chapter of Concrete Mathematics is devoted
Summation
Mathematical identities
The following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)}
Vector_calculus_identities
Generalization of the product rule in calculus
ISBN 9780387950006. Spivey, Michael Zachary (2019). The Art of Proving Binomial Identities. Boca Raton: CRC Press, Taylor & Francis Group. ISBN 9781351215817
General_Leibniz_rule
Generalization of Vandermonde's identity
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its
Rothe–Hagen_identity
Doubtful name in taxonomy
In binomial nomenclature, a nomen dubium (Latin for "doubtful name", plural nomina dubia) is a scientific name that is of unknown or doubtful application
Nomen_dubium
Discrete probability distribution
binomial identity, ( n k ) = ( − 1 ) k ( k − n − 1 k ) , {\displaystyle {{n \choose k}=(-1)^{k}{k-n-1 \choose k}},} and the Chu–Vandermonde identity,
Negative hypergeometric distribution
Negative_hypergeometric_distribution
Mathematical expression with disputed status
ring. Defining 00 = 1 is necessary for many polynomial identities. For example, the binomial theorem ( 1 + x ) n = ∑ k = 0 n ( n k ) x k {\textstyle
Zero_to_the_power_of_zero
Relation between sine and cosine
binomial theorem. Consequently, sin 2 x + cos 2 x = 1 , {\displaystyle \sin ^{2}x+\cos ^{2}x=1,} which is the Pythagorean trigonometric identity.
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
Q-analog of hypergeometric series
q-binomial coefficient. The special case of a = 0 is closely related to the q-exponential.[citation needed] Srinivasa Ramanujan gave the identity 1 ψ
Basic_hypergeometric_series
Discrete probability distribution
Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson
Logarithmic_distribution
Mathematical function
special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral B ( z 1 , z 2 ) = ∫ 0 1 t z
Beta_function
Theorem about the constant term of certain Laurent polynomials
,a_{n}).} The case n = 3 of Dyson's conjecture follows from the Dixon identity. Sills & Zeilberger (2006) and (Sills 2006) used a computer to find expressions
Dyson_conjecture
Character stereotype used to represent primitive men
Keith. The term "caveman" has its taxonomic equivalent in the now-obsolete binomial classification of Homo troglodytes (Linnaeus, 1758). Cavemen are typically
Caveman
polynomial Quantum calculus LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity q-Bessel polynomials q-Charlier polynomials
List_of_q-analogs
Selection of items from a set
{\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomial coefficient: ( n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 , {\displaystyle
Combination
One or more words used to refer to something
conventions include: In astronomy, astronomical naming conventions In biology, binomial nomenclature In chemistry, chemical nomenclature In classics, Roman naming
Name
Mathematical set of all subsets of a set
numbers, in which case we cannot enumerate all irrational numbers. The binomial theorem is closely related to the power set. A k–elements combination from
Power_set
loricatobaicalensis is sometimes cited as the longest binomial name—it is a kind of amphipod. However, this name, proposed by B. Dybowski
Longest_word_in_English
{(-s)_{n}}{n!}}a_{n}} where ( s n ) {\displaystyle {s \choose n}} is the binomial coefficient and ( s ) n {\displaystyle (s)_{n}} is the falling factorial
Table_of_Newtonian_series
Probability distribution
conjugate prior probability distribution for the Bernoulli, binomial, negative binomial, and geometric distributions. The formulation of the beta distribution
Beta_distribution
Special case of the Euler-Lagrange equations
The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations. The Euler–Lagrange
Beltrami_identity
Mathematical set with repetitions allowed
{\displaystyle {\tbinom {n}{k}}.} Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset
Multiset
Theorem: (cos x + i sin x)^n = cos nx + i sin nx
