Meijer G-Function
Encyclopedia
In 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 G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series
; nowadays the accepted and more general definition is via a path integral
in the complex plane
, introduced firstly by Arthur Erdélyi
in 1953. With the current definition, the majority of the special functions can be represented in terms of the G-function and of the gamma function
.
A still more general function, which introduces additional parameters into Meijer's G-function is Fox's H-function.
in the complex plane
:
This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform
. The definition holds under the following assumptions:
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
:
Implementions of the G-function in computer algebra system
s typically employ separate vector arguments for the four (possibly empty) parameter groups a1 ... an, an+1 ... ap, b1 ... bm, and bm+1 ... bq, 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:
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
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 G-function was able to include those as particular cases as well. The first definition was made by Meijer using a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
; nowadays the accepted and more general definition is via a path integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, introduced firstly by Arthur Erdélyi
Arthur Erdélyi
Arthur Erdélyi FRS, FRSE was a Hungarian-born British mathematician. Erdélyi was a leading expert on special functions - especially orthogonal polynomials and hypergeometric functions.-Biography:...
in 1953. With the current definition, the majority of the special functions can be represented in terms of the G-function and of the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.
A still more general function, which introduces additional parameters into Meijer's G-function is Fox's H-function.
Definition of the Meijer G-function
A general definition of the Meijer G-function is given by the following line integralLine integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
:
This integral is of the so-called Mellin–Barnes type, and may be viewed as an inverse Mellin transform
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...
. The definition holds under the following assumptions:
- 0 ≤ m ≤ q and 0 ≤ n ≤ p, where m, n, p and q are integer numbers
- ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m, which implies that no pole of any Γ(bj − s), j = 1, 2, ..., m, coincides with any pole of any Γ(1 − ak + 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
Vector (mathematics and physics)
In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n...
:
Implementions of the G-function in computer algebra system
Computer algebra system
A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
s typically employ separate vector arguments for the four (possibly empty) parameter groups a1 ... an, an+1 ... ap, b1 ... bm, and bm+1 ... bq, 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(bj − s), j = 1, 2, ..., m, are on the right of the path, while all poles of Γ(1 − ak + 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(bj − s), j = 1, 2, ..., m, exactly once in the negative direction, but not encircling any pole of Γ(1 − ak + 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 − ak + s), k = 1, 2, ..., n, exactly once in the positive direction, but not encircling any pole of Γ(bj − 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 already stated for the second path, 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 approximationStirling's approximationIn mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...
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 functionAnalytic functionIn 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, categories that are similar in some ways, but different in others...
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 equationOrdinary differential equationIn 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....
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 singularityMathematical singularityIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
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 functions pFq−1 of argument (−1)p−m−n z that are multiplied by a power zbh, and with generalized hypergeometric functions qFp−1 of argument (−1)q−m−n 1⁄z that are multiplied by a power zah−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 appear among the Γ(bj − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as a sum of residueResidue (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...
s in terms of generalized hypergeometric functions pFq−1 (Slater's theorem):
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 bj, 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 − ak + 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 ak, 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 ap reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function diverges and the parameter sets of the G-functions violate the requirement ak − bj ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m from the definition above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series pFq(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 pFq(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:
where h = 0, 1, 2, ... equals the degree of the polynomial p+1Fq(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 ap and bq of the polynomial give rise to divergent gamma prefactors or to a conflict with the definition of the G-function. Again, the formula can be verified by expressing the two G-functions as sums 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...
; no cases of confluent poles 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 ap and bq 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 of the particular position of the parameters in question. Thus, if one of the ak, k = 1, 2, ..., n, equals one of the bj, j = m + 1, ..., q, the G-function lowers its orders p, q and n:
For the same reason, if one of the ak, k = n + 1, ..., p, equals one of the bj, 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 derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of the G-function, there are these relationships:
From these four, equivalent relations can be deduced by simply calculating the derivative on the left-hand side and manipulating a bit. One obtains for example:
Moreover, for derivatives of arbitrary order k, one has
which hold for k < 0 as well, thus allowing to obtain the antiderivativeAntiderivativeIn calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
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 ak − bj ≠ 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 k < 0.
From these relationships, corresponding properties of the Gauss hypergeometric function 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 a1, bq and b1, ap 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 expansionTaylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
about w = 1, with the help of the basic properties discussed above. The radii of convergenceRadius of convergenceIn mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
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 BesselBessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
, hypergeometric and confluent hypergeometricConfluent hypergeometric functionIn 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...
functions.
Definite integrals involving the G-function
Among definite integralsIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
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 transformMellin transformIn mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
of a G-function should lead back to the integrand appearing in the definition above.
EulerLeonhard EulerLeonhard 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...
-type integrals for the G-function are given by:
Here too, the restrictions under which the integrals exist have been omitted. Note that, in view of their effect on the G-function, these integrals can be used to define the operations of fractional differentiation and fractional integrationFractional calculusFractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
for a fairly large class of functions (Erdélyi–Kober operators).
A result of fundamental importance is that the definite integral over a product of two arbitrary G-functions can be represented by just another G-function (convolution theorem):
Again, the restrictions under which the integral exists have been omitted here. Note how the Mellin transform of the result merely assembles the gamma factors from the Mellin transforms of the two functions in the integrand. Many of the amazing definite integrals listed in tables or produced by computer algebra systemComputer algebra systemA computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
s are nothing but special cases of this formula.
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 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 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 using the G-function
In general, two functions k(z,y) and h(z,y) are called transform kernels if, for any suitable function f(z) or any suitable function g(z), the following two relationships hold simultaneously:
The two kernels are said to be symmetric if k(z,y) = h(z,y).
Narain transform
showed that the functions:
are two asymmetric transform kernels, where γ > 0, n − p = m − q > 0, and:
along with further convergence conditions. In particular, if p = q, m = n, aj + bj = 0 for j = 1, 2, ..., p and cj + dj = 0 for j = 1, 2, ..., m, then the two kernels become symmetric. The well-known Hankel transformHankel transformIn mathematics, the Hankel transform expresses any given function f as the weighted sum of an infinite number of Bessel functions of the first kind Jν. The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r-axis...
is a symmetric special case of the Narain transform (γ = 1, p = q = 0, m = n = 1, c1 = −d1 = ν⁄2).
Wimp transform
showed that these two functions are asymmetric transform kernels:
where the function A(·) is defined as:
Generalized Laplace transform
The Laplace transform can be generalized in close analogy with Narain's generalization of the Hankel transform:
where γ > 0 and p ≤ q, 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 b1 = ν may be written :
and the case obtained for γ = 1⁄2, ρ = −m − k, p = q = 1, a1 = m − k and b1 = 2m may be written :
Here Iν and Kν are the modified Bessel functionsBessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
of the first and second kind, respectively, Mk,m and Wk,m are the Whittaker functionWhittaker functionIn 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...
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 functions result as special cases of the Meijer G-function:
Here, H denotes the Heaviside step functionHeaviside step functionThe Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
.
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 functionIncomplete gamma functionIn mathematics, the gamma function is defined by a definite integral. The incomplete gamma function is defined as an integral function of the same integrand. There are two varieties of the incomplete gamma function: the upper incomplete gamma function is for the case that the lower limit of...
s, Jν and Yν are the Bessel functionBessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s of the first and second kind, respectively, Iν and Kν are the corresponding modified Bessel functions, and Φ is the Lerch transcendent.
- 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