AI & ChatGPT searches , social queries for VANDERMONDES IDENTITY
Search references for VANDERMONDES IDENTITY. Phrases containing VANDERMONDES IDENTITY
See searches and references containing VANDERMONDES IDENTITY!
Search references for VANDERMONDES IDENTITY. Phrases containing VANDERMONDES IDENTITY
See searches and references containing VANDERMONDES IDENTITY!VANDERMONDES IDENTITY
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
Identity in mathematical combinatorics
the q-Vandermonde identity is a q-analogue of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states
Q-Vandermonde_identity
French mathematician, musician and chemist
et Métiers. Knight's Tour Knot theory Vandermonde's identity Vandermonde polynomial Vandermonde matrix Vandermonde : secret society of the Conservatoire
Alexandre-Théophile Vandermonde
Alexandre-Théophile_Vandermonde
identity Vandermonde's identity Woodbury matrix identity Exterior calculus identities Fibonacci identities: Combinatorial Fibonacci identities and Other
List of mathematical identities
List_of_mathematical_identities
Matrix of geometric progressions
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:
Vandermonde_matrix
Type of symmetric polynomials in mathematics
variables. Since they are alternating, they are all divisible by the Vandermonde determinant a ( n − 1 , n − 2 , … , 0 ) ( x 1 , x 2 , … , x n ) = det
Schur_polynomial
Number of subsets of a given size
(twice for the latter) and then substituting x = y = 1. The Chu–Vandermonde identity, which holds for any complex values m and n and any non-negative
Binomial_coefficient
Generalization of Vandermonde's identity
n-k}={\frac {x+y}{x+y+nz}}{x+y+nz \choose n}.} It is a generalization of Vandermonde's identity, and is named after the independent work of Heinrich August Rothe
Rothe–Hagen_identity
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
calculus LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity q-Bessel polynomials q-Charlier polynomials q-Hahn polynomials q-Jacobi
List_of_q-analogs
Subfactorial Table of Newtonian series Taylor series Trinomial expansion Vandermonde's identity Wilson prime Wilson's theorem Wolstenholme prime This list page
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Combinatorial identity
combinatorics, the Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century
Li_Shanlan_identity
Discrete probability distribution
\choose n-k} \over {N \choose n}}=1,} which essentially follows from Vandermonde's identity from combinatorics. Also note that ( K k ) ( N − K n − k ) ( N n
Hypergeometric_distribution
Mathematical functions
F_{n}^{(r)}(t):=\sum _{k\leq n}{\frac {t^{k}}{f(k)^{r}}}\,.} Pochhammer k-symbol Vandermonde identity Here the parts are distinct; for example, when x = n = 2, the (2)(2)
Falling_and_rising_factorials
Branch of discrete mathematics
coefficient · Pascal's triangle · Central binomial coefficient · Vandermonde's identity · Binomial theorem · Binomial series · Binomial transform · Binomial
Combinatorics
German mathematician (1765–1825)
analysis, his treatise on analysis, a paper independently deriving Vandermonde's identity and proving Saalschütz's theorem, and a paper solving for the largest
Johann_Friedrich_Pfaff
Type of proof technique
{\displaystyle k} trees, for any k {\displaystyle k} . Vandermonde's identity, another identity on sums of binomial coefficients that can be proven by
Double counting (proof technique)
Double_counting_(proof_technique)
n-k}={\alpha +\beta \choose n},{\text{where}}\ \alpha +\beta \geq n} (see Vandermonde identity) ∑ A ∈ P ( E ) 1 = 2 n , where E is a finite set, and card(
List_of_mathematical_series
Gaussian binomial coefficient q-derivative q-series q-theta function q-Vandermonde identity Rencontres numbers Rubik's Cube How to solve the Rubik's Cube Optimal
Index of combinatorics articles
Index_of_combinatorics_articles
Discrete analog of a derivative
result to Taylor's theorem. Historically, this, as well as the Chu–Vandermonde identity, ( x + y ) n = ∑ k = 0 n ( n k ) ( x ) n − k ( y ) k , {\displaystyle
Finite_difference
In mathematics, a statement that has been proven
holds for any value within its domain (e.g. Bézout's identity and Vandermonde's identity). A rule is a theorem that establishes a useful formula (e.g. Bayes'
Theorem
Function defined by a hypergeometric series
from Euler's integral formula by putting z = 1. It includes the Vandermonde identity as a special case. For the special case where a = − m {\displaystyle
Hypergeometric_function
Concept in combinatorics (part of mathematics)
