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general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's
Difference_polynomials
In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They
Q-difference_polynomial
Mathematical expression
polynomial can alternatively be constructed using Lagrange polynomials as an alternative basis, or generalized to Hermite interpolating polynomials when
Newton_polynomial
Automatic mechanical calculator
logarithmic and trigonometric functions, which can be approximated by polynomials, so a difference engine can compute many useful tables. English Wikisource has
Difference_engine
Polynomials used for interpolation
for indices , the Lagrange basis for polynomials of degree for those nodes is the set of polynomials each of degree which take values if
Lagrange_polynomial
Type of mathematical expression
polynomials, quadratic polynomials and cubic polynomials. For higher degrees, the specific names are not commonly used, although quartic polynomial (for
Polynomial
Product of consecutive factorial numbers
{\displaystyle n+1} integers are given, the product of their pairwise differences is always a multiple of s f ( n ) {\displaystyle {\mathit {sf}}(n)}
Superfactorial
Discrete analog of a derivative
polynomial of degree m − 1 where m ≥ 2 and the coefficient of the highest-order term be a ≠ 0. Assuming the following holds true for all polynomials of
Finite_difference
Polynomial interpolation using derivative values
via degree-0 basis polynomials, the full set of basis polynomials can be constructed entirely using only the degree-0 basis polynomial hi,0. For hi,0
Hermite_interpolation
Type of permutation of a set of elements
has the form ∫0∞Pn1(x)Pn2(x)⋯Pnr(x)e−xdx, for a certain sequence of polynomials Pn, where Pn has degree n. But the above answer for the case r = 2 gives
Derangement
mathematics, Macdonald-Koornwinder polynomials (also called Koornwinder polynomials) are a family of orthogonal polynomials in several variables, introduced
Koornwinder_polynomials
Mathematical theorem on convolved binomial coefficients
(n1+⋯+npm)=∑k1+⋯+kp=m(n1k1)(n2k2)⋯(npkp). In general, the product of two polynomials with degrees m and n, respectively, is given by (∑i=0maixi)(∑j=0nbjx
Vandermonde's_identity
Type of polynomial used in Numerical Analysis
Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves. A numerically stable way to evaluate polynomials in
Bernstein_polynomial
Mathematical concept
composition of two polynomials is strongly related to the degree of the input polynomials. The degree of the sum (or difference) of two polynomials is less than
Degree_of_a_polynomial
q-derivative q-difference polynomial Quantum calculus LLT polynomial q-binomial coefficient q-Pochhammer symbol q-Vandermonde identity q-Bessel polynomials q-Charlier
List_of_q-analogs
Algebraic structure
especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally
Polynomial_ring
Number theory expression
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Legendre's_formula
Number computed as a product of powers
factorial. The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation. Superfactorial Sloane, N. J. A. (ed
Hyperfactorial
Family of polynomials
coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients
Gaussian_binomial_coefficient
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis
Stirling_polynomials
Polynomial sequence
functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomials. The Bernoulli polynomials Bn can be defined by a
Bernoulli_polynomials
Number of subsets of a given size
combination of binomial coefficient polynomials is integer-valued too. Conversely, (4) shows that any integer-valued polynomial is an integer linear combination
Binomial_coefficient
Q-analog of the ordinary derivative
q\to 1} it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation
Q-derivative
{\displaystyle p_{n}(z)} is a polynomial of degree n {\displaystyle n} . Boas–Buck polynomials are a slightly more general class of polynomials. The choice of g (
Generalized Appell polynomials
Generalized_Appell_polynomials
Estimate of time taken for running an algorithm
in 2O(nlogn). However, at STOC 2016 a quasi-polynomial time algorithm was presented. It makes a difference whether the algorithm is allowed to be sub-exponential
Time_complexity
power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients
Power sum symmetric polynomial
Power_sum_symmetric_polynomial
Mathematical function
}u_{n}=\sum _{n=0}^{\infty }{\frac {p(n)}{q(n)}},} where p(n) and q(n) are polynomials of n. Performing partial fraction decomposition on un in the complex
Digamma_function
Mathematical function
elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Historical term in mathematics
composition of polynomial sequences Calculus of finite differences Pidduck polynomials Symbolic method in invariant theory Narumi polynomials Sheffer sequence
Umbral_calculus
Algebraic expansion of powers of a binomial
operators corresponding to the above "Pochhammer" families of polynomials are the backward difference I − E − c {\displaystyle I-E^{-c}} for c > 0 {\displaystyle
Binomial_theorem
Greatest common divisor of polynomials
GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Branch of discrete mathematics
and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, it is now considered
Combinatorics
Polynomial sequence
The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating
Bernoulli polynomials of the second kind
Bernoulli_polynomials_of_the_second_kind
Formal power series
Appell polynomials Chebyshev polynomials Difference polynomials Generalized Appell polynomials q-difference polynomials Other sequences generated by more
Generating_function
Probability distribution
