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Encyclopedia > Hypergeometric series

In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. The series, if convergent, will define a hypergeometric function which may then be defined over a wider domain of the argument by analytic continuation. Hypergeometric functions generalize many special functions, including the Bessel functions, the Gamma function, the error function, the elliptic integrals and the orthogonal polynomials. This is in part because the hypergeometric functions are solutions to the hypergeometric differential equation, which is a fairly general second-order ordinary differential equation. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, several functions are important enough to deserve their own name. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials... In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...


The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and less frequently studied series. The basic series is the q-analog of the ordinary hypergeometric series. There are several generalizations of the ordinary hypergeometric series, including a generalization to Riemann symmetric spaces. In mathematics, the basic hypergeometric series, also sometimes called the hypergeometric q-series, are q-analog generalizations of ordinary hypergeometric series. ... In mathematics, a q-series is defined as usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ... In mathematics, especially differential geometry, a Riemannian symmetric space is a connected Riemannian manifold, whose isometry group contains symmetries (defined below) about every point. ...

Contents

The series 2F1

The classical standard hypergeometric series is given by:

,_2F_1 (a,b;c;z) = sum_{n=0}^infty frac{(a)_n(b)_n}{(c)_n} , frac {z^n} {n!}

where (a)_n=a(a+1)(a+2)ldots(a+n-1) is the rising factorial or Pochhammer symbol. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence. This series is one of 24 closely related solutions, the Kummer solutions, of the hypergeometric differential equation. In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...   (23 April 1776 – 29 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including integral number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. ...


Special cases and applications

The classic orthogonal polynomials can all be expressed as special cases of {;}_2F_1 with one or both a and b being (negative) integers. Many other special cases are listed in the Category:Special hypergeometric functions. In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...


The function 2F1 has several integral representations, including the Euler hypergeometric integral. In mathematics, the Euler hypergeometric integral is a representation of the hypergeometric function by means of an integral. ...


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. In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. ... In complex analysis, a discipline within mathematics, a Schwarz-Christoffel mapping is a transformation of the complex plane that maps the upper half-plane conformally to a polygon. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... A rendering of the Riemann Sphere. ...


A wide range of integrals of simple functions can be expressed using the hypergeometric function, e.g.:

intsqrt{1+x^alpha},mathrm{d}x = frac{xleft(alpha,{}_2F_1left(frac{1}{alpha},frac{1}{2};1+frac{1}{alpha};-x^alpharight)+2sqrt{x^alpha+1},right)}{2+alpha} ,   alphaneq0

A limiting case of 2F1 is the Kummer function 1F1(a,b;z), known as the confluent hypergeometric function. In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...


The series pFq

In the general case, the hypergeometric series is written as:

,_pF_q(a_1,ldots,a_p;b_1,ldots,b_q;z)=sum_{n=0}^infty frac {alpha_n z^n}{n!},

where α0 = 1 and

frac{alpha_{n+1}}{alpha_n} = frac{(n+a_1)(n+a_2)cdots(n+a_p)}{(n+b_1)(n+b_2)cdots(n+b_q)}

The series may also be written:

,_pF_q(a_1,ldots,a_p;b_1,ldots,b_q;z)=sum_{n=0}^infty frac{(a_1)_n(a_2)_nldots(a_p)_n}{(b_1)_n(b_2)_nldots(b_q)_n},frac{z^n}{n!},

where (a)n = a(a + 1)(a + 2)...(a + n − 1) is the rising factorial or Pochhammer symbol. In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ... In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...


Identities

A number of hypergeometric function identities were discovered in the nineteenth and twentieth centuries; one classical list of such identities is Bailey's list. In mathematics, hypergeometric identies are equalities involving sums over hypergeometric terms. ... In mathematics, the hypergeometric series is involved in a number of hypergeometric function identities. ...


It is currently understood that there is a very large number of such identities, and several algorithms are now known to generate and prove these identities. Some mathematicians research the various patterns that emerge from these algorithms.


Formal definition

A hypergeometric series is formally defined as any formal power series In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

 sum_n beta_n z^n

in which the ratio of successive coefficients

frac{beta_{n+1}}{beta_n}

is a rational function of n. That is, In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

frac{beta_{n+1}}{beta_n} = frac{A(n)}{B(n)}

for some polynomials A(n) and B(n). Thus, for example, in the case of a geometric series, this ratio is a constant. Another example is the series for the exponential function, for which In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... The exponential function is one of the most important functions in mathematics. ...

frac{beta_{n+1}}{beta_n} =frac{1}{n+1}.

In practice the series is written as an exponential generating function, modifying the coefficients so that a general term of the series takes the form 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. ...

beta_n = frac{alpha_n}{n!},

and α0 = 1. One uses the exponential function as a 'baseline' for discussion.


Many interesting series in mathematics have the property that the ratio of successive terms is a rational function. However, when expressed as an exponential generating function, such series have a non-zero radius of convergence only under restricted conditions. 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. Such a function, and its analytic continuations, is called the hypergeometric function. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...


