Hypergeometric differential equation
Encyclopedia
In mathematics
, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear
ordinary differential equation
(ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by
,
, and .
Hypergeometric series were studied by Euler
, but the first full systematic treatment was given by ,
Studies in the nineteenth century included those of , and the fundamental characterisation by Bernhard Riemann
of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for the 2F1, examined in the complex plane, could be characterised (on the Riemann sphere
) by its three regular singularities.
The cases where the solutions are algebraic function
s were found by H. A. Schwarz (Schwarz's list
).
provided that c is not 0, −1, −2, … Notice that the series terminates if either "a" or "b" is a negative integer. The Pochhammer symbol
is defined by
For other complex values of z it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1.
.
The confluent hypergeometric function
(or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
Legendre function
s are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example
Several orthogonal polynomials, including Jacobi polynomials
P and their special cases Legendre polynomials, Chebyshev polynomials
, Gegenbauer polynomials
can be written in terms of hypergeometric functions using
Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.
Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0. For examples, if
then
is an elliptic modular function of τ.
Incomplete beta functions Bx(p,q) are related by
The complete elliptic integrals K and E are given by
which has three regular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation
. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.
. The equation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at the regular singular point. This gives 3 × 2 = 6 special solutions, as follows.
Around the point z = 0, two independent solutions are, if c is not an integer,
and
Around z = 1, if c − a − b is not an integer, one has two independent solutions
and
Around z = ∞, if a − b is not an integer, one has two independent solutions
and
Any 3 of these 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving = 20 linear relations between them called connection formulas.
Dn of order 2n-1×n!. For the hypergeometric equation n=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by
Kummer
. The isomorphism with the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a, b; c; z) to one of
which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4.
Applying Kummer's 24=6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities
(Euler transformation)
(Pfaff transformation)
(Pfaff transformation)
by making the substitution w = uv and eliminating the first-derivative term. One finds that
and v is given by the solution to
The Q-form is significant in its relation to the Schwarzian derivative
.
where k is one of the points 0, 1, ∞. The notation
is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.
Note that each triangle map is regular at z ∈ {0, 1, ∞} respectively, with
and
In the special case of λ, μ and ν real, with
then the s-maps are conformal map
s of the upper half-plane H to triangles on the Riemann sphere
, bounded by circular arcs. This mapping is a special case of a Schwarz–Christoffel mapping. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.
Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, and the s-maps are inverse functions of automorphic function
s for the triangle group
That is, when the path winds around a singularity of
, the value of the solutions at the endpoint will differ from the starting point.
Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation;
thus the monodromy is a mapping (group homomorphism):
where
is the fundamental group
.
In other words the monodromy is a two dimensional linear representation of the fundamental group.
The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices.
provided |z| < 1 or |z| = 1 and both sides converge, and can be proved by expanding (1 − zx)−a using the binomial theorem and then integrating term by term. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations.
Other representations, corresponding to other branches
, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy
action.
to evaluate the Barnes integral
as
where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a - 1, ..., −b, −b − 1, ... .
, 
, and 
are called contiguous to 
. Gauss showed that 
can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives =15 relations, given by identifying any two lines on the right hand side of
In the notation above,
and so on.
Repeatedly applying these relations gives a linear relation between any three functions of the form
, where m, n, and l are integers.
It follows by combining the two Pfaff transformations
which in turn follow from Euler's integral representation.
There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers, when the hypergeometric function becomes algebraic.
, is the identity
which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity, first found by Zhu Shijie
(= Chu Shi-Chieh), as a special case.
Dougall's formula
generalizes this to the bilateral hypergeometric series at z = 1.
A typical example is Kummer's theorem, named for Ernst Kummer
:
which follows from Kummer's quadratic transformations
and Gauss's theorem by putting z = −1 in the first identity.
Bailey's theorem is
and . Some typical examples are given by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....
ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
(ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by
,
, and .
History
The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.Hypergeometric series were studied by Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
, but the first full systematic treatment was given by ,
Studies in the nineteenth century included those of , and the fundamental characterisation by Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation (in z) for the 2F1, examined in the complex plane, could be characterised (on the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
) by its three regular singularities.
The cases where the solutions are algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s were found by H. A. Schwarz (Schwarz's list
Schwarz's list
In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically...
).
The hypergeometric series
The hypergeometric function is defined for |z| < 1 by the seriesprovided that c is not 0, −1, −2, … Notice that the series terminates if either "a" or "b" is a negative integer. The Pochhammer symbol
Pochhammer symbol
In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer is the notation ', where is a non-negative integer. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which...
is defined by
For other complex values of z it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1.
Special cases
Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are.The confluent hypergeometric function
Confluent hypergeometric function
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...
(or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
Legendre function
Legendre function
In mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions P, Q are generalizations of Legendre polynomials to non-integer degree.-Differential equation:...
s are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example
Several orthogonal polynomials, including Jacobi polynomials
Jacobi polynomials
In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
P and their special cases Legendre polynomials, Chebyshev polynomials
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
, Gegenbauer polynomials
Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C are orthogonal polynomials on the interval [−1,1] with respect to the weight function α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials...
can be written in terms of hypergeometric functions using
Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.
Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0. For examples, if
then
is an elliptic modular function of τ.
Incomplete beta functions Bx(p,q) are related by
The complete elliptic integrals K and E are given by
The hypergeometric differential equation
The hypergeometric function is a solution of Euler's hypergeometric differential equationwhich has three regular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation
Riemann's differential equation
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0,1, and ∞....
. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.
