For locally approximating a function with a polynomial, see Taylor series . In mathematics , an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributive property distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a sum of repeated products. During the expansion, simplifications such are grouping of like terms or cancellations of terms may also be applied. Instead of multiplications, the expansion steps could also involve replacing powers of a sum of terms by the equivalent expression obtained from the binomial formula this is a shortened form of what would happen if the power were treated as a repeated multiplication, and expanded repeatedly. It is customary to reintroduce powers in the final result when terms involve products of identical symbols. Simple examples of polynomial expansions are the well known rules ... process of trying to write an expanded polynomial as a product is called polynomial factorization . Expansion of a polynomial written in factored form To multiply two factors, each term of the first ... Calculator with Symbolic Calculations , livephysics.com DEFAULTSORT PolynomialExpansion Category ... single step expansion will introduce all products of a term of one of the sums being multiplied with a term of the other math a b c d x y z ax ay az bx by bz cx cy cz dx dy dz math An expansion which involves multiple nested rewrite steps is that of working out a Horner scheme to the expanded polynomial ... 10. math Expansion of x y sup n sup main Binomial theorem When expanding math x y n math , a special ... color red 1 y 6 , math See also Polynomial factorization Factorization External links Discussion http www.math.uakron.edu dpstory tutorial mptii lesson05.pdf Review of Algebra Expansion , University of Akron ... more details
wiktionary expand expansionExpansion may refer to tocright Physics Metric expansion of space Thermal expansion Prandtl Meyer expansion fan Computer programming Expand Unix Inline expansion Mathematics Expansion geometry Polynomialexpansion Taylor expansion Expander graph expansion of a graph The converse of a reduct , in model theory or universal algebra Economics Economic expansion Expansionary policies Other uses Expansion card , in computer hardware Expansion pack , in gaming Expansion team , in sports Trinucleotide repeat disorder Trinucleotide repeat expansion disorder , in medicine, the result of expansion during DNA translation Expansion album Expansion album , by Dave Burrell Expansions album , an album by jazz pianist McCoy Tyner Expansion joint , in engineering Audio level expansion, in audio engineering Expansion, the spelled out meaning of an abbreviation or acronym initialism disambig cs Expanze da Ekspansion de Expansion eu Dilatazio fr Expansion hr Ekspanzija it Espansione nl Expansie ru sk Expanzia sv Expansion ... more details
In mathematics , a polynomial is an expression mathematics expression of Finite set finite length constructed ... negative integer Exponentiation exponents . For example, nowrap x sup 2 sup &minus x 4 7 is a polynomial ... 3 2 . The term polynomial can also be used as an adjective, for quantities that can be expressed as a polynomial of some parameter, as in polynomial time , which is used in computational complexity theory . Polynomial comes from the Greek poly , many and medieval Latin binomium , binomial ... http www.cnrtl.fr etymologie bin C3 B4me ref ref Etymology of polynomial Compact Oxford English Dictionary .... For example, they are used to form polynomial equations, which encode a wide range of problems, from ... they are used to define polynomial functions, which appear in settings ranging from basic chemistry ... other functions. In advanced mathematics, polynomials are used to construct polynomial ring s, a central concept in abstract algebra and algebraic geometry . Overview A polynomial is either ..., variable should be used only when considering the function defined by the polynomial. In practice ... of a polynomial degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any ... the degree is 2 1 3. Forming a sum of several terms produces a polynomial. For example, the following is a polynomial math underbrace ,3x 2 begin smallmatrix mathrm term mathrm 1 end smallmatrix ... first, or in ascending powers of x . The polynomial in the example above is written in descending ... nowrap is 5 . The third term is a constant. Since the degree of a non zero polynomial is the largest degree of any one term, this polynomial has degree two. Two terms with the same variables ..., the distributive law is repeatedly applied, which results in each term of one polynomial being ... 2 21xy 2x 2y 12x 15y 2 3xy 2 28y 5 ,. math The sum or product of two polynomials is always a polynomial ... more details