Convergence conditions were given by Carl Friedrich Gauss, who examined the case of   (23 April 1776 – 29 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including integral number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...

frac{alpha_{n+1}}{alpha_n} = frac{(n+a)(n+b)}{(n+c)},

leading to the classical standard hypergeometric series

,_2F_1(a,b,c;z).

Notation

The standard notation for the general hypergeometric series is

,_mF_p.

Here, the integers m and p refer to the degree of the polynomials P and Q, respectively, referring to the ratio

frac{alpha_{n+1}}{alpha_n} = frac{P(n)}{Q(n)}.

If m>p+1, the radius of convergence is zero and so there is no analytic function. The series naturally terminates in case P(n) is ever 0 for n a natural number. If Q(n) were ever zero, the coefficients would be undefined.


The full notation for F assumes that P and Q are monic and factorised, so that the notation for F includes an m-tuple that is the list of the negatives of the zeroes of P and a p-tuple of the negatives of the zeroes of Q. This is not much of a restriction: the fundamental theorem of algebra applies, and we can also absorb a leading coefficient of P or Q by redefining z. As a result of the factorisation, a general term in the series then takes the form of a ratio of products of Pochhammer symbols. Since Pochhammer notation for rising factorials is traditional it is neater to write F with the negatives of the zeros. Thus, to complete the notational example, one has In mathematics, a monic can refer to monic morphism – a special kind of morphism in category theory, monic polynomial – a polynomial whose leading coefficient is one. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used... In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

,_2F_1 (a,b;c;z) = sum_{n=0}^infty frac{(a)_n(b)_n}{(c)_n} , frac {z^n} {n!}

where (a)n = a(a + 1)(a + 2)...(a + n − 1) is the rising factorial or Pochhammer symbol. Here, the zeros of P were −a and −b, while the zero of Q was −c. In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...


History and generalizations

Studies in the nineteenth century included those of Ernst Kummer, and the fundamental characterisation by Bernhard Riemann of the F-function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for the (2,1) variable F, examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities: that effectively the entire algorithmic side of the theory was a consequence of basic facts and the use of Möbius transformations as a symmetry group. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Bernhard Riemann. ... A rendering of the Riemann Sphere. ... In mathematics, in the theory of ordinary differential equations in the complex plane C, the points of C are classified into ordinary points, at which the equations coefficients are analytic functions, and singular points, at which some coefficient has a singularity. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...


The cases where the solutions are algebraic functions were found by H. A. Schwarz (Schwarz's list). This article or section does not cite its references or sources. ... Karl Hermann Amandus Schwarz (25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. ...


Subsequently the hypergeometric series were generalised to several variables, for example by Paul Emile Appell; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratio of successive terms, instead of being a rational function of n, are considered to be a rational function of qn. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n. Paul Émile Appell (September 27, 1855, Strasbourg - October 23, 1930) was the French mathematician. ... In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ... In mathematics, the basic hypergeometric series, also sometimes called the hypergeometric q-series, are q-analog generalizations of ordinary hypergeometric series. ... Heinrich Eduard Heine (March 15, 1821 in Berlin - October 21, 1881 in Halle (Saale)) was a German mathematician. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...


During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of hypergeometric series, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes). (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... Israel Moiseevich Gelfand (Russian: ) (born in 1913) is a prolific mathematician in the field of functional analysis, which he interprets in a broad sense as the mathematics of quantum mechanics. ... A hyperplane is a concept in geometry. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...


Hypergeometric series can be developed on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through a special case: the hypergeometric series 2F1 is closely related to the Legendre polynomials, and when used in the form of spherical harmonics, it expresses, in a certain sense, the symmetry properties of the two-sphere or equivalently the rotations given by the Lie group SO(3). Concrete representations are analogous to the Clebsch-Gordan coefficients. In mathematics, including applications to general relativity, a (Riemannian) symmetric space in differential geometry is a certain kind of homogeneous space in the theory of Lie groups. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ... In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... A rendering of the Riemann Sphere. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... This article may be too technical for most readers to understand. ...


References

  • Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. 
    • Chapter 13. - ,_1F_1 and ,_2F_0
    • Chapter 15. - ,_2F_1
  • Gerrit Heckman and Henrik Schlichtkrull (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. Academic Press, San Diego. ISBN 0-12-336170-2.  (Part 1 treats hypergeometric functions on Lie groups.)
  • Masaaki Yoshida (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Friedrick Vieweg & Son. ISBN 3-528-06925-2. 

  Results from FactBites:
 
Hypergeometric series - Wikipedia, the free encyclopedia (1030 words)
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.
Hypergeometric series - definition of Hypergeometric series in Encyclopedia (469 words)
In the case of geometric series the ratio is constant.
In practice it is preferred to write the series as an exponential generating function, modifying the coefficients to assume the general term of the series is
For the standard hypergeometric series denoted by F(a, b, c; z), the convergence conditions were given by Gauss.
  More results at FactBites »

 
 

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