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# Topic: Generating function

 Generating Functions from Interactive Mathematics Miscellany and Puzzles A generating function is a clothesline on which we hang up a sequence of numbers for display. In this sense, the term generating function is not quite consistent in that the operation of substituting a number for x in order to establish the value of the series at that point, is not legal. is the generating function of the sequence of the binomial coefficients www.cut-the-knot.org /blue/GeneratingFunctions.shtml   (757 words)

 JSSM- 2006, Vol.5, Issue 4, 567 - 574 The mathematical method of generating functions is used to show that the likelihood of long matches can be substantially reduced by using the tiebreak game in the fifth set, or more effectively by using a new type of game, the 50-40 game, throughout the match. The cumulant generating function (taking the natural logarithm of the moment generating function), can also be used to calculate the parameters of the distribution in a tennis match. The cumulant generating function is particularly useful for calculating the parameters of distributions for the number of points in a tiebreaker match, since the critical property of cumulant generating functions is that they are additive for linear combinations of independent random variables. www.jssm.org /vol5/n4/14/v5n4-14text.php   (2943 words)

 Generating Functions That was a proof by generating functions, another of the popular tools used by the species Homo sapiens for the proof of identities before the computer era. Generating functions have numerous applications in mathematics, especially in combinatorics, probability theory, statistics, the theory of Markov chains, and number theory. The generating function of a finite sequence is a polynomial in a single variable. www.maa.org /editorial/knot/GeneratingFunctions.html   (1083 words)

 Generating Functions Ordinary (pointwise) convergence of a sequence of generating functions corresponds to convergence of the corresponding distributions. By contrast, recall that the probability density function of a sum of independent variables is the convolution of the individual density functions, a much more complicated operation. The characteristic function of a random variable can be obtained from the moment generating function, under the basic existence condition that we saw earlier. www.math.uah.edu /statold/expect/Generating.xhtml   (1314 words)

 GeneratingFunctions - PineWiki is the generating function for the sequence (a The generating function for { 0, 1 } is 2z, so the generating function for sequences of zeros and ones is F = 1/(1-2z) by the repetition rule. The multivariate generating function for sequences of zeroes and ones is 1/(1-x-y) by the repetition rule. pine.cs.yale.edu /pinewiki/GeneratingFunctions   (4438 words)

 RSolve The exponential generating function of the sequence is the function and the exponential generating function of the constant sequence This confirms that the coefficients of the series expansion of the function are the squares of the integers. documents.wolfram.com /v5/Add-onsLinks/StandardPackages/DiscreteMath/RSolve.html   (288 words)

 Example generating functions A generating function for a given sequence of integers is simply the function defined by the power series whose coefficients are the integers in the sequence. A functional equation for a generating function G(z) is an equation which the function G(z) satisfies. Functional equations can often be derived from properties of the terms of the generating function without knowing a closed form for the coefficients or knowing a closed form for G(z). www.cs.drexel.edu /~jjohnson/2006-07/fall/cs300/lectures/gen.mw   (1001 words)

 PlanetMath: moment generating function th-derivative of the moment generating function evaluated at zero is the moment generating function of the sum of independent random variables This is version 5 of moment generating function, born on 2001-10-26, modified 2006-09-18. planetmath.org /encyclopedia/MomentGeneratingFunction.html   (83 words)

 Global Generating Function For Palindromic Products of Consecutive Integers This paper shows how to develop a general generating function for palindromic products of consecutive integers which is based the rather cryptic formula of base x (base+1). One of the earliest applications of a mixed mode filter (hereafter referred to by its acronym of MMF) was in the development of the sequence algebraic generating function of the prime sequence Prime(z) where a finite expansion in Laurent series form is given by equation (1) [1,3 to 6]. Comments: The numerical scans are brief as the objective is to develop and test the general algebraic generating function for this type of sequences as given by equations (2a,b). web.singnet.com.sg /~huens/paper56.htm   (1687 words)

