In mathematics, the G-function was introduced by as a very general
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
intended to include most of the known
special function
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined b ...
s as particular cases. This was not the only attempt of its kind: the
generalized hypergeometric function
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 ...
and the
MacRobert E-function had the same aim, but Meijer's G-function was able to include those as particular cases as well. The first definition was made by Meijer using a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
; nowadays the accepted and more general definition is via a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, introduced in its full generality by
Arthur Erdélyi
Arthur Erdélyi FRS, FRSE (2 October 1908 – 12 December 1977) was a Hungarian-born British mathematician. Erdélyi was a leading expert on special functions, particularly orthogonal polynomials and hypergeometric functions.
Biography
He ...
in 1953.
With the modern definition, the majority of the established special functions can be represented in terms of the Meijer G-function. A notable property is the
closure of the set of all G-functions not only under differentiation but also under indefinite integration. In combination with a
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
that allows to liberate from a G-function ''G''(''z'') any factor ''z''
''ρ'' that is a constant power of its argument ''z'', the closure implies that whenever a function is expressible as a G-function of a constant multiple of some constant power of the function argument, ''f''(''x'') = ''G''(''cx''
''γ''), the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
and the
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of this function are expressible so too.
The wide coverage of special functions also lends power to uses of Meijer's G-function other than the representation and manipulation of derivatives and antiderivatives. For example, the
definite integral over the
positive real axis
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
of any function ''g''(''x'') that can be written as a product ''G''
1(''cx''
''γ'')·''G''
2(''dx''
''δ'') of two G-functions with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
''γ''/''δ'' equals just another G-function, and generalizations of
integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...
s like the
Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
and the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
and their inverses result when suitable G-function pairs are employed as transform kernels.
A still more general function, which introduces additional parameters into Meijer's G-function, is
Fox's H-function and is used for Matrix transform by
Ram Kishore Saxena
Ram Kishore Saxena D.Sc , FNASc (11 November 1936) is an Indian mathematician and Emeritus professor, UGC Jai Narain Vyas University
Jai Narain Vyas University (JNVU, formerly known as University of Jodhpur) is an educational institution in ...
One application of the Meijer G-function has been the particle spectrum of radiation from an inertial horizon in the moving mirror model of the dynamical Casimir effect .
Definition of the Meijer G-function
A general definition of the Meijer G-function is given by the following
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
:
:
where Γ denotes the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. This integral is of the so-called
Mellin–Barnes type, and may be viewed as an inverse
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
. The definition holds under the following assumptions:
* 0 ≤ ''m'' ≤ ''q'' and 0 ≤ ''n'' ≤ ''p'', where ''m'', ''n'', ''p'' and ''q'' are integer numbers
* ''a''
''k'' − ''b''
''j'' ≠ 1, 2, 3, ... for ''k'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''m'', which implies that no
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
of any Γ(''b''
''j'' − ''s''), ''j'' = 1, 2, ..., ''m'', coincides with any pole of any Γ(1 − ''a''
''k'' + ''s''), ''k'' = 1, 2, ..., ''n''
* ''z'' ≠ 0
Note that for historical reasons the ''first'' lower and ''second'' upper index refer to the ''top'' parameter row, while the ''second'' lower and ''first'' upper index refer to the ''bottom'' parameter row. One often encounters the following more synthetic notation using
vectors:
:
Implementations of the G-function in
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s typically employ separate vector arguments for the four (possibly empty) parameter groups ''a''
1 ... ''a''
''n'', ''a''
''n''+1 ... ''a''
''p'', ''b''
1 ... ''b''
''m'', and ''b''
''m''+1 ... ''b''
''q'', and thus can omit the orders ''p'', ''q'', ''n'', and ''m'' as redundant.