Solutions at the singular points
Solutions to the hypergeometric differential equation are built out of the hypergeometric series. The equation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at the regular singular point. This gives 3 × 2 = 6 special solutions, as follows.Around the point z = 0, two independent solutions are, if c is not an integer,
and
Around z = 1, if c − a − b is not an integer, one has two independent solutions
and
Around z = ∞, if a − b is not an integer, one has two independent solutions
and
Any 3 of these 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving = 20 linear relations between them called connection formulas.
Kummer's 24 solutions
A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter groupCoxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
Dn of order 2n-1×n!. For the hypergeometric equation n=3, so the group is of order 24 and is isomorphic to the symmetric group on 4 points, and was first described by
Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
. The isomorphism with the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a, b; c; z) to one of
which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4.
Applying Kummer's 24=6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities
(Euler transformation) (Pfaff transformation) (Pfaff transformation)Q-form
The hypergeometric differential equation may be brought into the Q-formby making the substitution w = uv and eliminating the first-derivative term. One finds that
and v is given by the solution to
The Q-form is significant in its relation to the Schwarzian derivative
Schwarzian derivative
In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and...
.
Schwarz triangle maps
The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.where k is one of the points 0, 1, ∞. The notation
is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.
Note that each triangle map is regular at z ∈ {0, 1, ∞} respectively, with
and
In the special case of λ, μ and ν real, with
then the s-maps are conformal mapConformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
s of the upper half-plane H to triangles on the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
, bounded by circular arcs. This mapping is a special case of a Schwarz–Christoffel mapping. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.
Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, and the s-maps are inverse functions of automorphic function
Automorphic function
In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....
s for the triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
Monodromy group
The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point.That is, when the path winds around a singularity of
, the value of the solutions at the endpoint will differ from the starting point.Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation;
thus the monodromy is a mapping (group homomorphism):
where
is the fundamental groupFundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
.
In other words the monodromy is a two dimensional linear representation of the fundamental group.
The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices.
Euler type
If B is the beta function thenprovided |z| < 1 or |z| = 1 and both sides converge, and can be proved by expanding (1 − zx)−a using the binomial theorem and then integrating term by term. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations.
Other representations, corresponding to other branches
Principal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....
, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
action.
Barnes integral
Barnes used the theory of residuesResidue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
to evaluate the Barnes integral
Barnes integral
In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by...
as
where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a - 1, ..., −b, −b − 1, ... .
John transform
The Gauss hypergeometric function can be written as a John transform .Gauss' contiguous relations
The six functions, , and are called contiguous to . Gauss showed that can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a,b,c, and z. This gives =15 relations, given by identifying any two lines on the right hand side ofIn the notation above,
and so on.Repeatedly applying these relations gives a linear relation between any three functions of the form
, where m, n, and l are integers.Gauss's continued fraction
Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:Transformation formulas
Transformation formulas relate two hypergeometric functions at different values of the argument z.Fractional linear transformations
Euler's transformation isIt follows by combining the two Pfaff transformations
which in turn follow from Euler's integral representation.
Quadratic transformations
If two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by , and a complete list was given by . A typical example isHigher order transformations
If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by . A typical example isThere are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers, when the hypergeometric function becomes algebraic.
Values at special points z
See for a list of summation formulas at special points, most of which also appear in . gives further evaluations at more points. shows how most of these identities can be verifed by computer algorithms.Special values at z = 1
Gauss's theorem, named for Carl Friedrich GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, is the identity
which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity, first found by Zhu Shijie
Zhu Shijie
Zhu Shijie , courtesy name Hanqing , pseudonym Songting , was one of the greatest Chinese mathematicians lived during the Yuan Dynasty....
(= Chu Shi-Chieh), as a special case.
Dougall's formula
Bilateral hypergeometric series
In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratioof two terms is a rational function of n...
generalizes this to the bilateral hypergeometric series at z = 1.
Kummer's theorem (z = −1)
There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformation to change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result.A typical example is Kummer's theorem, named for Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
:
which follows from Kummer's quadratic transformations
and Gauss's theorem by putting z = −1 in the first identity.
Values at z = 1/2
Gauss's second summation theorem isBailey's theorem is
Other points
There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed inand . Some typical examples are given by
Generalizations
Generalizations of the hypergeometric function include:- Appell seriesAppell seriesIn mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by and that generalize Gauss's hypergeometric series 2F1 of one variable...
, a 2-variable generalization of hypergeometric series - Basic hypergeometric seriesBasic hypergeometric seriesIn mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series....
where the ratio of terms is a periodic function of the index - Bilateral hypergeometric seriesBilateral hypergeometric seriesIn mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratioof two terms is a rational function of n...
pHp are similar to generalized hypergeometric series, but summed over all integers - Elliptic hypergeometric seriesElliptic hypergeometric seriesIn mathematics, an elliptic hypergeometric series is a series Σcn such that the ratiocn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the...
where the ratio of terms is an elliptic function of the index - Fox H-function, an extension of the Meijer G-function
- Generalized hypergeometric series pFq where the ratio of terms is a rational function of the index
- Heun function, solutions of second order ODE's with four regular singular points
- Horn functionHorn functionIn the theory of special functions in mathematics, the Horn functions are the 34 distinct convergent hypergeometric series of order two , enumerated by . They are listed in ....
, 34 distinct convergent hypergeometric series in two variables - Hypergeometric function of a matrix argumentHypergeometric function of a matrix argumentIn mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals....
, the multivariate generalization of the hypergeometric series - Lauricella hypergeometric series, hypergeometric series of three variables
- MacRobert E-function, an extension of the generalized hypergeometric series pFq to the case p>q+1.
- Meijer G-functionMeijer G-FunctionIn mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...
, an extension of the generalized hypergeometric series pFq to the case p>q+1.
External links
- The book "A = B", this book is freely downloadable from the internet.