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Mott Polynomials
In mathematics the Mott polynomials ''s''''n''(''x'') are polynomials introduced by who applied them to a problem in the theory of electrons. They are given by the exponential generating function : e^=\sum_n s_n(x) t^n/n!. Because the factor in the exponential has the power series : \frac = -\sum_ C_k \left(\frac\right)^ in terms of Catalan numbers C_k, the coefficient in front of x^k of the polynomial can be written as : ^ks_n(x) =(-1)^k\frac\sum_C_C_\cdots C_, according to the general formula for generalized Appell polynomials, where the sum is over all compositions n=l_1+l_2+\cdots+l_k of n into k positive odd integers. The empty product appearing for k=n=0 equals 1. Special values, where all contributing Catalan numbers equal 1, are : ^n_n(x) = \frac. : ^_n(x) = \frac. By differentiation the recurrence for the first derivative becomes : s'(x) =- \sum_^ \frac C_k s_(x). The first few of them are :s_0(x)=1; :s_1(x)=-\fracx; :s_2(x)=\fracx^2; :s_3(x)=-\fracx-\fracx^3; :s_4( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan (1814–1894). The ''n''th Catalan number can be expressed directly in terms of binomial coefficients by :C_n = \frac = \frac = \prod\limits_^\frac \qquad\textn\ge 0. The first Catalan numbers for ''n'' = 0, 1, 2, 3, ... are :1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... . Properties An alternative expression for ''C''''n'' is :C_n = - for n\ge 0, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that ''C''''n'' is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula. The Catalan numbers satisfy the recurrence relations :C_0 = 1 \quad \text \quad C_=\sum_^C_i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Appell Polynomials
In mathematics, a polynomial sequence \ has a generalized Appell representation if the generating function for the polynomials takes on a certain form: :K(z,w) = A(w)\Psi(zg(w)) = \sum_^\infty p_n(z) w^n where the generating function or kernel K(z,w) is composed of the series :A(w)= \sum_^\infty a_n w^n \quad with a_0 \ne 0 and :\Psi(t)= \sum_^\infty \Psi_n t^n \quad and all \Psi_n \ne 0 and :g(w)= \sum_^\infty g_n w^n \quad with g_1 \ne 0. Given the above, it is not hard to show that p_n(z) is a polynomial of degree n. Boas–Buck polynomials are a slightly more general class of polynomials. Special cases * The choice of g(w)=w gives the class of Brenke polynomials. * The choice of \Psi(t)=e^t results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials. * The combined choice of g(w)=w and \Psi(t)=e^t gives the Appell sequence of polynomials. Explicit representation The generalized Appell polynomials ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composition (combinatorics)
In mathematics, a composition of an integer ''n'' is a way of writing ''n'' as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer ''n'' has distinct compositions. A weak composition of an integer ''n'' is similar to a composition of ''n'', but allowing terms of the sequence to be zero: it is a way of writing ''n'' as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the ''end'' of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheffer Sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer. Definition Fix a polynomial sequence (''p''''n''). Define a linear operator ''Q'' on polynomials in ''x'' by :Qp_n(x) = np_(x)\, . This determines ''Q'' on all polynomials. The polynomial sequence ''p''''n'' is a ''Sheffer sequence'' if the linear operator ''Q'' just defined is ''shift-equivariant''; such a ''Q'' is then a delta operator. Here, we define a linear operator ''Q'' on polynomials to be ''shift-equivariant'' if, whenever ''f''(''x'') = ''g''(''x'' + ''a'') = ''T''''a'' ''g''(''x'') is a "shift" of ''g''(''x''), then (''Qf'')(''x'') = (''Qg'')(''x'' + ''a''); i.e., ''Q'' commutes with every shift operator: ''T''''a''''Q'' = ''QT''''a''. Properties The set of all Sheffer sequences is a group un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
