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Meixner Polynomial
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) \,. \e ... 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] |
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. Listeratur *{{Citation , last ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) 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 . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , 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 with yet another meaning, namely to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kravchuk 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 :\mathcal_k(x; n,q) = \mathcal_k(x) = \sum_^(-1)^j (q-1)^ \binom \binom, \quad k=0,1, \ldots, n. Properties The Kravchuk polynomial has the following alternative expressions: :\mathcal_k(x; n,q) = \sum_^(-q)^j (q-1)^ \binom \binom. :\mathcal_k(x; n,q) = \sum_^(-1)^j q^ \binom \binom. Symmetry relations For integers i,k \ge 0, we have that :\begin (q-1)^ \mathcal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |