Generalized hypergeometric function

Last updated
Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D.svg
Plot of the generalized hypergeometric function pFq(a b z) with a=(2,4,6,8) and b=(2,3,5,7,11) in the complex plane from -2-2i to 2+2i created with Mathematica 13.1 function ComplexPlot3D

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.

Contents

Notation

A hypergeometric series is formally defined as a power series

in which the ratio of successive coefficients is a rational function of n. That is,

where A(n) and B(n) are polynomials in n.

For example, in the case of the series for the exponential function,

we have:

So this satisfies the definition with A(n) = 1 and B(n) = n + 1.

It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are complex numbers.

For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.

The ratio between consecutive coefficients now has the form

,

where c and d are the leading coefficients of A and B. The series then has the form

,

or, by scaling z by the appropriate factor and rearranging,

.

This has the form of an exponential generating function. This series is usually denoted by

or

Using the rising factorial or Pochhammer symbol

this can be written

(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)

Terminology

When all the terms of the series are defined and it has a non-zero radius of convergence, then the series defines an analytic function. Such a function, and its analytic continuations, is called the hypergeometric function.

The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion

which could be written za−1e−z 2F0(1−a,1;;−z−1). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function.

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.

The series without the factor of n! in the denominator (summed over all integers n, including negative) is called the bilateral hypergeometric series.

Convergence conditions

There are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0.

Excluding these cases, the ratio test can be applied to determine the radius of convergence.

The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z = 1 if

.

Further, if p=q+1, and z is real, then the following convergence result holds Quigley et al. (2013):

.

Basic properties

It is immediate from the definition that the order of the parameters aj, or the order of the parameters bk can be changed without changing the value of the function. Also, if any of the parameters aj is equal to any of the parameters bk, then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. For example,

.

This cancelling is a special case of a reduction formula that may be applied whenever a parameter on the top row differs from one on the bottom row by a non-negative integer. [1] [2]

Euler's integral transform

The following basic identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones [3]

Differentiation

The generalized hypergeometric function satisfies

and

Additionally,

Combining these gives a differential equation satisfied by w = pFq:

.

Take the following operator:

From the differentiation formulas given above, the linear space spanned by

contains each of

Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent: [4] [5]


These dependencies can be written out to generate a large number of identities involving .

For example, in the simplest non-trivial case,

,
,
,

So

.

This, and other important examples,

,
,
,
,
,

can be used to generate continued fraction expressions known as Gauss's continued fraction.

Similarly, by applying the differentiation formulas twice, there are such functions contained in

which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way.

A function obtained by adding ±1 to exactly one of the parameters aj, bk in

is called contiguous to

Using the technique outlined above, an identity relating and its two contiguous functions can be given, six identities relating and any two of its four contiguous functions, and fifteen identities relating and any two of its six contiguous functions have been found. (The first one was derived in the previous paragraph. The last fifteen were given by Gauss in his 1812 paper.)

Identities

A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries. A 20th century contribution to the methodology of proving these identities is the Egorychev method.

Saalschütz's theorem

Saalschütz's theorem [6] ( Saalschütz 1890 ) is

For extension of this theorem, see a research paper by Rakha & Rathie.

Dixon's identity

Dixon's identity, [7] first proved by Dixon (1902), gives the sum of a well-poised 3F2 at 1:

For generalization of Dixon's identity, see a paper by Lavoie, et al.

Dougall's formula

Dougall's formula ( Dougall   1907 ) gives the sum of a very well-poised series that is terminating and 2-balanced.

Terminating means that m is a non-negative integer and 2-balanced means that

Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.

Generalization of Kummer's transformations and identities for 2F2

Identity 1.

where

;

Identity 2.

which links Bessel functions to 2F2; this reduces to Kummer's second formula for b = 2a:

Identity 3.

.

Identity 4.

which is a finite sum if b-d is a non-negative integer.

Kummer's relation

Kummer's relation is

Clausen's formula

Clausen's formula

was used by de Branges to prove the Bieberbach conjecture.

Special cases

Many of the special functions in mathematics are special cases of the confluent hypergeometric function or the hypergeometric function; see the corresponding articles for examples.

The series 0F0

As noted earlier, . The differential equation for this function is , which has solutions where k is a constant.

The series 0F1

The functions of the form are called confluent hypergeometric limit functions and are closely related to Bessel functions.

