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Topic: Hypergeometric series

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	Hypergeometric series - Wikipedia, the free encyclopedia
		In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k.
		Thus, by convention, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function with a non-zero radius of convergence.
		Applications of hypergeometric series includes the inversion of elliptic integrals; these are constructed by taking the ratio of the two linearly independent solutions of the hypergeometric differential equation to form Schwarz-Christoffel maps of the fundamental domain to the complex projective line or Riemann sphere.
	en.wikipedia.org /wiki/Hypergeometric_series (1030 words)

	Encyclopedia: Hypergeometric series (Site not responding. Last check: 2007-11-03)
		In mathematics, a hypergeometric series is the sum of a sequence of terms in which the ratios of successive coefficients k is a rational function of k.
		Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
	www.nationmaster.com /encyclopedia/Hypergeometric-series (528 words)

	Hypergeometric function identities - Wikipedia, the free encyclopedia
		In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e.
		There are two definitions of hypergeometric terms, both used in different cases as explained below.
		These algorithms first find a simple expression for a sum over hypergeometric terms and then provide a certificate which anyone could use to easily check and prove the correctness of the identity.
	en.wikipedia.org /wiki/Hypergeometric_function_identities (266 words)

	Basic Hypergeometric Series - Cambridge University Press (Site not responding. Last check: 2007-11-03)
		The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, orthogonal polynomials in several variables, and generating functions.
		Chapters 9-11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series.
		After a gentle introduction to basic series and some special cases (such as the ‘q’-binomial theorem) the authors bring the reader up to the latest results on the general theory and its extensions, many such results are due to them.
	www.cup.cam.ac.uk /catalogue/print.asp?isbn=0521833574&print=y (673 words)

	[No title]
		The notation used in this paper is the standard notation of Gasper and Rahman~\cite{6} for hypergeometric series ${}_pF_q$ and basic hypergeometric series ${}_p\Phi_q$; for double hypergeometric series (or their basic analogues) we shall use a notation which is close to that of Srivastava and Karlsson~\cite{24}.
		\section{Other hypergeometric series and basic analogues} The invariance groups of series transformations are thus obtained by iteration of known transformation formulas, together with the trivial numerator and denominator permutations.
		The $q$-analogue of Thomae's transformation is Sears's transformation~\cite{17} for the ${}_3\Phi_2$ series.
	allserv.rug.ac.be /~jvdjeugt/files/tex/LucknowProceedings.tex (3522 words)

	Lesson 2. Recognizing hypergeometric terms. (Site not responding. Last check: 2007-11-03)
		So hypergeometric term and series are generalizations of geometric term and series respectively.
		A rational function of n is a special case of a hypergeometric term.
		We have seen that hypergeometric terms and series are generalizations of geometric terms and series.
	www.webpearls.com /hypergeo/hgf_l02.html (195 words)

	Lesson 3. Putting hypergeometric series into normal form. (Site not responding. Last check: 2007-11-03)
		A hypergeometric function is a hypergeometric series in which the evaluation point is a variable.
		In this new series, the value of the term at index 0 is 2/3.
		The introduced F-notation is fundamental in the theory of hypergeometrics.
	www.webpearls.com /hypergeo/hgf_l03.html (657 words)

	Hypergeometric Function
		The indicial equation of the hypergeometric differential equation is:
		The sum of the hypergeometric series denoted by
		Gamma Function: A hypergeometric function can be expressed in terms of gamma functions.
	www.efunda.com /math/hypergeometric/hypergeometric.cfm (50 words)

	Geoff Campbell's Mathematics Homepage
		which enable me to write down new Dirichlet series summation formulas which are demonstrably analogues for known basic hypergeometric series summations.
		analogue of the Gauss hypergeometric sum, and analogues to Dixon's theorem.
		Professor Askey is an expert on Hypergeometric Series as they apply to the theory of Special Functions.
	www.geocities.com /CapeCanaveral/Launchpad/9416 (551 words)

	Untitled Document (Site not responding. Last check: 2007-11-03)
		Hypergeometric series and the Riemann Zeta function, Acta Arithmetica LXXXII:2 (1997), 103-118.
		The extended Cesàro theorem and hypergeometric asymptotics, Analysis 16:4 (1996), 379-384.
		A new proof for a terminating "Strange" hypergeometric evaluation of Gasper and Rahman conjectured by Gosper, C.
	www.dm.unile.it /ricerca/pubbl.asp?N=3 (1349 words)