The ''L'' in the integral represents the path to be followed while integrating. Three choices are possible for this path:
:1. ''L'' runs from −''i''∞ to +''i''∞ such that all poles of Γ(''b''
''j'' − ''s''), ''j'' = 1, 2, ..., ''m'', are on the right of the path, while all poles of Γ(1 − ''a''
''k'' + ''s''), ''k'' = 1, 2, ..., ''n'', are on the left. The integral then converges for , arg ''z'', < ''δ'' ''π'', where
::
:an obvious prerequisite for this is ''δ'' > 0. The integral additionally converges for , arg ''z'', = ''δ'' ''π'' ≥ 0 if (q − p) (''σ'' +
1⁄
2) > Re(''ν'') + 1, where ''σ'' represents Re(''s'') as the integration variable ''s'' approaches both +''i''∞ and −''i''∞, and where
::
:As a corollary, for , arg ''z'', = ''δ'' ''π'' and ''p'' = ''q'' the integral converges independent of ''σ'' whenever Re(''ν'') < −1.
:2. ''L'' is a loop beginning and ending at +∞, encircling all poles of Γ(''b''
''j'' − ''s''), ''j'' = 1, 2, ..., ''m'', exactly once in the negative direction, but not encircling any pole of Γ(1 − ''a''
''k'' + ''s''), ''k'' = 1, 2, ..., ''n''. Then the integral converges for all ''z'' if ''q'' > ''p'' ≥ 0; it also converges for ''q'' = ''p'' > 0 as long as , ''z'', < 1. In the latter case, the integral additionally converges for , ''z'', = 1 if Re(''ν'') < −1, where ''ν'' is defined as for the first path.
:3. ''L'' is a loop beginning and ending at −∞ and encircling all poles of Γ(1 − ''a''
''k'' + ''s''), ''k'' = 1, 2, ..., ''n'', exactly once in the positive direction, but not encircling any pole of Γ(''b''
''j'' − ''s''), ''j'' = 1, 2, ..., ''m''. Now the integral converges for all ''z'' if ''p'' > ''q'' ≥ 0; it also converges for ''p'' = ''q'' > 0 as long as , ''z'', > 1. As noted for the second path too, in the case of ''p'' = ''q'' the integral also converges for , ''z'', = 1 when Re(''ν'') < −1.
The conditions for convergence are readily established by applying
Stirling's asymptotic approximation to the gamma functions in the integrand. When the integral converges for more than one of these paths, the results of integration can be shown to agree; if it converges for only one path, then this is the only one to be considered. In fact, numerical path integration in the complex plane constitutes a practicable and sensible approach to the calculation of Meijer G-functions.
As a consequence of this definition, the Meijer G-function is an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
of ''z'' with possible exception of the origin ''z'' = 0 and of the unit circle , ''z'', = 1.
Differential equation
The G-function satisfies the following linear
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
of order max(''p'',''q''):
:
For a fundamental set of solutions of this equation in the case of ''p'' ≤ ''q'' one may take:
:
and similarly in the case of ''p'' ≥ ''q'':
:
These particular solutions are analytic except for a possible
singularity at ''z'' = 0 (as well as a possible singularity at ''z'' = ∞), and in the case of ''p'' = ''q'' also an inevitable singularity at ''z'' = (−1)
''p''−''m''−''n''. As will be seen presently, they can be identified with
generalized hypergeometric function
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 ...
s
''p''''F''
''q''−1 of argument (−1)
''p''−''m''−''n'' ''z'' that are multiplied by a power ''z''
''b''''h'', and with generalized hypergeometric functions
''q''''F''
''p''−1 of argument (−1)
''q''−''m''−''n'' ''z''
−1 that are multiplied by a power ''z''
''a''''h''−1, respectively.
Relationship between the G-function and the generalized hypergeometric function
If the integral converges when evaluated along the
second path introduced above, and if no confluent
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
appear among the Γ(''b''
''j'' − ''s''), ''j'' = 1, 2, ..., ''m'', then the Meijer G-function can be expressed as a sum of
residue
Residue may refer to:
Chemistry and biology
* An amino acid, within a peptide chain
* Crop residue, materials left after agricultural processes
* Pesticide residue, refers to the pesticides that may remain on or in food after they are applied ...
s in terms of
generalized hypergeometric function
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 ...
s
''p''''F''
''q''−1 (Slater's theorem):
:
:
The star indicates that the term corresponding to ''j'' = ''h'' is omitted.
