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122 relations: Abraham de Moivre, Absolute convergence, Analytic function, Andrew Odlyzko, Annals of Statistics, Appell sequence, Arithmetic function, Asymptotic analysis, Bell number, Bell series, Bernoulli number, Betti number, Binomial coefficient, Binomial transform, Binomial type, Canadian Journal of Mathematics, Catalan number, Cauchy product, Cauchy's integral formula, Chebyshev polynomials, Closed-form expression, Codomain, Coefficient, Combinatorial principles, Combinatorics, Concrete Mathematics, Constant-recursive sequence, Contingency table, Continued fraction, Contour integration, Convergent series, Convolution, Cut-the-Knot, Cyclic sieving, Difference polynomials, Dirichlet character, Dirichlet L-function, Dirichlet series, Discrete Fourier transform, Divergent series, Divisor function, Domain of a function, Donald Knuth, Dover Publications, Ed Pegg Jr., Entire function, Enumeration, Enumerative combinatorics, Euler number, Euler product, ..., Euler–Maclaurin formula, Eulerian number, Even and odd functions, Experimental mathematics, Falling and rising factorials, Fibonacci number, Finite difference, Formal power series, Fractional calculus, Functional equation, Gamma function, Generalized Appell polynomials, Generalized continued fraction, Generating function transformation, Geometric progression, Geometric series, George Pólya, Harmonic number, Herbert Wilf, I. J. Good, Indeterminate (variable), Integer sequence, Laguerre polynomials, Lambert series, Leonhard Euler, List of mathematical series, Mathematics, Mathematics and plausible reasoning, Moment-generating function, Multiplicative function, Multiplicative inverse, Multiset, Number theory, Order of accuracy, Partial fraction decomposition, Partition (number theory), Partition function (mathematics), Periodic function, Pierre-Simon Laplace, Polylogarithm, Polynomial sequence, Probability mass function, Probability-generating function, Q-difference polynomial, Q-Pochhammer symbol, Radius of convergence, Ramanujan's congruences, Random variable, Rational function, Recurrence relation, Residue (complex analysis), Richard P. Stanley, Riemann zeta function, Rook polynomial, Root of unity, Sequence, Series expansion, Sheffer sequence, Spanning tree, Spence's function, Square number, Stanley's reciprocity theorem, Stirling number, Stirling numbers of the first kind, Stirling numbers of the second kind, Stirling polynomials, Stirling transform, Taylor series, The Art of Computer Programming, Triangular number, Wolfram Demonstrations Project, Z-transform. Expand index (72 more) »
Abraham de Moivre
Abraham de Moivre (26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
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Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Andrew Odlyzko
Andrew Michael Odlyzko (born 23 July 1949) is a mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute.
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Annals of Statistics
The Annals of Statistics is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics.
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Appell sequence
In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence \_ satisfying the identity and in which p_0(x) is a non-zero constant.
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Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers.
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Asymptotic analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
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Bell number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set.
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Bell series
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions.
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Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.
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Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.
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Binomial coefficient
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.
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Binomial transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences.
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Binomial type
In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities Many such sequences exist.
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Canadian Journal of Mathematics
The Canadian Journal of Mathematics (Journal canadien de mathématiques; print:, online) is a bimonthly mathematics journal published by the Canadian Mathematical Society.
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Catalan number
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects.
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Cauchy product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.
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Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
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Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.
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Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.
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Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.
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Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.
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Combinatorial principles
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used.
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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
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Concrete Mathematics
Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, first published in 1989, is a textbook that is widely used in computer-science departments as a substantive but light-hearted treatment of the analysis of algorithms.
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Constant-recursive sequence
In mathematics, a constant-recursive sequence or C-finite sequence is a sequence satisfying a linear recurrence with constant coefficients.
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Contingency table
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables.
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Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.
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Contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.
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Convergent series
In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
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Convolution
In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.
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Cut-the-Knot
Cut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics.
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Cyclic sieving
In combinatorial mathematics, cyclic sieving is a phenomenon by which evaluating a generating function for a finite set at roots of unity counts symmetry classes of objects acted on by a cyclic group.
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Difference polynomials
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.
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Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.
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Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.
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Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.
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Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.
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Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
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Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.
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Domain of a function
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Donald Knuth
Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.
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Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.
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Ed Pegg Jr.