 Springer Online Reference Works If the generating function is known, then properties of the Taylor coefficients of analytic functions are used in the study of the sequence Generating functions in the sense of formal power series are also often used. Usually it is possible to justify manipulations with such functions regardless of convergence. eom.springer.de /g/g043900.htm   (188 words)

 Encyclopedia of Combinatorial Structures (Introduction) A sequence of integers: the (n+1)th term of this sequence is the number of objects of size n defined by the specification. In the unlabelled universe, ordinary generating functions are used. If the objects are unlabelled (ordinary generating functions), these coefficients are the number of objects, otherwise, in the labelled case (exponential generating functions), they are the number of objects divided by n!. algo.inria.fr /encyclopedia   (366 words)

 Generating Functions The moment generating function shares many of the important properties of the probability generating function, but is defined for a larger collection of random variables. In the last exercise, we considered a distribution for which only some of the moments are finite; naturally, the moment generating function was infinite. Naturally, the results for bivariate moment generating functions have analogues in the general multivariate case. www.ds.unifi.it /VL/VL_EN/expect/expect4.html   (556 words)

 Moment Generating Function Since the exponential function is positive, the moment generating function of X always exists, either as a real number or as positive infinity. The most important fact is that if the moment generating function of X is finite in an open interval about 0, then this function completely determines the distribution of X. The result in Exercise 5 is one of the most important properties of moment generating functions, and is frequently used to determine the distribution of a sum of independent variables. www.fmi.uni-sofia.bg /vesta/Virtual_Labs/expect/expect5.html   (375 words)

 Generating Function Order Lists A function order list is text that specifies the order in which the linker should link the non-static functions of your program. A function order list, created with proforder, lets you optimize the layout of non-static functions; that is, external and library functions whose names are exposed to the linker. The linker cannot directly affect the layout order for static functions because the names of these functions are not available in the object files. www.intel.com /software/products/compilers/clin/docs/main_cls/mergedProjects/optaps_cls/common/optaps_pgo_fctn.htm   (435 words)

 [No title] This paper defines a canonical generating function called CGF(z) for short which is nicknamed the Swiss-knife function. For the uninitiated, the axiom simply defines the natural number sequence holistically by a generating function z/(z- 1) which is in fact a restatement of the order principle from number theory in sequence algebraic format. In the next section, we will define a canonical generating function called CGF(z), which is holistic and yet generates a single integer output for a single integer input. web.singnet.com.sg /~huens/paper2.htm   (2707 words)

 QNX Community Resources However, you can also generate some C and C++ stub files on the spot when using various dialogs to develop your application; use the icons that are located next to function or filename fields: This means you're free to edit a callback function while still in the process of attaching it to the widget. PhAB generates function prototypes that are used by the compiler to check that your functions are called correctly. www.qnx.com /developers/docs/qnx_4.25_docs/photon114/prog_guide/generating.html   (2555 words)

 Generating Function Prototypes For example: a C function which takes a float as an argument has two possible calling conventions: the argument may be promoted to double (used by the C compiler when no prototype is in scope), or it may be passed as a float (used by the C compiler when there is a prototype). Note that I am trying to make > it quite clear that an ffi declaration is not portable unless you > provide function prototypes except in the special case that your C > compiler generates the same code with or without a prototype. If a particular compiler requires prototypes in order to generate correct code (such as GHC when going via C) then this is a matter for that compiler's documentation. www.haskell.org /pipermail/ffi/2002-July/000579.html   (662 words)

 Generating Function Order Lists A function order list specifies the order in which the linker should link the non-static functions of your program. Profile-Guided Optimizations supports generating a function order list to be used by the linker; the compiler determines the order using profile information. To generate a function order list, use the profmerge and proforder Tools. www.intel.com /software/products/compilers/docs/fmac/doc_files/source/extfile/optaps_for/common/optaps_pgo_fctn.htm   (432 words)