	[No title]
		This inherent lack of symmetry is now turned to advantage, as we show that it is the inspiring source for obtaining new relations between hypergeometric series.
		\section{The triple sum series and an application} Apart from the well known expressions for 9-$j$ coefficients, a new expression (the triple sum series) was obtained by Ali\v sauskas, Jucys and Bandzaitis~\cite{Alisauskas,Jucys}.
		\section{Hypergeometric series} Due to the double summation in the above expression, we recall the definition of a certain type of double hypergeometric series.
	allserv.rug.ac.be /~jvdjeugt/files/tex/proc-mex.tex (861 words)

	Amazon.com: Basic Hypergeometric Series and Applications (Mathematical Surveys and Monographs): Books (Site not responding. Last check: 2007-11-03)
		These series were first studied systematically by Heine, but many early results are attributed to Euler, Gauss, and Jacobi.
		A bilateral series, formed from this function, is summed as an infinite product, thereby providing an elegant and fruitful result which goes back to Ramanujan.
		He also touches on general transformation theory for basic series in many variables and the basic multinomial, which is a generalization of a finite sum.
	www.amazon.com /exec/obidos/tg/detail/-/0821815245?v=glance (629 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Date: Mon, 22 Nov 1999 01:16:46 -0500 Newsgroups: sci.math Does there exist some C code for the computation of the complex hypergeometric series 2F1 (a, b; c, z) -- including the analytic continuation for z
		I need to calculate many such series as part of some new gravitational lens modeling software I'm writing -- and I've hit a bit of a wall in my calculations of the analytic continuation.
		TOMS algorithm 191 and algorithm 707 (for the confluent hypergeometric function).
	www.math.niu.edu /~rusin/papers/known-math/99/drusin (233 words)

	Citebase - Theta hypergeometric series
		Authors: Spiridonov, V. We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series.
		A characterization theorem for a single variable totally elliptic hypergeometric series is proved.
		W.J. Hollman III, L.C. Biedenharn, and J.D. Louck, On hypergeometric series well-poised in SU(n), SIAM J. Math.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0303204 (796 words)

	4.2 Horizon solution in series of hypergeometric functions
		This implies that it cannot be represented in the form of a single hypergeometric equation.
		However, if we focus on the solution near the horizon, it may be approximated by a hypergeometric equation.
		) is in the form of a hypergeometric equation.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2003-6/articlesu13.html (683 words)

	Algorithms Seminar 23/02/2005 (Site not responding. Last check: 2007-11-03)
		Hypergeometric series are used to approximate many important constants, such as $e$ and Apery's constant $\zeta (3)$.
		One traditional method to evaluate such series is binary splitting, a divide-and-conquer algorithm followed by integer division.
		The space complexity of our algorithm is the same as a bound on size of the reduced numerator and denominator of the series approximation.
	www.cs.mcgill.ca /~beezer/Seminars/23_02_05.html (164 words)

	I P M - Bulletin Board (Site not responding. Last check: 2007-11-03)
		We present two major (pure mathematical!) methods in the high-precision evaluation of such constants, namely, (1) accelerated convergence series, and (2) polynomial recurrences.
		We also explain interaction of the methods with the study of arithmetic properties of the constants and indicate some open problems in this respect.
		The aim of the talk is to show that hypergeometric integrals appear quite naturally in the corresponding irrationality proofs.
	www.ipm.ac.ir /IPM/news/ViewNewsInfo.jsp?NTID=184 (222 words)

	Hypergeometric generating functions for values of Dirichlet and other L functions -- Lovejoy and Ono 100 (12): 6904 -- ... (Site not responding. Last check: 2007-11-03)
		Hypergeometric generating functions for values of Dirichlet and other L functions -- Lovejoy and Ono 100 (12): 6904 -- Proceedings of the National Academy of Sciences
		the argument above to the series that is obtained by differentiating
		Andrews, G. Problems and Prospects for Basic Hypergeometric Series: The Theory and Application of Special Functions (Academic, New York).
	www.pnas.org /cgi/content/full/100/12/6904 (1080 words)

	Logarithm-free -hypergeometric series, Mutsumi Saito
		We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a Gel’fand-Kapranov-Zelevinskiĭ (GKZ) hypergeometric) system.
		In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set.
		[3] I. Gel'fand, A. Zelevinskiĭ, and M. Kapranov, Equations of hypergeometric type and Newton polyhedra, Soviet Math.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1085598118 (259 words)

	Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) \| Bill's Outbursts
		The Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) is part of our discount Book catalog.
		Used Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) are in stock for only $71.98.
		Discount pricing is subject to change, in order to get the Book Basic Hypergeometric Series Encyclopedia of Mathematics and its Applications at this reduced price, you must buy now!
	billcookmusic.com /amazon/asin.0521833574.Book_Basic_Hypergeometric_Series_Encyclopedia_of_Mathematics_and_its_Applications_.html (332 words)

