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Krawtchouk Polynomials
Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname ) are discrete orthogonal polynomials associated with the binomial distribution, introduced by . The first few polynomials are (for ''q'' = 2): : \mathcal_0(x; n) = 1, : \mathcal_1(x; n) = -2x + n, : \mathcal_2(x; n) = 2x^2 - 2nx + \binom, : \mathcal_3(x; n) = -\fracx^3 + 2nx^2 - (n^2 - n + \frac)x + \binom. The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind. Definition For any prime power ''q'' and positive integer ''n'', define the Kravchuk polynomial \begin \mathcal_k(x; n,q) = \mathcal_k(x) =& \sum_^(-1)^j (q-1)^ \binom \binom \\ =& \sum_^k (-1)^j (q-1)^ \frac \frac \end for k=0,1, \ldots, n . In the second line, the factors depending on x have been rewritten in terms of falling factorials, to aid readers uncomfortable with non-integer arguments of binomial coefficients. Properties The Kravc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Orthogonal Polynomials
In mathematics, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure. Examples include the discrete Chebyshev polynomials, Charlier polynomials, Krawtchouk polynomials, Meixner polynomials, dual Hahn polynomials, Hahn polynomials, and Racah polynomials. If the measure has finite support, then the corresponding sequence of discrete orthogonal polynomials has only a finite number of elements. The Racah polynomials give an example of this. Definition Consider a discrete measure \mu on some set S=\ with weight function \omega(x). A family of orthogonal polynomials \ is called discrete if they are orthogonal with respect to \omega (resp. \mu), i.e., :\sum\limits_ p_n(x)p_m(x)\omega(x)=\kappa_n\delta_, where \delta_ is the Kronecker delta. Remark Any discrete measure is of the form : \mu = \sum_ a_i \delta_, so one can define a weight function by \omega(s_i) = a_i. Literature *{{Citation , las ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory), experiments, each asking a yes–no question, and each with its own Boolean-valued function, Boolean-valued outcome (probability), outcome: ''success'' (with probability ) or ''failure'' (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . If the sampling is carried out without replacement, the draws ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meixner Polynomials
In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end ... by :M_n(x,\beta,\gamma) = \sum_^n (-1)^kk!(x+\beta)_\gamma^ See also * Kravchuk polynomials References * * * * * * * * * * * * * *{{cite journal , first1= Xiang-Sheng , last1=Wang , first2=Roderick , last2=Wong , title= Global asymptotics of the Meixner polynomials , journal = Asymptot. Anal. , year=2011 , volume=75 , number=3–4 , pages=211–231 , doi=10.3233/ASY-2011-1060 , arxiv=1101.4370 Orthogonal polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (that is, the group of units of the ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Falling Factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial \begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) . \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, — A reprint of the 1950 edition by Chelsea Publishing. rising sequential product, or upper factorial) is defined as \begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) . \end The value of each is taken to be 1 (an empty product) when n=0. These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n, where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used (x)_n with yet another meaning, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Series
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krawtchouk Matrix
In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix ''K''(''N'') is an matrix. The first few Krawtchouk matrices are: : K^ = \begin 1 \end, \qquad K^ = \left \begin 1 & 1 \\ 1 & -1 \end \right \qquad K^ = \left \begin 1 & 1 & 1 \\ 2 & 0 & -2 \\ 1 & -1 & 1 \end \right \qquad K^ = \left \begin 1 & 1 & 1 & 1 \\ 3 & 1 & -1 & -3 \\ 3 & -1 & -1 & 3 \\ 1 & -1 & 1 & -1 \end \right : K^ = \left \begin 1 & 1 & 1 & 1 & 1 \\ 4 & 2 & 0 & -2 & -4 \\ 6 & 0 & -2 & 0 & 6 \\ 4 & -2 & 0 & 2 & -4 \\ 1 & -1 & 1 & -1 & 1 \end \right \qquad K^ = \left[ \begin 1 & 1 & 1 & 1 & 1 & 1 \\ 5 & 3 & 1 & -1 & -3 & -5 \\ 10 & 2 & -2 & -2 & 2 & 10 \\ 10 & -2 & -2 & 2 & 2 & -10 \\ 5 & -3 & 1 & 1 & -3 & 5 \\ 1 & -1 & 1 & -1 & 1 & -1 \end \right]. Definition In general, for positive integer N, the entries K^_ are given by the generating functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well as in connection with Brownian motion; * combinatorics, as an example of an Appell sequence, obeying the umbral calculus; * numerical analysis as Gaussian quadrature; * physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term \beginxu_\end is present); * systems theory in connection with nonlinear operations on Gaussian noise. * random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled ''CRC Concise Encyclopedia of Mathematics''. The free online version became only partially accessible to the public. In 1999 Weisstein we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