Pentagonal number theorem q-derivative q-theta function q-Vandermonde identity Rogers–Ramanujan identities Rogers–Ramanujan continued fraction Berndt, B. C. "What
Q-Pochhammer_symbol
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, ∑ j
Negative hypergeometric distribution
Negative_hypergeometric_distribution
over the Vandermonde polynomial is a polynomial. Schur polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial
Alternating_polynomial
Mathematics concept
\choose q}.} The dimensions add up correctly as a consequence of Vandermonde's identity. The space of (p,q)-forms Λp,q VJ* is the space of (complex) multilinear
Linear_complex_structure
Polynomial sequence
+G_{n}} It can be shown using the second integral representation and Vandermonde's identity. The Bernoulli polynomials of the second kind satisfy the recurrence
Bernoulli polynomials of the second kind
Bernoulli_polynomials_of_the_second_kind
In mathematics, invariant of square matrices
for computing the determinants of highly symmetric matrix such as the Vandermonde matrix | 1 1 1 ⋯ 1 x 1 x 2 x 3 ⋯ x n x 1 2 x 2 2 x 3 2 ⋯ x n 2 ⋮ ⋮ ⋮
Determinant
Family of power series in mathematics
Dougall-Ramanujan identity. It is a special case of Jackson's identity, and it gives Dixon's identity and Saalschütz's theorem as special cases. Identity 1. e −
Generalized hypergeometric function
Generalized_hypergeometric_function
Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant matrix Hankel matrix
Outline_of_linear_algebra
Vector satisfying some of the criteria of an eigenvector
where I {\displaystyle I} is the n {\displaystyle n} × n {\displaystyle n} identity matrix and 0 {\displaystyle \mathbf {0} } is the zero vector of length
Generalized_eigenvector
Square matrix where a[i,j]=1/(i+j-1)
Vol. II. Beckermann, Bernhard (2000). "The condition number of real Vandermonde, Krylov and positive definite Hankel matrices". Numerische Mathematik
Hilbert_matrix
Matrix equal to its conjugate-transpose
in complex coordinate space is the Hermitian form associated with the identity matrix. For any Hermitian matrix A {\displaystyle A} , v H A v {\displaystyle
Hermitian_matrix
Algorithm used for frequency estimation and radio direction finding
\cdots ,\mathbf {a} (\omega _{p})]} is an M × p {\displaystyle M\times p} Vandermonde matrix of steering vectors a ( ω ) = [ 1 , e j ω , e j 2 ω , … , e j
MUSIC_(algorithm)
Mathematical formula for the number of Young tableaux
− x j ) {\displaystyle \Delta (x)=\prod _{i<j}(x_{i}-x_{j})} is the Vandermonde determinant. For the partition λ = ( λ 1 ≥ ⋯ ≥ λ k ) {\displaystyle \lambda
Hook_length_formula
Matrix operation generalizing exponentiation of scalar numbers
}{\frac {1}{k!}}X^{k}} where X 0 {\displaystyle X^{0}} is defined to be the identity matrix I {\displaystyle I} with the same dimensions as X {\displaystyle
Matrix_exponential
Stochastic process
{\displaystyle \Delta _{n}:=\prod _{i<j}(\lambda _{i}-\lambda _{j})} is the Vandermonde determinant, then the time-evolution of eigenspectrum is equivalent to
Dyson_Brownian_motion
Property in group theory
indeed well-defined and equivalent. Proof 2 An alternative proof uses the Vandermonde polynomial P ( x 1 , … , x n ) = ∏ i < j ( x i − x j ) . {\displaystyle
Parity_of_a_permutation
Function in discrete mathematics
the above discussion, the DFT can be expressed as the DFT matrix, a Vandermonde matrix, introduced by Sylvester in 1867, F = [ ω N 0 ⋅ 0 ω N 0 ⋅ 1 ⋯
Discrete_Fourier_transform
Sequence in computer science
(confluent) Hermite interpolation as well as for parallel algorithms for Vandermonde systems. Parallel prefix algorithms can also be used for temporal parallelization
Prefix_sum
Polynomials used for interpolation
also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant. But, as can be seen from the construction
Lagrange_polynomial
Families of matrices in mathematics, physics, and quantum information
{\displaystyle n} th qubit and the identity on all other qubits. We can also use a = 0 {\displaystyle a=0} for the identity, i.e., for any n {\displaystyle
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Determinant of the matrix of first derivatives of a set of functions
differential equation, the Wrońskian can be found explicitly using Abel's identity, even if the functions themselves are not known explicitly. (See below
Wronskian
Expression for sums of powers
problem have taken the matrix path, leveraging useful tools such as the Vandermonde vector. Other researchers continue to explore through the traditional
Faulhaber's_formula
Error-correcting codes
Specialized forms of Reed–Solomon codes, specifically Cauchy-RS and Vandermonde-RS, can be used to overcome the unreliable nature of data transmission