also cos x and sin x, are real numbers, then the identity of these parts can be written using binomial coefficients. This formula was given by 16th century
De_Moivre's_formula
Algebra in algebraic topology
0 {\displaystyle i,j>0} such that i < 2 j {\displaystyle i<2j} . (The binomial coefficients are to be interpreted mod 2.) The Adem relations allow one
Steenrod_algebra
Discrete probability distribution
k}{{N-n} \choose {K-k}}} \over {N \choose K}};} This identity can be shown by expressing the binomial coefficients in terms of factorials and rearranging
Hypergeometric_distribution
Mathematical polynomial formula
according to the identity a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) {\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})} in elementary algebra. Binomial numbers generalize
Sum_of_two_cubes
Constant equal to twice pi
2024. Harremoës, Peter (2017). "Bounds on tail probabilities for negative binomial distributions". Kybernetika. 52 (6): 943–966. arXiv:1601.05179. doi:10
Tau_(mathematics)
Type of proof technique
n} . Double counting can also be used to prove the following identity related to binomial coefficient ( n k ) = ( n n − k ) {\displaystyle {\binom {n}{k}}={\binom
Double counting (proof technique)
Double_counting_(proof_technique)
Identity expressing an integral as a sum
In mathematics, the sophomore's dream is the pair of identities (especially the first) ∫ 0 1 x − x d x = ∑ n = 1 ∞ n − n ∫ 0 1 x x d x = ∑ n = 1 ∞ ( −
Sophomore's_dream
Probability theorem
S_{n}=X_{1}+\cdots +X_{n}.} (i.e. S n {\displaystyle S_{n}} follows a Poisson binomial distribution) Then ∑ k = 0 ∞ | Pr ( S n = k ) − λ n k e − λ n k ! | < 2
Le_Cam's_theorem
Formula computing the inverse of the sum of a matrix and the outer product of two vectors
performs a rank-1 update to a determinant. Woodbury matrix identity Quasi-Newton method Binomial inverse theorem Bunch–Nielsen–Sorensen formula Maxwell stress
Sherman–Morrison_formula
Type of polynomial sequence
differentiation, and the group of sequences of binomial type, which are those that satisfy the identity p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x )
Sheffer_sequence
Concept in combinatorics (part of mathematics)
_{n=0}^{\infty }{\frac {x^{n}}{(q;q)_{n}}},} which are both special cases of the q-binomial theorem: ( a x ; q ) ∞ ( x ; q ) ∞ = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n
Q-Pochhammer_symbol
Technique for proving sets have equal size
cones. Problems that admit bijective proofs are not limited to binomial coefficient identities. As the complexity of the problem increases, a bijective proof
Bijective_proof
hierarchical generalized linear models. If y ∣ u {\displaystyle y\mid u} follows binomial distribution with certain mean, u {\displaystyle u} has the conjugate beta
Hierarchical generalized linear model
Hierarchical_generalized_linear_model
Number of partitions of an integer
of p ( N , M , n ) {\displaystyle p(N,M,n)} is the following Gaussian binomial coefficient: ∑ n = 0 ∞ p ( N , M , n ) q n = ( N + M M ) q = ( 1 − q N
Partition function (number theory)
Partition_function_(number_theory)
Wife of Odysseus in Greek mythology
arbitrarily identified with the Eurasian wigeon, to which Linnaeus gave the binomial Anas penelope), where -elōps (-έλωψ) is a common Pre-Greek suffix for predatory
Penelope
Mathematical identity of polynomials
{\displaystyle {\tbinom {n-1}{k}}} . Sum of two cubes Binomial number Sophie Germain's identity Aurifeuillean factorization Congruum, the shared difference
Difference_of_two_squares
Species of fish
fish species Weber, Claude; Covain, Raphaël; Fisch-Muller, Sonia (2012). "Identity of Hypostomus plecostomus (Linnaeus, 1758), with an overview of Hypostomus
Hypostomus_plecostomus
Pair of functions in combinatorics
combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients
Wilf–Zeilberger_pair
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
Species of beetle
11: 69–75. doi:10.1016/j.japb.2017.12.005. Boudreaux HB (1969). "The Identity of Sitophilus oryzae". Annals of the Entomological Society of America.
