Egorychev Method

	Egorychev Method The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Catalan numbers and other combinatorial numbers. The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the origin). The sought-for identity can now be found using manipulations of integrals. Some of these manipulations are not clear from the generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero, yielding a new expression for the original sum. The residue at infinity is particularly important in these consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Georgy Egorychev Georgy Petrovich Egorychev (or Yegorychev) (Георгий Петрович Егорычев, born 1938) is a Russian mathematician, known for the Egorychev method. Biography He graduated in mathematics from Ural State University and in 1960 became a teacher of mathematics in secondary school. In 1982 G. P. Egorychev and D. I. Falikman shared the Fulkerson Prize for (independently) proving van der Waerden's conjecture that the matrix with all entries equal has the smallest permanent of any doubly stochastic matrix.D. I. Falikman, "A proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix," ''Matematicheskie Zametki'' 29: 931–938, 1981. Egorychev is now a professor in the Department of Mathematical Support of Discrete Devices and Systems, Institute of Mathematics and Fundamental Informatics at Siberian Federal University (Russian abbreviation is SFU, SibFU, or СФУ), founded in 2006. He was an Invited Speaker of the ICM in 1986 in Berkeley, Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Residue At Infinity In complex analysis, a branch of mathematics, the residue at infinity is a Residue (complex analysis), residue of a holomorphic function on an Annulus (mathematics), annulus having an infinite external radius. The ''infinity'' \infty is a point added to the local space \mathbb C in order to render it compact space, compact (in this case it is a Alexandroff extension, one-point compactification). This space denoted \hat is isomorphism, isomorphic to the Riemann sphere.Michèle Audin, ''Analyse Complexe'', lecture notes of the University of Strasbouravailable on the web pp. 70–72 One can use the residue at infinity to calculate some integrals. Definition Given a holomorphic function ''f'' on an Annulus (mathematics), annulus A(0, R, \infty) (centered at 0, with inner radius R and infinite outer radius), the residue at infinity of the function ''f'' can be defined in terms of the usual residue (mathematics), residue as follows: : \operatorname(f,\infty) = -\operatorname\left( f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Newton Binomial In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x)^ centered at x = 0, where \alpha \in \Complex and , x, 0 and diverges to +\infty if \operatorname\alpha<0. If $\operatorname\alpha=0$ , then $n^ = e^$ converges if and only if the sequence $\operatorname\alpha\log n$ converges $\bmod$ , which is certainly true if $\alpha=0$ but false if $\operatorname\alpha \neq0$ : in the latter case the sequence is dense $\bmod$ , due to the fact that $\log n$ diverges and $\log (n+1)-\log n$ converges to zero). Summation of the binomial series The usual argument to compute the sum of the binomial series goes as follows. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Stirling Number 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 \left\. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind (or Stirling cycle numbers). Mutually inverse (finite or infinite) triangular matrices can be formed from the Stirling numbers of each kind according to the parameters ''n'', ''k''. Definition The Stirling numbers of the second kind, written S(n,k) or \lbrace\textstyle\rbrace or with other notations, count the number of ways to partition a set of n labelled objects into k nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations with p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Stirling Number Of The First Kind In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers in general. Definitions The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s(n,k) in the expansion of the falling factorial :(x)_n = x(x-1)(x-2)\cdots(x-n+1) into powers of the variable x: :(x)_n = \sum_^n s(n,k) x^k, For example, (x)_3 = x(x-1)(x - 2) = 1x^3 - 3x^2 + 2x, leading to the values s(3, 3) = 1, s(3, 2) = -3, and s(3, 1) = 2. Subsequently, it was discovered that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
$Click for more on -> K \le N$ picture info	K \le N K, or k, is the eleventh letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''kay'' (pronounced ), plural ''kays''. The letter K usually represents the voiceless velar plosive. History The letter K comes from the Greek letter Κ (kappa), which was taken from the Semitic kaph, the symbol for an open hand. This, in turn, was likely adapted by Semitic tribes who had lived in Egypt from the hieroglyph for "hand" representing /ḏ/ in the Egyptian word for hand, ⟨ ḏ-r-t⟩ (likely pronounced in Old Egyptian). The Semites evidently assigned it the sound value instead, because their word for hand started with that sound. K was brought into the Latin alphabet with the name ''ka'' /kaː/ to differentiate it from C, named ''ce'' (pronounced /keː/) and Q, named ''qu'' and pronounced /kuː/. In the earliest Latin inscriptions, the letters C, K and Q were all used t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Iverson Bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets: = \begin 1 & \text P \text \\ 0 & \text \end In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true. The Iverson bracket allows using capital-sigma notation without summation index. That is, for any property P(k) of the integer k, \sum_kf(k)\, (k)= \sum_f(k). By convention, f(k) does not need to be defined for the values of for which the Iverson bracket equals ; that is, a summand f(k) textbf/math> must evaluate to 0 regardless of whether f(k) is de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Exponential Integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of ''x'', the exponential integral Ei(''x'') is defined as : \operatorname(x) = -\int_^\infty \fract\,dt = \int_^x \fract\,dt. The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of ''x'', but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and Instead of Ei, the following notation is used, :E_1(z) = \int_z^\infty \frac\, dt,\qquad, (z), 0. Properties Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Rational Function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field ''K''. In this case, one speaks of a rational function and a rational fraction ''over K''. The values of the variables may be taken in any field ''L'' containing ''K''. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is ''L''. The set of rational functions over a field ''K'' is a field, the field of fractions of the ring of the polynomial functions over ''K''. Definitions A function f(x) is called a rational function if and only if it can be written in the form : f(x) = \frac where P\, and Q\, are polynomial functions of x\, and Q\, is not the zero function. The domain of f\, is the set of all values of x\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Identity (mathematics) In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometric ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cauchy Residue Theorem In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem. Statement The statement is as follows: Let be a simply connected open subset of the complex plane containing a finite list of points , , and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented simple closed curve, if is i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Generating Function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

Summation of the binomial series