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• Generating_function abstract "In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. 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 arrays of numbers indexed by several 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. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal power series. These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal power series as its Taylor series; this explains the designation \"generating functions\". However such interpretation is not required to be possible, because formal power series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal power series; negative and fractional powers of x are examples of this.Generating functions are not functions in the formal sense of a mapping from a domain to a codomain; the name is merely traditional, and they are sometimes more correctly called generating series.".
• Generating_function wikiPageExternalLink book.pdf.
• Generating_function wikiPageExternalLink 1200514223.
• Generating_function wikiPageExternalLink GeneratingFunctions.
• Generating_function wikiPageExternalLink GeneratingFunctions.shtml.
• Generating_function wikiPageExternalLink DownldGF.html.
• Generating_function wikiPageExternalLink DownldGF.html.
• Generating_function wikiPageExternalLink search?q=1%2C1%2C2%2C2%2C3%2C5%2C5%2C7%2C10%2C15%2C15&sort=&language=&go=Search.
• Generating_function wikiPageID "160993".
• Generating_function wikiPageLength "25805".
• Generating_function wikiPageOutDegree "86".
• Generating_function wikiPageRevisionID "702247486".
• Generating_function wikiPageWikiLink Abraham_de_Moivre.
• Generating_function wikiPageWikiLink Absolute_convergence.
• Generating_function wikiPageWikiLink Analytic_function.
• Generating_function wikiPageWikiLink Andrew_Odlyzko.
• Generating_function wikiPageWikiLink Appell_sequence.
• Generating_function wikiPageWikiLink Asymptotic_analysis.
• Generating_function wikiPageWikiLink Bell_series.
• Generating_function wikiPageWikiLink Betti_number.
• Generating_function wikiPageWikiLink Binomial_coefficient.
• Generating_function wikiPageWikiLink Binomial_type.
• Generating_function wikiPageWikiLink Category:Generating_functions.
• Generating_function wikiPageWikiLink Cauchy_product.
• Generating_function wikiPageWikiLink Chebyshev_polynomials.
• Generating_function wikiPageWikiLink Closed-form_expression.
• Generating_function wikiPageWikiLink Codomain.
• Generating_function wikiPageWikiLink Coefficient.
• Generating_function wikiPageWikiLink Combinatorial_principles.
• Generating_function wikiPageWikiLink Combinatorics.
• Generating_function wikiPageWikiLink Concrete_Mathematics.
• Generating_function wikiPageWikiLink Contingency_table.
• Generating_function wikiPageWikiLink Convergent_series.
• Generating_function wikiPageWikiLink Convolution.
• Generating_function wikiPageWikiLink Cut-the-Knot.
• Generating_function wikiPageWikiLink Cyclic_sieving.
• Generating_function wikiPageWikiLink Difference_polynomials.
• Generating_function wikiPageWikiLink Dirichlet_L-function.
• Generating_function wikiPageWikiLink Dirichlet_character.
• Generating_function wikiPageWikiLink Dirichlet_series.
• Generating_function wikiPageWikiLink Domain_of_a_function.
• Generating_function wikiPageWikiLink Dover_Publications.
• Generating_function wikiPageWikiLink Ed_Pegg,_Jr..
• Generating_function wikiPageWikiLink Entire_function.
• Generating_function wikiPageWikiLink Enumeration.
• Generating_function wikiPageWikiLink Enumerative_combinatorics.
• Generating_function wikiPageWikiLink Euler_product.
• Generating_function wikiPageWikiLink Fibonacci_number.
• Generating_function wikiPageWikiLink Formal_power_series.
• Generating_function wikiPageWikiLink Gamma_function.
• Generating_function wikiPageWikiLink Generalized_Appell_polynomials.
• Generating_function wikiPageWikiLink Geometric_progression.
• Generating_function wikiPageWikiLink George_Pólya.
• Generating_function wikiPageWikiLink Harmonic_number.
• Generating_function wikiPageWikiLink Herbert_Wilf.
• Generating_function wikiPageWikiLink I._J._Good.
• Generating_function wikiPageWikiLink Indeterminate_(variable).
• Generating_function wikiPageWikiLink Indexed_family.
• Generating_function wikiPageWikiLink Lambert_series.
• Generating_function wikiPageWikiLink Leonhard_Euler.
• Generating_function wikiPageWikiLink Mathematics.
• Generating_function wikiPageWikiLink Mathematics_and_plausible_reasoning.
• Generating_function wikiPageWikiLink Moment-generating_function.
• Generating_function wikiPageWikiLink Multiplicative_function.
• Generating_function wikiPageWikiLink Multiplicative_inverse.
• Generating_function wikiPageWikiLink Natural_number.
• Generating_function wikiPageWikiLink Number_theory.
• Generating_function wikiPageWikiLink Partition_(number_theory).
• Generating_function wikiPageWikiLink Pierre-Simon_Laplace.
• Generating_function wikiPageWikiLink Polynomial_sequence.
• Generating_function wikiPageWikiLink Probability-generating_function.
• Generating_function wikiPageWikiLink Probability_distribution.
• Generating_function wikiPageWikiLink Probability_mass_function.
• Generating_function wikiPageWikiLink Q-difference_polynomial.
• Generating_function wikiPageWikiLink Radius_of_convergence.
• Generating_function wikiPageWikiLink Rational_function.
• Generating_function wikiPageWikiLink Recurrence_relation.
• Generating_function wikiPageWikiLink Richard_P._Stanley.
• Generating_function wikiPageWikiLink Riemann_zeta_function.
• Generating_function wikiPageWikiLink Rook_polynomial.
• Generating_function wikiPageWikiLink Sequence.
• Generating_function wikiPageWikiLink Sheffer_sequence.
• Generating_function wikiPageWikiLink Square_number.
• Generating_function wikiPageWikiLink Stanleys_reciprocity_theorem.
• Generating_function wikiPageWikiLink Taylor_series.
• Generating_function wikiPageWikiLink Triangular_number.
• Generating_function wikiPageWikiLink Wolfram_Demonstrations_Project.
• Generating_function wikiPageWikiLinkText "Exponential generating function".
• Generating_function wikiPageWikiLinkText "Generating Functions".
• Generating_function wikiPageWikiLinkText "Generating function".
• Generating_function wikiPageWikiLinkText "Generating function#Bivariate and multivariate generating functions".
• Generating_function wikiPageWikiLinkText "Generating function#Dirichlet series generating functions".
• Generating_function wikiPageWikiLinkText "Generating function#Exponential generating function".
• Generating_function wikiPageWikiLinkText "Generating function#Ordinary generating function".
• Generating_function wikiPageWikiLinkText "Ordinary generating function".
• Generating_function wikiPageWikiLinkText "asymptotic behavior".
• Generating_function wikiPageWikiLinkText "exponential generating function".
• Generating_function wikiPageWikiLinkText "g.f.".
• Generating_function wikiPageWikiLinkText "generate".
• Generating_function wikiPageWikiLinkText "generated".

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