Rice_weevil
Formula that provides the solutions to a quadratic equation
constant k 2 {\displaystyle \textstyle k^{2}} to obtain a squared binomial x 2 + 2 k x + k 2 = {\displaystyle \textstyle x^{2}+2kx+k^{2}={}}
Quadratic_formula
2.71828...; base of natural logarithms
characterizations using the limit and the infinite series can be proved via the binomial theorem. Jacob Bernoulli discovered this constant in 1683, while studying
E_(mathematical_constant)
Mathematical series, portmanteau of "Fibonacci" and "factorial"
coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients. The series of
Fibonorial
Data structure for priority queue operations
many other priority queue data structures including the binary heap and binomial heap. Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps
Fibonacci_heap
Extinct genus of theropod dinosaurs
Theropoda Family: †Abelisauridae Genus: †Kryptops Sereno & Brusatte, 2008 Species: †K. palaios Binomial name †Kryptops palaios Sereno & Brusatte, 2008
Kryptops
Species of bird
it with all the other nightjars in the genus Caprimulgus and coined the binomial name Caprimulgus carolinensis. Gmelin based his description on those of
Chuck-will's-widow
Product of numbers from 1 to n
1 , {\textstyle {\tbinom {n}{n}}={\tfrac {n!}{n!0!}}=1,} a binomial coefficient identity that would only be valid with 0 ! = 1 {\displaystyle 0!=1}
Factorial
Algebraic structure with addition and multiplication
contains the zero ring as a subring, then R itself is the zero ring. The binomial formula holds for any x and y satisfying xy = yx. Equip the set Z / 4 Z
Ring_(mathematics)
American author (born 1979)
Knoxville. His doctoral thesis is titled "Combinatorial Unification of Binomial-Like Arrays", and his advisor was Carl G. Wagner. After completing his
James_A._Lindsay
Mathematical approximation of a function
convergent for |x| < 1. These are special cases of the binomial series given in the next section. The binomial series is the power series ( 1 + x ) α = ∑ n =
Taylor_series
Sum of inverse squares of natural numbers
x}{\sin x}}\right)^{n}\\[4pt]&=(\cot x+i)^{n}.\end{aligned}}} From the binomial theorem, we have ( cot x + i ) n = ( n 0 ) cot n x + ( n 1 ) ( cot
Basel_problem
Type of mathematical generalization
move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: ( n k ) q
Q-analog
Species of arachnid (type of jumping spider)
but due to the loss of specimens there was much confusion about their identities. In 1846, Carl Ludwig Koch created the genus Phidippus in which Phidippus
Phidippus_audax
Set of quantities in probability theory
sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined
Cumulant
Instantaneous rate of change (mathematics)
Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz
Derivative
Deciding between multiple options
choices poses issues for ethics and for jurisprudence Mathematics: the binomial coefficient is also known as the choice function Politics: a political
Choice
Mathematical functions
\\[6pt]{\frac {x^{(n)}}{n!}}&={\binom {x+n-1}{n}}.\end{aligned}}} Thus many identities on binomial coefficients carry over to the falling and rising factorials. The
Falling_and_rising_factorials
Mathematical function for the probability a given outcome occurs in an experiment
distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, the negative binomial distribution and categorical distribution
Probability_distribution
Approximation method in statistics
family with identity as its natural sufficient statistics and mild-conditions are satisfied (e.g. for normal, exponential, Poisson and binomial distributions)
Least_squares
Species of flowering plant
are stored in vacuoles in the plant. Cultivars of G. globosa vary in the identity of floral volatiles but the volatile compounds of nonanal, decanal, geranyl
Gomphrena_globosa
Type of polynomial used in Numerical Analysis
\ \ldots \ ,n,} where ( n ν ) {\displaystyle {\tbinom {n}{\nu }}} is a binomial coefficient. So, for example, b 2 , 5 ( x ) = ( 5 2 ) x 2 ( 1 − x )
Bernstein_polynomial
Why British people voted to leave the EU
the idea of being British as a part of their identity were more likely to vote leave. However, a Binomial logit analysis was conducted to determine the
Causes of the vote in favour of Brexit
Causes_of_the_vote_in_favour_of_Brexit
Decomposition of an integer as a sum of positive integers
partition yields a partition of n − M into at most M parts. The Gaussian binomial coefficient is defined as: ( k + ℓ ℓ ) q = ( k + ℓ k ) q = ∏ j = 1 k +
Integer_partition
Method for partial-fraction expansion
has fractional expressions where some factors may repeat as powers of a binomial. In integral calculus we would want to write a fractional algebraic expression
Heaviside_cover-up_method
Matrix with a multiplicative inverse
is invertible, it can be multiplied by its inverse matrix to yield the identity matrix. Invertible matrices are the same size as their inverse. The inverse
Invertible_matrix
Branch of mathematics
derivatives and integrals in alternative calculi List of differentiation identities Publications in calculus Table of integrals Real Analysis Mathematical
Calculus
BINOMIAL IDENTITY
BINOMIAL IDENTITY
Girl/Female
Indian
Identity
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Punjabi, Sikh, Sindhi, Tamil, Telugu
Glories; Love; Identity; Pride
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from a place in northern France, of which the identity is not clear. It is probably Sainville in Eure-et-Loire, so called from Old French saisne ‘Saxon’ + ville ‘settlement’.