this topic. Interactive graphic: Univariate Distribution Relationships Difference of two binomial variables: X-Y or |X-Y| Querying the binomial probability
Binomial_distribution
(Mathematical) decomposition into a product
factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials). A commutative ring possessing the
Factorization
Graphical aid for deriving some concepts in combinatorics
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Knot invariant that is a polynomial
the Alexander polynomial. Alexander–Briggs notation organizes knots by their crossing number. Alexander polynomials and Conway polynomials can not recognize
Knot_polynomial
Class of numerical techniques
factorial of n, and Rn(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. Following is the
Finite_difference_method
Polynomial with negative exponents
polynomials in X {\displaystyle X} form a ring denoted F [ X , X − 1 ] {\displaystyle \mathbb {F} [X,X^{-1}]} . They differ from ordinary polynomials
Laurent_polynomial
Recurrence relations of binomial coefficients in Pascal's triangle
triangle Vandermonde's identity Faulhaber's formula, for sums of arbitrary polynomials. CH Jones (1996) Generalized Hockey Stick Identities and N-Dimensional
Hockey-stick_identity
Family of orthogonal polynomials
mathematics, Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty
Hahn_polynomials
Triangular array of the binomial coefficients
triangle Star of David theorem Trinomial expansion Trinomial triangle Polynomials calculating sums of powers of arithmetic progressions Maurice Winternitz
Pascal's_triangle
Expression for sums of powers of integers
Some authors call the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because
Faulhaber's_formula
Polynomial whose roots are the eigenvalues of a matrix
applies to matrices and polynomials over complex numbers (or any algebraically closed field). In that case, the characteristic polynomial of any square matrix
Characteristic_polynomial
Generalization of the binomial distribution
interval for the difference of proportions from the marginals of the following (sampled) contingency table: In this case, checking the difference in marginal
Multinomial_distribution
Mathematical approximation of a function
of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function
Taylor_series
Generalization of the binomial theorem to other polynomials
"Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution", Annales Mathematicae et Informaticae
Multinomial_theorem
\mathbb {K} [x]\longrightarrow \mathbb {K} [x]} on the vector space of polynomials in a variable x {\displaystyle x} over a field K {\displaystyle \mathbb
Delta_operator
Recursive mathematical formula
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Exponential_factorial
certain families of sparse polynomials than it is for other polynomials. The algebraic varieties determined by sparse polynomials have a simple structure
Sparse_polynomial
Polynomial whose coefficients are all 1 or −1
1966 that flat Littlewood polynomials exist. An influential earlier contribution was József Beck's 1991 paper on flat polynomials and Littlewood's problem
Littlewood_polynomial
Form of interpolation
polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials. The original use of interpolation polynomials was
Polynomial_interpolation
Sequence of numbers ((2n) choose (n))
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Central_binomial_coefficient
Mathematical functions
falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ f ( x ) := f ( x + 1 ) − f ( x ) , {\displaystyle
Falling_and_rising_factorials
Mathematical identity of polynomials
a^{2}-b^{2}=(a+b)(a-b)} . The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity
Difference_of_two_squares
Formula for solving a quadratic equation
q are symmetric polynomials in α and β. Specifically, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can
Quadratic_formula
In mathematics, Meixner polynomials, also called discrete Laguerre polynomials, are a family of discrete orthogonal polynomials introduced by Josef Meixner
Meixner_polynomials
Concept in mathematics
cos(nx) are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers
Trigonometric_polynomial
Combination Combinatorial number system De Polignac's formula Difference operator Difference polynomials Digamma function Egorychev method Erdős–Ko–Rado theorem
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Coefficient used in numerical approximation
latter definition. The theory of Lagrange polynomials provides explicit formulas for the finite difference coefficients. For the first six derivatives
Finite_difference_coefficient
Operator for offsetting time series elements
_{i=1}^{q}\theta _{i}L^{i}.\,} Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example
Lag_operator
Type of polynomial sequence
on the space of polynomials in x that reduces degrees of polynomials by 1. The most obvious examples of delta operators are difference operators and differentiation
Binomial_type
multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
designed by Ali Akansu and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals. Akansu and his fellow
Binomial_QMF
is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials. The basic alternating polynomial is the
Alternating_polynomial
Theorem on prime numbers
trivial for p = 2, so suppose p is an odd prime, p ≥ 3. Consider the polynomial g ( x ) = ( x − 1 ) ( x − 2 ) ⋯ ( x − ( p − 1 ) ) . {\displaystyle g(x)=(x-1)(x-2)\cdots
Wilson's_theorem
Conjecture in combinatorial number theory
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Singmaster's_conjecture
Generalized chain rule in calculus
formula expressed in terms of partial (or incomplete) exponential Bell polynomials Bn,k(x1,…,xn−k+1): dndxnf(g(x))=∑k=0nf(k)(g(x))⋅Bn,k(g′(x),g″(x),…,g(n−k+1)(x))