In mathematics , a Hurwitz polynomial , named after Adolf Hurwitz , is a polynomial whose coefficients are positive real number s and whose zeros are located in the left half plane of the complex number complex plane , that is, the real part of every zero is negative. One sometimes uses the term Hurwitz polynomial simply as a real or complex polynomial with all zeros in the left half plane i.e., a Hurwitz stable polynomial . A polynomial is said to be Hurwitz if the following conditions are satisfied 1. P s is real when s is real. 2. The roots of P s have real parts which are zero or negative. Note Here P s is any polynomial in s. Examples A simple example of a Hurwitz polynomial is the following math x 2 2x 1. math The only real solution is &minus 1, as it factors to math x 1 2. math Properties For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial s roots to lie in the left half plane, it is necessary and sufficient that the polynomial in question pass the Routh Hurwitz stability criterion . A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique. 1. All the poles and zeros of a function are in the left half plane or on its boundary the imaginary axis. 2. Any poles and zeros on the imaginary axis are simple have a multiplicity of one . 3. Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative. 4. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle . 5. there have no any missing term of s but it possible after the testing the prf stability DEFAULTSORT Hurwitz Polynomial Category Polynomials mathanalysis stub de Hurwitzpolynom fr Polyn me ... more details
In mathematics, Conway polynomial can refer to the Alexander polynomial Alexander Conway polynomial Alexander Conway polynomial in knot theory the Conway polynomial finite fields disambig ... more details
In mathematics, Carlitz polynomial, named for Leonard Carlitz , may refer to Al Salam Carlitz polynomials Carlitz cyclotomic polynomial Carmichael Carlitz polynomial Dedekind Carlitz polynomial Stieltjes Carlitz polynomial Tricomi Carlitz polynomial mathdab ... more details
A polynomial is palindromic, if the sequence of its coefficients are a palindrome . Let math P x sum i 0 n a ix i math be a polynomial of degree n, then P is palindromic if math a i a n i math for i 0...n. Similarly, P is called antipalindromic if math a i a n i math for i 0...n. Examples Some examples of palindromic polynomials are math x 1 2 x 2 2x 1 math math x 1 3 x 3 3x 2 3x 1. math Generally, the expansion of math x 1 n math is palindromic for all n can see this from binomial expansion It also follows that if P is of even degree so has odd number of terms in the polynomial , then it can only be antipalindromic when the middle term is 0, i.e. math a i a i math , where math n 2i math . See also Reciprocal polynomial External links MathPages id home kmath294 kmath294 title The Fundamental Theorem for Palindromic Polynomials algebra stub Category Polynomials Category Palindromes de Symmetrische Gleichungen ru ... more details
disambig Minimal polynomial may refer to Minimal polynomial linear algebra Minimal polynomial of a square matrix A , the monic polynomial p x of least degree such that p A 0. Minimal polynomial field theory Minimal polynomial of an algebraic element over a field F , the monic polynomial p x over F of least degree such that p 0. fr Polyn me minimal ru ... more details
Polynomial chaos PC , also called Wiener Chaos expansion , is a non sampling based method to determine evolution of uncertainty in dynamical system, when there is probabilistic uncertainty in the system parameters. PC was first introduced by Wiener where Hermite polynomials were used to model stochastic processes with Gaussian random variables. It can be thought of as an extension of Volterra s theory of nonlinear functionals for stochastic systems. According to Cameron and Martin such an expansion converges in the math mathcal L 2 math sense for any arbitrary stochastic process with finite second moment. This applies to most physical systems. Xiu generalized the result of Cameron Martin to various continuous and discrete distributions using orthogonal polynomials from the so called Askey scheme and demonstrated math mathcal L 2 math convergence in the corresponding Hilbert functional space. This is popularly known as the generalized polynomial chaos gPC framework. The gPC framework has been applied to applications including stochastic fluid dynamics , stochastic finite elements, solid mechanics , nonlinear estimation, and probabilistic robust control. It has been