Reed–Solomon_error_correction
Generalisation of Fourier transform to any ring
{\displaystyle \alpha } . Let A {\displaystyle A} be the above DFT matrix, a Vandermonde matrix with entries A i j = α i j {\displaystyle A_{ij}=\alpha ^{ij}}
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
German mathematician
the more famous mathematician Johann Friedrich Pfaff. The Rothe–Hagen identity, a summation formula for binomial coefficients, appeared in Rothe's 1793
Heinrich_August_Rothe
Study of mathematical knots
mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing
Knot_theory
Measures the size of the ring of integers of the algebraic number field
^{n-1}} . Then, the matrix B {\displaystyle B} in the definition is the Vandermonde matrix associated to α i = σ i ( α ) {\displaystyle \alpha _{i}=\sigma
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Polynomial invariant under variable permutations
of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial
Symmetric_polynomial
Differential equation that is linear with respect to the unknown function
solutions can be shown to be linearly independent, by considering the Vandermonde determinant of the values of these solutions at x = 0, ..., n – 1. Together
Linear_differential_equation
Discrete fourier transform expressed as a matrix
{\displaystyle x} we have the identity ω x = ω x mod N . {\displaystyle \omega ^{x}=\omega ^{x{\bmod {N}}}.} This is the Vandermonde matrix for the roots of
DFT_matrix
Algorithm to smooth data points
^{\mathsf {T}}\mathbf {y} ,} where J {\displaystyle \mathbf {J} } is a Vandermonde matrix, that is i {\displaystyle i} -th row of J {\displaystyle \mathbf
Savitzky–Golay_filter
American mathematician
JSTOR 2306394. Some generalizations of Vandermonde's convolution, Amer. Math. Monthly, 63(1956), 84–91. Final analysis of Vandermonde's convolution, Amer. Math. Monthly
Henry_W._Gould
Matrix class
Karla (1993). "An Inversion Formula and Fast Algorithms for Cauchy-Vandermonde Matrices" (PDF). Linear Algebra and Its Applications. 183 (1): 179–191
Cauchy_matrix
Algebraic structure with addition, multiplication, and division
multiplication: a + b = b + a, and a ⋅ b = b ⋅ a. Additive and multiplicative identity: there exist distinct elements 0 and 1 in F such that a + 0 = a and a ⋅
Field_(mathematics)
Function of the coefficients of a polynomial that gives information on its roots
_{i\neq j}(r_{i}-r_{j}).} It is thus the square of the Vandermonde polynomial times a n 2 n − 2 {\displaystyle a_{n}^{2n-2}} . This expression
Discriminant
Type of mathematical equation
n\times 1} constant vector. By use of the Cayley–Hamilton theorem and Vandermonde-type matrices, this formal matrix exponential solution may be reduced
Matrix_differential_equation
Representation theory
X_{n}^{\sigma (n)-1}=\prod _{1\leq i<j\leq n}(X_{j}-X_{i})} for the Vandermonde determinant. By evaluating the character at H = 0 {\displaystyle H=0}
Weyl_character_formula
the entries, including constant matrices. Important examples include the identity matrix given by I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] . {\displaystyle
List_of_named_matrices
Dutch-American mathematician
1090/s0002-9904-1928-04560-3. MR 1561587. Dresden, Arnold (1933). "On the generalized Vandermonde determinant and symmetric functions". Bull. Amer. Math. Soc. 39 (6):
Arnold_Dresden
Algorithm for multiplying large numbers
evaluation points were chosen suitably, this matrix is invertible (see also Vandermonde matrix), and so: ( r 0 r 1 r 2 r 3 r 4 ) = ( 1 0 0 0 0 1 1 1 1 1 1 −
Toom–Cook_multiplication
American mathematician
Beckenbach Book Prize in 2012. Kalman, Dan (January 1984). "The generalized Vandermonde matrix". Mathematics Magazine. 57 (1): 15–21. doi:10.1080/0025570X.1984
Dan_Kalman
Signal processing method
{a} (\omega _{2}),\ ...,\ \mathbf {a} (\omega _{K})} are put into a Vandermonde matrix A = [ a ( ω 1 ) a ( ω 2 ) . . . a ( ω K ) ] {\displaystyle
Estimation of signal parameters via rotational invariance techniques
Estimation_of_signal_parameters_via_rotational_invariance_techniques
Matrix whose determinant is a resultant
Bezoutians can be reduced to block diagonal form with the help of confluent Vandermonde matrix. An important application of Bézout matrices can be found in control
Bézout_matrix
interpolation by polynomials Linear interpolation Runge's phenomenon Vandermonde matrix Chebyshev polynomials Chebyshev nodes Lebesgue constants Different
List of numerical analysis topics
List_of_numerical_analysis_topics
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY
VANDERMONDES IDENTITY