For the integral to converge along the second path one must have either ''p'' < ''q'', or ''p'' = ''q'' and , ''z'', < 1, and for the poles to be distinct no pair among the ''b''
''j'', ''j'' = 1, 2, ..., ''m'', may differ by an integer or zero. The asterisks in the relation remind us to ignore the contribution with index ''j'' = ''h'' as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function, if we recall the meaning of the vector notation,
:
this amounts to shortening the vector length from ''q'' to ''q''−1.
Note that when ''m'' = 0, the second path does not contain any pole, and so the integral must vanish identically,
:
if either ''p'' < ''q'', or ''p'' = ''q'' and , ''z'', < 1.
Similarly, if the integral converges when evaluated along the
third path above, and if no confluent poles appear among the Γ(1 − ''a''
''k'' + ''s''), ''k'' = 1, 2, ..., ''n'', then the G-function can be expressed as:
:
:
For this, either ''p'' > ''q'', or ''p'' = ''q'' and , ''z'', > 1 are required, and no pair among the ''a''
''k'', ''k'' = 1, 2, ..., ''n'', may differ by an integer or zero. For ''n'' = 0 one consequently has:
:
if either ''p'' > ''q'', or ''p'' = ''q'' and , ''z'', > 1.
On the other hand, any generalized hypergeometric function can readily be expressed in terms of the Meijer G-function:
:
where we have made use of the vector notation:
:
This holds unless a nonpositive integer value of at least one of its parameters a
p reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function vanishes and the parameter sets of the G-functions violate the requirement ''a''
''k'' − ''b''
''j'' ≠ 1, 2, 3, ... for ''k'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''m'' from the
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series
''p''''F''
''q''(''z'') converges, i. e. for any finite ''z'' when ''p'' ≤ ''q'', and for , ''z'', < 1 when ''p'' = ''q'' + 1. In the latter case, the relation with the G-function automatically provides the analytic continuation of
''p''''F''
''q''(''z'') to , ''z'', ≥ 1 with a branch cut from 1 to ∞ along the real axis. Finally, the relation furnishes a natural extension of the definition of the hypergeometric function to orders ''p'' > ''q'' + 1. By means of the G-function we can thus solve the generalized hypergeometric differential equation for ''p'' > ''q'' + 1 as well.
Polynomial cases
To express polynomial cases of generalized hypergeometric functions in terms of Meijer G-functions, a linear combination of two G-functions is needed in general:
:
:
where ''h'' = 0, 1, 2, ... equals the degree of the polynomial
''p''+1''F''
''q''(''z''). The orders ''m'' and ''n'' can be chosen freely in the ranges 0 ≤ ''m'' ≤ ''q'' and 0 ≤ ''n'' ≤ ''p'', which allows to avoid that specific integer values or integer differences among the parameters a
p and b
q of the polynomial give rise to divergent gamma functions in the prefactor or to a conflict with the
definition of the G-function. Note that the first G-function vanishes for ''n'' = 0 if ''p'' > ''q'', while the second G-function vanishes for ''m'' = 0 if ''p'' < ''q''. Again, the formula can be verified by expressing the two G-functions as sums of
residues; no cases of confluent
poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
permitted by the definition of the G-function need be excluded here.
Basic properties of the G-function
As can be seen from the
definition of the G-function, if equal parameters appear among the a
p and b
q determining the factors in the numerator and the denominator of the integrand, the fraction can be simplified, and the order of the function thereby be reduced. Whether the order ''m'' or ''n'' will decrease depends on the particular position of the parameters in question. Thus, if one of the ''a''
''k'', ''k'' = 1, 2, ..., ''n'', equals one of the ''b''
''j'', ''j'' = ''m'' + 1, ..., ''q'', the G-function lowers its orders ''p'', ''q'' and ''n'':
:
For the same reason, if one of the ''a''
''k'', ''k'' = ''n'' + 1, ..., ''p'', equals one of the ''b''
''j'', ''j'' = 1, 2, ..., ''m'', then the G-function lowers its orders ''p'', ''q'' and ''m'':
:
Starting from the definition, it is also possible to derive the following properties:
:
:
:
:
:
:
The abbreviations ''ν'' and ''δ'' were introduced in the
definition of the G-function above.