Ed Pegg Jr. (born December 7, 1963) is an expert on mathematical puzzles and is a self-described recreational mathematician.
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Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
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Enumeration
An enumeration is a complete, ordered listing of all the items in a collection.
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Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.
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Euler number
In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where is the hyperbolic cosine.
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Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
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Euler–Maclaurin formula
In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums.
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Eulerian number
In combinatorics, the Eulerian number A(n, m), is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents").
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Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.
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Experimental mathematics
Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.
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Falling and rising factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as 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 The value of each is taken to be 1 (an empty product) when n.
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Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones: Often, especially in modern usage, the sequence is extended by one more initial term: By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.
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Finite difference
A finite difference is a mathematical expression of the form.
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Formal power series
In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.
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Fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator and of the integration operator and developing a calculus for such operators generalizing the classical one.
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Functional equation
In mathematics, a functional equation is any equation in which the unknown represents a function.
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Gamma function
In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.
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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: where the generating function or kernel K(z,w) is composed of the series and and 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.
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Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values.
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Generating function transformation
In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another.
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Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
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Geometric series
In mathematics, a geometric series is a series with a constant ratio between successive terms.
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George Pólya
George Pólya (Pólya György; December 13, 1887 – September 7, 1985) was a Hungarian mathematician.
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Harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.
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Herbert Wilf
Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was a mathematician, specializing in combinatorics and graph theory.
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I. J. Good
Irving John ("I. J."; "Jack") Good (9 December 1916 – 5 April 2009) The Times of 16-apr-09, http://www.timesonline.co.uk/tol/comment/obituaries/article6100314.ece was a British mathematician who worked as a cryptologist at Bletchley Park with Alan Turing.
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Indeterminate (variable)
In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.
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Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
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Laguerre polynomials
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation: which is a second-order linear differential equation.
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Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n).
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Leonhard Euler
Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.
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List of mathematical series
This list of mathematical series contains formulae for finite and infinite sums.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematics and plausible reasoning
Mathematics and plausible reasoning is a two-volume book by the mathematician George Pólya describing various methods for being a good guesser of new mathematical results.
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Moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.
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Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1).
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Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.
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Multiset
In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements.
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Number theory
Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.
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Order of accuracy
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution.
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Partial fraction decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
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Partition (number theory)
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.
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Partition function (mathematics)
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics.
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Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.
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Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a French scholar whose work was important to the development of mathematics, statistics, physics and astronomy.
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Polylogarithm
In mathematics, the polylogarithm (also known as '''Jonquière's function''', for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or rational functions.
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Polynomial sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3,..., in which each index is equal to the degree of the corresponding polynomial.
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Probability mass function
In probability and statistics, a probability mass function (pmf) is a function that gives the probability that a discrete random variable is exactly equal to some value.
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Probability-generating function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable.
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Q-difference polynomial
In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the ''q''-derivative.
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Q-Pochhammer symbol
In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a ''q''-analog of the Pochhammer symbol.
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Radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges.
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Ramanujan's congruences
In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n).
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Random variable
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.
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Rational function
In mathematics, a rational function is any function which can be defined by a rational fraction, i.e. an algebraic fraction such that both the numerator and the denominator are polynomials.
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Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.
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Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.
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Richard P. Stanley
Richard Peter Stanley (born June 23, 1944 in New York City, New York) is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts.
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
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Rook polynomial
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column.
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Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Series expansion
In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
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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.
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Spanning tree
In the mathematical field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G, with minimum possible number of edges.
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Spence's function
In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm.
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Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.
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Stanley's reciprocity theorem
In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
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Stirling number
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems.
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Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations.
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Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by S(n,k) or \textstyle \lbrace\rbrace.
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Stirling polynomials
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials.
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Stirling transform
In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by where \left\ is the Stirling number of the second kind, also denoted S(n,k) (with a capital S), which is the number of partitions of a set of size n into k parts.
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Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
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The Art of Computer Programming
The Art of Computer Programming (sometimes known by its initials TAOCP) is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis.
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Triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right.
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Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.
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Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation.
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Bivariate generating function, Dirichlet generating function, Exponential generating function, Generating Function, Generating functional, Generating functions, Generating polynomial, Generating series, Ordinary generating function, Ordinary generating functions, Poisson generating function, Snake oil method.
References
[1] https://en.wikipedia.org/wiki/Generating_function