Faà_di_Bruno's_formula
Expression in calculus
In single-variable calculus, the difference quotient is usually the name for the expression f ( x + h ) − f ( x ) h {\displaystyle {\frac {f(x+h)-f(x)}{h}}}
Difference_quotient
Formula in mathematics
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Trinomial_expansion
Generalization of the mathematical factorial
their pairwise differences is a multiple of 0! 1! ... n!. Let ℤ be the set of integers and n any integer. Then the number of polynomial functions from
Bhargava_factorial
Algorithm for computing polynomial coefficients
Taylor polynomial into a Taylor series. Let f {\displaystyle f} be a function which corresponds to a power series. You can compute the divided difference scheme
Divided_differences
Solved prime-number problem
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Proof_of_Bertrand's_postulate
Function of a polynomial's coefficients that gives information about its roots
polynomials of higher degree, the sign alone generally provides less information about the number of real roots. Discriminants are used in polynomial
Discriminant
Amount left over after computation
(integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation
Remainder
Theorem in p-adic analysis
continuous p-adic function as an infinite series of certain special polynomials. This series is referred to as a Mahler series or Mahler expansion, or
Mahler's_theorem
Mathematical result on arithmetic properties of binomial coefficients
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Star_of_David_theorem
Mathematical polynomial formula
like terms, a 3 + b 3 . {\displaystyle a^{3}+b^{3}.} Similarly for the difference of two cubes, ( a − b ) ( a 2 + a b + b 2 ) = a ( a 2 + a b + b 2 ) −
Sum_of_two_cubes
Mathematical series
each contains a factor of (n − n). In this case, the series is a finite polynomial, equivalent to the binomial formula. Whether (1) converges depends on
Binomial_series
Recursive integer sequence
binomial topics Lobb numbers Motzkin number Narayana number Narayana polynomials Schröder number Schröder–Hipparchus number Semiorder Tamari lattice
Catalan_number
Roots of multiple multivariate polynomials
of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in
System of polynomial equations
System_of_polynomial_equations
In mathematics, a polynomial with two terms
a2 + b2 = c2. Binomials that are sums or differences of cubes can be factored into smaller-degree polynomials as follows: x 3 + y 3 = ( x + y ) ( x 2 −
Binomial_(polynomial)
Probability distribution
M.D.; Smyth, G.K. (2007). "Moderated statistical tests for assessing differences in tag abundance". Bioinformatics. 23 (21): 2881–2887. doi:10
Negative binomial distribution
Negative_binomial_distribution
Complexity class
solutions can be verified quickly. NP is short for "nondeterministic polynomial-time" and a problem is NP-complete when: It is a decision problem, meaning
NP-completeness
Pattern defining an infinite sequence of numbers
hypergeometric series. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. For these specific recurrence
Recurrence_relation
Algebraic study of differential equations
number of polynomials remains true for differential polynomials. In particular, greatest common divisors exist, and a ring of differential polynomials is a
Differential_algebra
Numerical method for calculating the gamma function
the (n, m)th element of the matrix of coefficients for the Chebyshev polynomials, which can be calculated recursively from these identities:[citation
Lanczos_approximation
Upper bound on intersecting set families
Alexander (1981), "Association schemes and the Shannon capacity: Eberlein polynomials and the Erdős–Ko–Rado theorem", in Lovász, László; Sós, Vera T. (eds
Erdős–Ko–Rado_theorem
Algorithms for zeros of functions
However, for polynomials specifically, the study of root-finding algorithms belongs to computer algebra, since algebraic properties of polynomials are fundamental
Root-finding_algorithm
Numbering of combinations of items
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Combinatorial_number_system
Method for solving one problem using another
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine
Polynomial-time_reduction
Theorem
Among all the polynomials of degree ≤ n {\displaystyle \leq n} , the polynomial g {\displaystyle g} minimizes the uniform norm of the difference ‖ f − g ‖
Equioscillation_theorem
Computational method
mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the
Factorization_of_polynomials
Characteristic polynomial whose associated linear system is stable
not sufficient. Just as stable polynomials are crucial for assessing the stability of systems described by polynomials, stability matrices play a vital
Stable_polynomial
Approximation of a function by a polynomial
Similarly, we might get still better approximations to f if we use polynomials of higher degree, since then we can match even more derivatives with
Taylor's_theorem
Mathematics concept
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski
Romanovski_polynomials
Expectation or average of the falling factorial of a random variable
entries in Pascal's triangle Operators & calculus links Difference operator · Difference polynomials · Table of Newtonian series · Faà di Bruno's formula
Factorial_moment
Statistics concept
(0, 1). Although the correlation can be reduced by using orthogonal polynomials, it is generally more informative to consider the fitted regression function
Polynomial_regression
Measure of algorithmic complexity
polynomial only in the Turing machine model. The difference between strongly- and weakly-polynomial time is when the inputs to the algorithms consist
Strongly-polynomial_time
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