demonstrated that gPC based methods are computationally superior to Monte Carlo based methods in a number of applications. However, the method has a notable limitation. For large numbers of random variables, polynomial ... systems using polynomial chaos expansion K. Sepahvand, S. Marburg and H. J. Hardtke, International ... State Trajectories in Bayesian Framework with Polynomial Chaos P. Dutta, R. Bhattacharya, Journal ... System Uncertainty Using Polynomial Chaos J. Fisher, R. Bhattacharya, Journal of Dynamic ..., and C. Sandu Polynomial Chaos Based Parameter Estimation Methods for Vehicle Systems . Journal ... with the Polynomial Chaos Method for Stiff Systems . Computers and Mathematics with Applications ..., 184 p., Softcover ISBN 978 0 387 95015 0 DEFAULTSORT Polynomial Chaos Category Stochastic processes ... more details
Refimprove date February 2008 In mathematics , especially in the field of abstract algebra , a polynomial ring is a ring mathematics ring formed from the Set mathematics set of polynomial s in one or more variables with coefficients in another ring mathematics ring . Polynomial rings have influenced ... of a linear operator . Many important conjectures involving polynomial rings, such as Quillen ... power series . Polynomials in one variable over a field Polynomials A polynomial in X with coefficients ... sup 0 sup 1, and the sum defining the polynomial p may be viewed as the linear combination of the symbols ... , , p sub 1 sub , p sub 0 sub . Using the summation symbol, the same polynomial is expressed more ... limits are frequently omitted, so that the same polynomial is written as math p sum k p kX k, , math ... large enough values of k , in our case, for k m . The degree of a polynomial is the largest k such that the coefficient of X &thinsp sup k sup is not zero. In the special case of zero polynomial, all ... The polynomial ring K X The set of all polynomials with coefficients in the field K forms a commutative ..., which may be alternatively viewed as real or complex polynomial functions . However, in general .... In the case of the polynomial ring K X , these operations are explicitly given by the following ... . Since a polynomial from K X can be multiplied by a scalar mathematics scalar k from K to yield a new polynomial, K X actually constitute an associative algebra over K . Viewed as a vector space, K ... R , giving rise to the polynomial ring over R , which is denoted R X . Properties of K X The polynomial ... Dedekind . K X is an integral domain The first property of the polynomial ring is elementary and says that a product of two non zero polynomials is also a non zero polynomial. Indeed, the product of a polynomial p of degree m starting with p sub m sub X &thinsp sup m sup , p sub m sub 0, and a polynomial q of degree n starting with q sub n sub X &thinsp sup n sup , q sub n sub 0, is the polynomial ... more details
Image Wilkinson polynomial.png 250px thumb right Plot of Wilkinson s polynomial Image log Wilkinson polynomial.png ... s polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when root finding algorithm finding the root of a polynomial the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial. The polynomial is math w x prod i 1 20 x i x 1 x 2 cdots x 20 . math Sometimes, the term Wilkinson s polynomial is also used to refer to some other polynomials appearing in Wilkinson s discussion. Background Wilkinson s polynomial arose in the study of algorithms for finding the roots of a polynomial math p x sum i 0 n ..., that is not the case here. The problem is ill conditioned when the polynomial has a multiple root. For instance, the polynomial x sup 2 sup has a double root at x 0. However, the polynomial x sup 2 sup ... natural to expect that ill conditioning also occurs when the polynomial has zeros which ... separated zeros. Wilkinson used the polynomial w x to illustrate this point Wilkinson 1963 . In 1984 ... polynomial ref Wilkinson s polynomial is often used to illustrate the undesirability of naively ... polynomial and then finding its roots, since using the coefficients as an intermediate step may ... Algebra publisher SIAM year 1997 ref Conditioning of Wilkinson s polynomial Wilkinson s polynomial math ... are far apart. However, the polynomial is still very ill conditioned. Expanding the polynomial ... &minus 23 sup to 210.0000001192, then the polynomial value w 20 decreases from 0 to &minus 2 sup &minus ... is insufficient for separating the roots of Wilkinson s polynomial. Stability analysis Suppose that we perturb a polynomial p x x &minus sub j sub with roots sub j sub by adding a small multiple t c x of a polynomial c x , and ask how this affects the roots sub j sub . To first ... unstable. For small values of t the perturbed root is given by the power series expansion in t math ... more details
