Examples of "hypergeometric"
In mathematics, Heine's basic hypergeometric series, or hypergeometric "q"-series, are "q"-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series.
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ.
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by "n" is a rational function of "n". The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as
and many of the properties of the confluent hypergeometric function are limiting cases of properties of the hypergeometric function.
Confluent Hypergeometric Functions can be used to solve the Extended Confluent Hypergeometric Equation whose general form is given as:
Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.
The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function
The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as
There are two definitions of hypergeometric terms, both used in different cases as explained below. See also hypergeometric series.
In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in solutions to combinatorial problems, and also in the analysis of algorithms.
In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by . The general hypergeometric function is a function that is (more or less) defined on a Grassmannian, and depends on a choice of some complex numbers and signs.
The solution of the hypergeometric differential equation is very important. For instance, Legendre's differential equation can be shown to be a special case of the hypergeometric differential equation. Hence, by solving the hypergeometric differential equation, one may directly compare its solutions to get the solutions of Legendre's differential equation, after making the necessary substitutions. For more details, please check the hypergeometric differential equation.
The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the q-analog of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from zonal spherical functions on Riemannian symmetric spaces.
Historically, the most important are the functions of the form formula_46. These are sometimes called Gauss's hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function is used for the functions "F" if there is risk of confusion. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.
Given this recurrence, the algorithm does not return any hypergeometric solution, which proves that formula_8 does not simplify to a hypergeometric term.
In mathematics, Clausen's formula, found by , expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states
Fisher's noncentral hypergeometric distribution was first given the name extended hypergeometric distribution (Harkness, 1965), and some authors still use this name today.
The Pochhammer symbol is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for |"z"| < 1 by the power series
where formula_22 is a hypergeometric function. This function is also known as "Barnes's extended hypergeometric function". The definition of formula_23 is