 Math Forum Discussions - Re: Extracting coefficients from Generating Function Also, the generating function is just a good tool for counting things. function to find other sequences with are related to it in some way with The Math Forum is a research and educational enterprise of the Drexel School of Education. www.mathforum.org /kb/thread.jspa?forumID=13&threadID=84286&messageID=407243   (504 words)

 XSLT: Chapter 5: Creating Links and Cross-References function to retrieve the text of the referenced term; if the name of a term changes (as buzzwords go in and out of fashion, some marketing genius might want to rename the "pattern-matching character," for example), we can rerun our stylesheet and be confident that all references to the new term contain the correct text. functions to find the first token and the rest of the string, it's important that there be at least one space in the string. Be aware that using extension functions limits the portability of your stylesheets. oreilly.com /catalog/xslt/chapter/ch05.html   (5169 words)

 The Central Limit Theorem A way of proof can be seen through the fact (that I didn't prove) that if the generating functions of a sequence of random variables converges to the limiting generating function of a random variable Z then the distribution functions converge to the distribution function of Z. We show through use of the fact that the moment generating function of a sum of independent rv's is the product of their mgf's and the fact that they have the same distribution then The theorem is also true for independent variables who do not have the same distributions, along as they are bounded and the means and variances are finite. www-stat.stanford.edu /~susan/courses/s116/node120.html   (144 words)

 egfh.com EGFH and EGF Homepage In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural 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. The particular generating function that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. egfh.com   (688 words)

 Generating Function Prototypes What I believe I'm hearing from Manuel is that the presence of function prototypes is allowed to affect code generation. I'd much rather have a spec where its presence doesn't matter but failing that I'd like a spec which is clear about what you need to do to write portable code. If specifying a header file with a function prototype is allowed to affect the generated code (i.e., the calling convention), then it is clear that the Haskell type does not completely determine the calling convention. www.haskell.org /pipermail/ffi/2002-July/000582.html   (873 words)

 [No title] We have earlier defined generating functions for a sequence of numbers a However for some sequences whose terms count permutations it is more useful to consider generating functions using the set of monomials 1,x,x Determine the number of ways of colouring the squares of a 1-by-n chessboard using the colours red, white, blue if an even number of squares are to be coloured white. www.cs.iitm.ernet.in /theory/mfcs98page/mfcshtml/notes3/expgenfun.html   (155 words)

 Function Squashing Manipulations of what we give to a function are a little more complicated. What we're doing when we do this is manipulating where the function is generating from. This is like stretching or shrinking the domain from which we're generating the function. library.thinkquest.org /2647/algebra/squash.htm   (444 words)

 Generating XML Data from the Database As part of generating a valid XML element name from an SQL identifier, characters that are disallowed in an XML element name are escaped. function, the name of the element is not escaped in any way and hence this function can be used to transport SQL columns and values without escaped names. When generating the XML, the number of rows indicated by the setSkipRows call are skipped, then the maximum number of rows as specified by the setMaxRows call (or the entire result if not specified) is fetched and converted to XML. download-west.oracle.com /docs/cd/B10501_01/appdev.920/a96620/xdb12gen.htm   (6172 words)

 UMTYMP Advanced Topics – Introduction to Combinatorics Method #3 Using the combinatorial interpretation of the product of exponential generating functions. The exponential generating function for the number of permutations is Every permutation consists of a certain set of fixed points, and a derangement on the remaining points. www.math.umn.edu /~jhall/courses/2474/hw11sol.htm   (220 words)

 Amazon.com: "ordinary generating function": Key Phrase page Some general properties of generating functions are listed as Theorem 3.1. (i) If g(x) is the ordinary generating function of (a), then (1 - x)g(x) is the ordinary generating function of (ar - ar-,). As this section discusses ordinary generating functions only, we will sometimes omit the word "ordinary"... www.amazon.com /phrase/ordinary-generating-function   (333 words)

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