polynomial is completely understood. In the expansion math P G x a 1 x a 2x 2 dots a nx n math ...Image Chromatic polynomial of all 3 vertex graphs BW.png thumb 250px right All nonisomorphic graphs on 3 ... math k k 1 k 2 math . The chromatic polynomial is a polynomial studied in algebraic graph theory , a branch ... to the Tutte polynomial by H. Whitney and W. T. Tutte , linking it to the Potts model of statistical physics . History George David Birkhoff introduced the chromatic polynomial in 1912, defining ... to the combinatorial coloring problem. Hassler Whitney generalised Birkhoff s polynomial from the planar ... ref Several chapters harvtxt Biggs 1993 ref Definition Image Chromatic polynomial of all 3 vertex graphs ... using math k math colors for math k 0,1,2,3 math . The chromatic polynomial of each graph interpolates through the number of proper colorings. The chromatic polynomial of a graph counts the number ... math G math the function is a polynomial in math k math , the number of colors. For example, the path ... math k math 0 1 2 3 Number of colorings math P G k math 0 0 2 12 The chromatic polynomial is defined as the unique interpolating polynomial of degree math n math through the points math k,P G k math ... graph, math P G k k k 1 2 math , and indeed math P G 3 12 math . The chromatic polynomial includes ... polynomial, math chi G min k colon P G k 0 . math Examples class wikitable Chromatic polynomials for certain ... n math vertices, the chromatic polynomial math P G,t math is in fact a polynomial it has degree math n math . Nonisomorphic graphs may have the same chromatic polynomial. By definition, evaluating the chromatic polynomial in math P G,k math yields the number of math k math colorings of math G math ... of every chromatic polynomial alternate in signs. The absolute values of coefficients of every chromatic polynomial form a Logarithmically concave sequence log concave sequence . ref harvtxt ... equivalent graphs.svg thumb right 250px center The three graphs with a chromatic polynomial equal ... more details
Auxiliary polynomial is a term in mathematics which may refer to The auxiliary function argument in transcendence theory The characteristic polynomial of a recurrence relation mathdab ... more details
disambig In different branches of mathematics , primitive polynomial has different meanings In field theory mathematics field theory , a primitive polynomial field theory primitive polynomial is the minimal polynomial field theory minimal polynomial of a primitive element finite field primitive element of the finite field finite extension field GF p sup m sup . In algebra and specifically ring theory , a Primitive polynomial ring theory primitive polynomial is a polynomial over an integral domain R such as the integer s such that no non invertible element of R divides all its coefficient s at once. ru zh ... more details
In the mathematics mathematical field of knot theory , a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot mathematics knot . History The first knot polynomial, the Alexander polynomial , was introduced by J. W. Alexander in 1923, but other knot polynomials were not found until almost 60 years later. In the 1960s, John Horton Conway John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander&ndash Conway polynomial . The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial . This led to the discovery of more knot polynomials, such as the so called HOMFLY polynomial . Soon after Jones discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a state sum model , which involved the bracket polynomial , an invariant of framing knot framed knots. This opened up avenues of research linking knot theory and statistical mechanics . In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones type invariants, had an interpretation in Chern&ndash Simons theory . Victor Anatolyevich Vasilyev Viktor Vassiliev and Mikhail Goussarov started the theory of finite type invariant s of knots. The coefficients of the previously named polynomials are known to be of finite type after perhaps a suitable change of variables . In recent years, the Alexander polynomial has been shown to be related to Floer homology . The graded Euler characteristic of the Heegaard Floer homology knot Floer homology of Ozsv th and Szab is the Alexander polynomial. References Colin Adams, The Knot Book , American ... knot polynomials Alexander polynomial Bracket polynomial HOMFLY polynomial Jones polynomial Kauffman polynomial Related topics skein relationship for a formal definition of the Alexander polynomial ... more details