Derivatives and antiderivatives
Concerning
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s of the G-function, one finds these relationships:
:
:
:
:
From these four, equivalent relations can be deduced by simply evaluating the derivative on the left-hand side and manipulating a bit. One obtains for example:
:
Moreover, for derivatives of arbitrary order ''h'', one has
:
:
which hold for ''h'' < 0 as well, thus allowing to obtain the
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of any G-function as easily as the derivative. By choosing one or the other of the two results provided in either formula, one can always prevent the set of parameters in the result from violating the condition ''a''
''k'' − ''b''
''j'' ≠ 1, 2, 3, ... for ''k'' = 1, 2, ..., ''n'' and ''j'' = 1, 2, ..., ''m'' that is imposed by the
definition of the G-function. Note that each pair of results becomes unequal in the case of ''h'' < 0.
From these relationships, corresponding properties of the
Gauss hypergeometric function
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and of other special functions can be derived.
Recurrence relations
By equating different expressions for the first-order derivatives, one arrives at the following 3-term recurrence relations among contiguous G-functions:
:
:
:
:
Similar relations for the diagonal parameter pairs ''a''
1, ''b''
''q'' and ''b''
1, ''a''
''p'' follow by suitable combination of the above. Again, corresponding properties of hypergeometric and other special functions can be derived from these recurrence relations.
Multiplication theorems
Provided that ''z'' ≠ 0, the following relationships hold:
:
:
:
:
These follow by
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
about ''w'' = 1, with the help of the
basic properties discussed above. The
radii of convergence will be dependent on the value of ''z'' and on the G-function that is expanded. The expansions can be regarded as generalizations of similar theorems for
Bessel,
hypergeometric and
confluent hypergeometric functions.
Definite integrals involving the G-function
Among
definite integrals involving an arbitrary G-function one has:
:
Note that the restrictions under which this integral exists have been omitted here. It is, of course, no surprise that the
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used i ...
of a G-function should lead back to the integrand appearing in the
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
above.
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
-type integrals for the G-function are given by:
:
:
Extensive restrictions under which these integrals exist can be found on p. 417 of "Tables of Integral Transforms", vol. II(1954), Edited by A. Erdelyi. Note that, in view of their effect on the G-function, these integrals can be used to define the operation of
fractional integration
In fractional calculus, an area of mathematical analysis, the differintegral (sometime also called the derivigral) is a combined differentiation/ integration operator. Applied to a function ƒ, the ''q''-differintegral of ''f'', here denoted ...
for a fairly large class of functions (
Erdélyi–Kober operators).
A result of fundamental importance is that the product of two arbitrary G-functions integrated over the positive real axis can be represented by just another G-function (convolution theorem):
:
:
:
Restrictions under which the integral exists can be found in Meijer, C. S., 1941: Nederl. Akad. Wetensch, Proc. 44, pp. 82–92. Note how the Mellin transform of the result merely assembles the gamma factors from the Mellin transforms of the two functions in the integrand.
The convolution formula can be derived by substituting the defining Mellin–Barnes integral for one of the G-functions, reversing the order of integration, and evaluating the inner Mellin-transform integral. The preceding Euler-type integrals follow analogously.
Laplace transform
Using the above
convolution integral and
basic properties one can show that:
:
where Re(''ω'') > 0. This is the
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of a function ''G''(''ηx'') multiplied by a power ''x''
−''α''; if we put ''α'' = 0 we get the Laplace transform of the G-function. As usual, the inverse transform is then given by:
:
where ''c'' is a real positive constant that places the integration path to the right of any
pole
Pole may refer to:
Astronomy
*Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets
*Pole star, a visible star that is approximately aligned with the ...
in the integrand.
Another formula for the Laplace transform of a G-function is:
:
where again Re(''ω'') > 0. Details of the restrictions under which the integrals exist have been omitted in both cases.
Integral transforms based on the G-function
In general, two functions ''k''(''z'',''y'') and ''h''(''z'',''y'') are called a pair of transform kernels if, for any suitable function ''f''(''z'') or any suitable function ''g''(''z''), the following two relationships hold simultaneously:
:
The pair of kernels is said to be symmetric if ''k''(''z'',''y'') = ''h''(''z'',''y'').