Girl/Female
Hindu
Higher, North the direction, Name of a start (Princess of Virata, pupil of Arjuna as Brihhannala (his disguised identity as the eunuch dance teacher during the Pandavas final year of exile).)
Girl/Female
Muslim
Identity
Girl/Female
African, American, Arabic, Australian, Gujarati, Indian, Jain, Japanese, Muslim, Sanskrit, Swahili, Tamil
Name; One's Self; The Victorious; Named Child; Identity
Boy/Male
Muslim
Identity
Surname or Lastname
English
English : metonymic occupational name for a felt maker, from Old English felt ‘felt’.Said to be an Americanized or Germanized spelling of a Hungarian name, of uncertain identity.
Girl/Female
Tamil
Higher, North the direction, Name of a start (Princess of Virata, pupil of Arjuna as Brihhannala (his disguised identity as the eunuch dance teacher during the Pandavas final year of exile).)
Boy/Male
Arabic, Gujarati, Hindu, Indian, Kannada, Muslim
Identity
BINOMIAL IDENTITY
BINOMIAL IDENTITY
Boy/Male
Australian, Hindu, Indian, Telugu
Good Person
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Wish; Desire; Kindness; Enjoyment
Boy/Male
Australian, French, German, Portuguese, Spanish
Brave Adviser; Honest Advisor; Brave; Bold Ruler; Counsel; Spanish Form of Conrad Able Counsel
Biblical
conversion; captivity
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Marathi, Oriya, Tamil, Traditional
Name of a Sage
Biblical
joy in tribulation; joy of the vintage
Boy/Male
Hindu
God of drama Lord Shiva
Boy/Male
Australian, German
Bold
Girl/Female
American, Australian, British, English
Carol and Ann; Feminine Variant of Charles
Female
Danish
, a rope with a noose, i.e., a young woman whose beauty ensnares men.
BINOMIAL IDENTITY
BINOMIAL IDENTITY
BINOMIAL IDENTITY
BINOMIAL IDENTITY
BINOMIAL IDENTITY
a.
Consisting of two terms; pertaining to binomials; as, a binomial root.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
n.
A rare metallic element of doubtful identity.
a.
Consisting of but a single term or expression.
a.
Having two names; -- used of the system by which every animal and plant receives two names, the one indicating the genus, the other the species, to which it belongs.
a.
Consisting of three terms; of or pertaining to trinomials; as, a trinomial root.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
A monomial.
n.
A quantity consisting of three terms, connected by the sign + or -; as, x + y + z, or ax + 2b - c2.
n.
Repetition of a theme or melody with fanciful embellishments or modifications, in time, tune, or harmony, or sometimes change of key; the presentation of a musical thought in new and varied aspects, yet so that the essential features of the original shall still preserve their identity.
n.
The condition of being the same with something described or asserted, or of possessing a character claimed; as, to establish the identity of stolen goods.
a.
Of or pertaining to two names; binomial.
n. & a.
Trinomial.
a.
Binominal.
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
Identity in pitch; coincidence of sounds proceeding from an equality in the number of vibrations made in a given time by two or more sonorous bodies. Parts played or sung in octaves are also said to be in unison, or in octaves.
n.
An expression consisting of two terms connected by the sign plus (+) or minus (-); as, a + b, or 7 - 3.
n.
The state of being the same; identity; absence of difference; near resemblance; correspondence; similarity; as, a sameness of person, of manner, of sound, of appearance, and the like.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
n.
A name or term.