In algebra, a multilinear polynomial is a polynomial that is linear in each of its variables. In other words, no variable occurs to a power of 2 or higher or alternatively, each monomial is a constant times a product of distinct variables. They are important in the study of polynomial identity testing , because if a multilinear polynomial is zero on a set of vectors that Linear span span the space, it will be zero everywhere. The Degree of a polynomial degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial. ref A. Giambruno, Mikhail Zaicev. Polynomial Identities and Asymptotic Methods. AMS Bookstore, 2005 ISBN 978 0 82183829 7. Section 1.3. ref References references Category Polynomials ... more details
In mathematics , in the realm of abstract algebra , a radical polynomial is a multivariate polynomial over a field that can be expressed as a polynomial in the sum of squares of the variables. That is, if math k x 1, x 2, ldots, x n math is a polynomial ring , the ring of radical polynomials is the subring generated by the polynomial math sum i 1 n x i 2. math Radical polynomials are characterized as precisely those polynomials that are invariant mathematics invariant under the action of the orthogonal group . The ring of radical polynomials is a graded algebra graded subalgebra of the ring of all polynomials. The standard separation of variables theorem asserts that every polynomial can be expressed as a finite sum of terms, each term being a product of a radical polynomial and a harmonic polynomial . This is equivalent to the statement that the ring of all polynomials is a free module over the ring of radical polynomials. References unreferenced date June 2008 Category Abstract algebra Category Polynomials Category Invariant theory Abstract algebra stub ... more details
In algebra, the Vandermonde polynomial of an ordered set of n variables math X 1, dots, X n math , named after Alexandre Th ophile Vandermonde , is the polynomial math V n prod 1 le i j le n X j X i . math ... . The value depends on the order of the terms it is an alternating polynomial , not a symmetric polynomial . Alternating The defining property of the Vandermonde polynomial is that it is alternating ..., while permuting them by an even permutation does not change the value of the polynomial in fact, it is the basic alternating polynomial, as will be made precise below. It thus depends on the order, and is zero ... polynomial is a factor of every alternating polynomial as shown above, an alternating polynomial vanishes ... i neq j math . Alternating polynomials main Alternating polynomial Thus, the Vandermonde polynomial together with the symmetric polynomial s generates the alternating polynomial s. Discriminant Its square is widely called the discriminant , though some sources call the Vandermonde polynomial itself the discriminant. The discriminant the square of the Vandermonde polynomial math Delta V n 2 math ... set of points. If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in n ... classes , the Vandermonde polynomial corresponds to the Euler class , and its square the discriminant ... polynomial and alternating polynomials generally is an unstable phenomenon, which corresponds ... stable or compatibly defined. However, this is not the case for the Vandermonde polynomial or alternating polynomials the Vandermonde polynomial in n variables is not obtained from the Vandermonde polynomial in math n 1 math variables by setting math X n 1 0 math . Vandermonde polynomial of a polynomial Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting field for a non monic polynomial, with leading coefficient a , one may define the Vandemonde polynomial ... with the discriminant. Generalizations Over arbitrary rings, one instead uses a different polynomial ... more details
In the mathematics mathematical field of knot theory , the bracket polynomial also known as the Kauffman bracket is a polynomial invariant of framed link s. Although it is not an invariant of knots or links as it is not invariant under type I Reidemeister move s , a suitably normalized version yields the famous knot invariant called the Jones polynomial . The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariant s. In particular, Kauffman s interpretation of the Jones polynomial allows generalization to invariants of 3 manifold s. The bracket polynomial was discovered by Louis Kauffman in 1987. Definition The bracket polynomial of any unoriented link diagram L , denoted L , is characterized by the three rules O 