Narain transform
showed that the functions:
:
:
are an asymmetric pair of transform kernels, where ''γ'' > 0, ''n'' − ''p'' = ''m'' − ''q'' > 0, and:
:
along with further convergence conditions. In particular, if ''p'' = ''q'', ''m'' = ''n'', ''a''
''j'' + ''b''
''j'' = 0 for ''j'' = 1, 2, ..., ''p'' and ''c''
''j'' + ''d''
''j'' = 0 for ''j'' = 1, 2, ..., ''m'', then the pair of kernels becomes symmetric. The well-known
Hankel transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling ...
is a symmetric special case of the Narain transform (''γ'' = 1, ''p'' = ''q'' = 0, ''m'' = ''n'' = 1, ''c''
1 = −''d''
1 =
''ν''⁄
2).
Wimp transform
showed that these functions are an asymmetric pair of transform kernels:
:
:
where the function ''A''(·) is defined as:
:
Generalized Laplace transform
The
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
can be generalized in close analogy with Narain's generalization of the Hankel transform:
:
:
where ''γ'' > ''0'', ''p'' ≤ ''q'', and:
:
and where the constant ''c'' > 0 places the second integration path to the right of any pole in the integrand. For ''γ'' =
1⁄
2, ''ρ'' = 0 and ''p'' = ''q'' = 0, this corresponds to the familiar Laplace transform.
Meijer transform
Two particular cases of this generalization were given by C.S. Meijer in 1940 and 1941. The case resulting for ''γ'' = 1, ''ρ'' = −''ν'', ''p'' = 0, ''q'' = 1 and ''b''
1 = ''ν'' may be written :
:
:
and the case obtained for ''γ'' =
1⁄
2, ''ρ'' = −''m'' − ''k'', ''p'' = ''q'' = 1, ''a''
1 = ''m'' − ''k'' and ''b''
1 = 2''m'' may be written :
:
:
Here ''I''
''ν'' and ''K''
''ν'' are the
modified Bessel functions
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
of the first and second kind, respectively, ''M''
''k'',''m'' and ''W''
''k'',''m'' are the
Whittaker function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced W ...
s, and constant scale factors have been applied to the functions ''f'' and ''g'' and their arguments ''s'' and ''t'' in the first case.
Representation of other functions in terms of the G-function
The following list shows how the familiar
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
s result as special cases of the Meijer G-function:
:
:
:
:
:
:
:
:
:
:
:
Here, ''H'' denotes the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
.
The subsequent list shows how some
higher functions can be expressed in terms of the G-function:
:
:
:
:
:
:
:
:
Even the
derivatives of γ(''α'',''x'') and Γ(''α'',''x'') with respect to ''α'' can be expressed in terms of the Meijer G-function. Here, γ and Γ are the lower and upper
incomplete gamma functions, ''J''
''ν'' and ''Y''
''ν'' are the
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary ...
s of the first and second kind, respectively, ''I''
''ν'' and ''K''
''ν'' are the corresponding modified Bessel functions, and Φ is the
Lerch transcendent In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who pub ...
.
See also
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Gradshteyn and Ryzhik
''Gradshteyn and Ryzhik'' (''GR'') is the informal name of a comprehensive table of integrals originally compiled by the Russian mathematicians I. S. Gradshteyn and I. M. Ryzhik. Its full title today is ''Table of Integrals, Series, and Products ...
References
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* (see § 5.3, "Definition of the G-Function", p. 206)
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* (see Chapter V, "The Generalized Hypergeometric Function and the G-Function", p. 136)
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* (see § 8.2, "The Meijer G-function", p. 617)
* (there is a 2008 paperback with )
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External links
* {{mathworld , urlname= MeijerG-Function , title= Meijer G-Function
hypergeomon
GitLab
GitLab Inc. is an open-core company that operates GitLab, a DevOps software package which can develop, secure, and operate software. The open source software project was created by Ukrainian developer Dmitriy Zaporozhets and Dutch developer S ...
Hypergeometric functions