	Pascal's Triangle - Modern Algorithmic Methods
		A geometric series is a sum of the form
		Many famous series in math are hypergeometric, including Bessel functions and the polynomial sequences named for Legendre, Chebyshev, Laguerre and Hermite, among others, including the binomial coefficient summations we are chiefly interested in.
		First published in 1945, Sister Celine’s Method is the oldest of the algorithmic methods now used and the progenitor of all those ideas that came afterwards.
	binomial.csuhayward.edu /MAM.html (1032 words)

	Geoff Campbell's Dirichlet series analogues of q-series
		In recent years I discovered a few Euler product transforms which apply to basic hypergeometric series (q-series).
		I found that applying the transforms to well known results such as the q-binomial theorem, or to results like Heine's q-series version of the Gauss hypergeometric series, led me to new sums of Dirichlet series.
		Some examples of cases of the D-analogue of the Gauss hypergeometric series summation (and also for the "Heine q-summation") are
	www.geocities.com /CapeCanaveral/Launchpad/9416/vpv1.html (198 words)

	HYP--A Package for Handling Hypergeometric Series -- from Mathematica Information Center
		The package HYP allows the handling of binomial and hypergeometric series.
		It provides tools for manipulating factorial expressions, transforming binomial sums into hypergeometric notation, summing hypergeometric series, transforming hypergeometric series, applying contiguous relations, doing formal limits of hypergeometric expressions, transforming hypergeometric Mathematica expressions into TeX-code, and applying Gosper's and Zeilberger's algorithms.
		HYP and HYPQ: Mathematica Packages for the Manipulation of Binomial Sums and Hypergeometric Series Respectively q-Binomial Sums and Basic Hypergeometric Series [in Articles]
	library.wolfram.com /infocenter/MathSource/656 (222 words)

	milne
		The purpose of this talk is to survey the transformation theory of $U(n+1)$ multiple basic hypergeometric series---starting with the $U(n+1)$ terminating very-well-poised $_6\phi_5$ summation theorems.
		These series were strongly motivated by L. Biedenharn and J. Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups $U(n+1)$, or equivalently $A_n$.
		Our $16$ and $24$ squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's $C_{\ell}$ nonterminating ${}_6\phi_5$ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's $2$, $4$, $6$, and $8$ squares identities.
	math.la.asu.edu /~sf2000/milne.html (660 words)

	Multidimensional matrix inversions and multiple basic hypergeometric series (Site not responding. Last check: 2007-11-03)
		As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series.
		Furthermore, we derive summation formulas for a different kind of multidimensional basic hypergeometric series associated to root systems of classical type.
		Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation and transformation theorems of the classical theory of basic hypergeometric series.
	www.mat.univie.ac.at /~schlosse/diss.html (211 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		A hypergeometric series $F$ depends on of two sets: parameters $A=(a_1, a_2,..., a_N)$ and variables $X=(x_1, x_2,..., x_n)$.
		A classical problem: For which sets of parameters the series $F$ can be written as a series $G$ with a lesser number of parameters $B=(b_1, b_2,..., b_M)$ and variables $Y=(y_1, y_2,..., y_m)$ where elements of $B$'s are rational expressions in $a_i$'s and elements of $Y$ are rational expressions in $y_j$'s.
		A series of reduction formulas was constructed in 1993 by Gelfand, Graev and Retakh as a by-product of a sophisticated geometrical machine.
	dimacs.rutgers.edu /Events/2005/abstracts/retakh.html (176 words)

	HYP-HYPQ
		HYP is a package, written in Mathematica, for the manipulation and identification of binomial and hypergeometric series and identities.
		There is also a ``q-analogue", the package HYPQ which allows you to manipulate and identify basic hypergeometric series.
		There is also the package HYP which allows you to manipulate and identify ordinary hypergeometric series.
	www.mat.univie.ac.at /~kratt/hyp_hypq/hyp.html (536 words)

	Gaussian Hypergeometric Series And Combinatorial Congruences (ResearchIndex)
		Abstract: this paper is to investigate similar phenomena for the hypergeometric series 3 F 2 (#) p.
		If p is an odd prime, then let F p be the field with p elements.
		2 A Gaussian hypergeometric series evaluation and Apery number..
	citeseer.ist.psu.edu /337880.html (193 words)

	Citebase - How can we escape Thomae's relations?
		It is well-known since then that there are 120 such relations (including the trivial ones which come from permutations of the parameters of the hypergeometric series).
		Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q-binomial sums and basic hypergeometric series, J. Symbol.
		[19] J. Whipple, A group of generalized hypergeometric series:relations between 120 allied series of type F[a; b; c; d; e], Proc.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0502276 (697 words)

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