1, where O is the standard diagram of the unknot Image kauffman bracket2.png 275px math langle O cup L rangle A 2 A 2 langle L rangle math The pictures in the second rule represent brackets of the link diagrams which differ inside a disc as shown but are identical outside. The third rule means that adding a circle disjoint from the rest of the diagram multiplies the bracket of the remaining diagram by A sup 2 sup A sup 2 sup . References Louis H. Kauffman, State models and the Jones polynomial. Topology 26 1987 , no. 3, 395 407. introduces the bracket polynomial External links http mathworld.wolfram.com BracketPolynomial.html mathworld Category Knot theory Category Polynomials knottheory stub ja ru ... more details
In knot theory , the Kauffman polynomial is a 2 variable knot polynomial due to Louis Kauffman . It is initially defined on a link knot theory link diagram as math F K a,z a w K L K , math where math w K math is the writhe of the link diagram and math L K math is a polynomial in a and z defined on link diagrams by the following properties math L O 1 math O is the unknot math L s r aL s , qquad L s ell a 1 L s . math L is unchanged under type II and III Reidemeister move s Here math s math is a strand and math s r math resp. math s ell math is the same strand with a right handed resp. left handed curl added using a type I Reidemeister move . Additionally L must satisfy Kauffman s skein relation Image Kauffman poly.png 400px The pictures represent the L polynomial of the diagrams which differ inside a disc as shown but are identical outside. Kauffman showed that L exists and is a regular isotopy invariant of unoriented links. It follows easily that F is an ambient isotopy invariant of oriented links. The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial . The Kauffman polynomial is related to Chern Simons theory Chern Simons gauge theories for SO N in the same way that the HOMFLY polynomial is related to Chern Simons gauge theories for SU N see Witten s article Quantum field theory and the Jones polynomial , in Commun. Math. Phys. References Louis Kauffman , On Knots , 1987 , ISBN 0 691 08435 1 External links http eom.springer.de k k120040.htm Springer EoM entry for Kauffman polynomial http katlas.math.toronto.edu wiki The Kauffman Polynomial Knot Atlas entry for Kauffman polynomial Category Knot theory Category Polynomials knottheory stub ... more details
In mathematics , in abstract algebra , a multivariate polynomial over a field whose Laplacian is zero is termed a harmonic polynomial . The harmonic polynomials form a vector space vector subspace of the vector space of polynomials over the field. In fact, they form a graded algebra graded subspace . The Laplacian is the sum of second partials with respect to all the variables, and is an invariant mathematics invariant differential operator under the action of the orthogonal group viz the Group mathematics group of rotations. The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials. References Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 July 1963 algebra stub Category Abstract algebra Category Polynomials ru uk ... more details
In mathematics, a Neumanns polynomial , introduced by Carl Neumann for the special case math alpha 0 math , is a polynomial in 1 z used to expand functions in term of Bessel function s. ref Abramowitz and Stegun, http www.math.sfu.ca cbm aands page 363.htm p. 363, 9.1.82 ff. ref The first few polynomials are math O 0 alpha t frac 1 t, math math O 1 alpha t 2 frac alpha 1 t 2 , math math O 2 alpha t frac 2 alpha t 4 frac 2 alpha 1 alpha t 3 , math math O 3 alpha t 2 frac 1 alpha 3 alpha t 2 8 frac 1 alpha 2 alpha 3 alpha t 4 , math math O 4 alpha t frac 1 alpha 4 alpha 2t 4 frac 1 alpha 2 alpha 4 alpha t 3 16 frac 1 alpha 2 alpha 3 alpha 4 alpha t 5 . math A general form for the polynomial is math O n alpha t frac alpha n 2 alpha sum k 0 lfloor n 2 rfloor 1 n k frac n k k alpha choose n k left frac 2 t right n 1 2k , math they have the generating function math frac left frac z 2 right alpha Gamma alpha 1 frac 1 t z sum n 0 O n alpha t J alpha n z , math where J are Bessel function s. To expand a function f in form math f z sum n 0 a n J alpha n z , math for math z c math compute math a n frac 1 2 pi i oint z c frac Gamma alpha 1 left frac z 2 right alpha f z O n alpha z mathrm d z, math where math c c math and c is the distance of the nearest singularity of math z alpha f z math from math z 0 math . Examples An example is the extension math left tfrac 1 2 z right s Gamma s cdot sum k 0 1 k J s 2k z s 2k s choose k math or the more general Sonine formula ref harvnb Erd lyi Magnus ... s k frac J s k z left frac z 2 right s . math where math C k s math is Gegenbauer polynomial Gegenbauer s polynomial . Then, fact date September 2011 or date September 2011 math frac left frac z 2 ... n z , math the Taylor expansion addition formula math frac J s left sqrt z 2 2uz right left sqrt z .... It does not directly support the identity here. and the expansion of the integral of the Bessel ... Lommel polynomial Hankel transform Fourier Bessel series Notes references Category Polynomials Category ... more details
In mathematics , a polynomial sequence is a sequence of polynomial s indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics , as well as applied mathematics . Examples Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equation s Laguerre polynomials Chebyshev polynomials Legendre polynomials Jacobi polynomials Others come from statistics Hermite polynomials Many are studied in algebra and combinatorics Monomial s Rising factorial s Falling factorial s Abel polynomials Bell polynomials Bernoulli polynomials Dickson polynomial s Fibonacci polynomials Lagrange polynomials Lucas polynomials Spread polynomials Touchard polynomials Rook polynomials Classes of polynomial sequences Polynomial sequences of binomial type Orthogonal polynomials Secondary polynomials Sheffer sequence Sturm sequence Generalized Appell polynomials See also Umbral calculus References Aigner, Martin. A course in enumeration , GTM Springer, 2007, ISBN 3 540 39032 4 p21. Roman, Steven The Umbral Calculus , Dover Publications, 2005, ISBN 0 486 44129 3 Please check ISBN reason Check digit 3 does not correspond to calculated figure. . Williamson, S. Gill Combinatorics for Computer Science , Dover Publications, 2002 p177. DEFAULTSORT Polynomial Sequence Category Polynomials Category Sequences and series ar fr Suite de polyn mes it Sequenza polinomiale ... more details
In mathematics , an invariant polynomial is a polynomial math P math that is invariant mathematics invariant under a group mathematics group math Gamma math group action acting on a vector space math V math . Therefore math P math is a math Gamma math invariant polynomial if math P gamma x P x math for all math gamma in Gamma math and math x in V math . Cases of particular importance are for &Gamma a finite group in the theory of Molien series , in particular , a compact group , a Lie group or algebraic group . For a basis independent definition of polynomial nothing is lost by referring to the symmetric power s of the given linear representation of &Gamma . References PlanetMath attribution id 4337 title Invariant polynomial Category Commutative algebra Category Invariant theory Category Polynomials algebra stub ... more details
In the mathematics mathematical field of knot theory , the HOMFLY polynomial , sometimes called the HOMFLY PT polynomial or the generalized Jones polynomial , is a 2 variable knot polynomial , i.e. a knot invariant in the form of a polynomial of variables m and l . A central question in the knot theory mathematical theory of knots is whether two knot diagram s represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an knot invariant invariant of the knot , i.e. if two diagrams have different polynomial s, they represent different knots. The converse may not be true. The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the Alexander polynomial and the Jones polynomial both of which can be obtained by appropriate substitutions from HOMFLY. The HOMFLY polynomial is also a quantum invariant . The name HOMFLY combines the initials of its co discoverers Jim Hoste, Adrian Ocneanu , Kenneth Millett, Peter J. Freyd , W. B. R. Lickorish , and David Nelson Yetter. ref cite journal last Freyd first P. coauthors Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., and Ocneanu, A. title A New Polynomial Invariant of Knots and Links journal Bulletin ... and Pawe Traczyk. Definition The polynomial is defined using skein relation s math P mathrm unknot ... 200px center The HOMFLY polynomial of a link L that is a split union of two links math L 1 math and math ... for an example of a computation using these relations. Other HOMFLY skein relations This polynomial ... polynomial. math Delta t P alpha 1,z t 1 2 t 1 2 , , math where math Delta t , math is the Alexander polynomial. math P L 1 L 2 P L 1 P L 2 , , math math P K ell,m P Mirror Image K ell 1 ,m , , math .... Weisstein, Eric W. HOMFLY Polynomial. From MathWorld A Wolfram Web Resource. http mathworld.wolfram.com HOMFLYPolynomial.html DEFAULTSORT Homfly Polynomial Category Knot theory Category Polynomials ... more details
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