The relationship is:

The differential equation for this function is

or

When a is not a positive integer, the substitution

gives a linearly independent solution

so the general solution is

where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)

A special case is:

The series 1F0

An important case is:

The differential equation for this function is

or

which has solutions

where k is a constant.

is the geometric series with ratio z and coefficient 1.
is also useful.

The series 1F1

The functions of the form are called confluent hypergeometric functions of the first kind, also written . The incomplete gamma function is a special case.

The differential equation for this function is

or

When b is not a positive integer, the substitution

gives a linearly independent solution

so the general solution is

where k, l are constants.

When a is a non-positive integer, −n, is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of 1F1 as well.

The series 1F2

Relations to other functions are known for certain parameter combinations only.

The function is the antiderivative of the cardinal sine. With modified values of and , one obtains the antiderivative of . [8]

The Lommel function is . [9]

The series 2F0

The confluent hypergeometric function of the second kind can be written as: [10]

The series 2F1

Historically, the most important are the functions of the form . 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 pFq if there is risk of confusion. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence.

The differential equation for this function is

or

It is known as the hypergeometric differential equation. When c is not a positive integer, the substitution

gives a linearly independent solution

so the general solution for |z| < 1 is

where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, known as the Kummer solutions, derivable using various identities, valid in different regions of the complex plane.

When a is a non-positive integer, −n,

is a polynomial. Up to constant factors and scaling, these are the Jacobi polynomials. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2F1 as well. This includes Legendre polynomials and Chebyshev polynomials.

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

The series 3F0

The Mott polynomials can be written as: [11]

The series 3F2

The function

is the dilogarithm [12]

The function

is a Hahn polynomial.

The series 4F3

The function

is a Wilson polynomial.

All roots of a quintic equation can be expressed in terms of radicals and the Bring radical, which is the real solution to . The Bring radical can be written as: [13]

The series q+1Fq

The functions

for and are the Polylogarithm.

For each integer n≥2, the roots of the polynomial xnx+t can be expressed as a sum of at most N−1 hypergeometric functions of type n+1Fn, which can always be reduced by eliminating at least one pair of a and b parameters. [13]

Generalizations

The generalized hypergeometric function is linked to the Meijer G-function and the MacRobert E-function. Hypergeometric series were generalised to several variables, for example by Paul Emile Appell and Joseph Kampé de Fériet; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the q-series analogues, called the basic hypergeometric series, were given by Eduard Heine in the late nineteenth century. Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function of qn. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n.

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of general hypergeometric functions, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes).

Special hypergeometric functions occur as zonal spherical functions on Riemannian symmetric spaces and semi-simple Lie groups. Their importance and role can be understood through the following example: the hypergeometric series 2F1 has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO(3). In tensor product decompositions of concrete representations of this group Clebsch–Gordan coefficients are met, which can be written as 3F2 hypergeometric series.

Bilateral hypergeometric series are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.

Fox–Wright functions are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.

See also

Notes

  1. Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1990). Integrals & Series Volume 3: More Special Functions. Gordon and Breach. p. 439.
  2. Karlsson, Per W. (1970). "Hypergeometric functions with integral parameter differences". J. Math. Phys. 12 (2): 270–271. doi:10.1063/1.1665587.
  3. ( Slater 1966 , Equation (4.1.2))
  4. Gottschalk, J. E.; Maslen, E. N. (1988). "Reduction formulae for generalised hypergeometric functions of one variable". J. Phys. A: Math. Gen. 21: 1983--1998. doi:10.1088/0305-4470/21/9/015.
  5. Rainville, D. (1945). "The contiguous function relations for pFq with application to Bateman's J and Rice's H". Bull. Amer. Math. Soc. 51 (10): 714--723. doi:10.1090/S0002-9904-1945-08425-0.
  6. See ( Slater 1966 , Section 2.3.1) or ( Bailey 1935 , Section 2.2) for a proof.
  7. See ( Bailey 1935 , Section 3.1) for a detailed proof. An alternative proof is in ( Slater 1966 , Section 2.3.3)
  8. Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)
  9. Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)
  10. "DLMF: §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions". dlmf.nist.gov.
  11. See Erdélyi et al. 1955.
  12. Candan, Cagatay. "A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x" (PDF).
  13. 1 2 Glasser, M. Lawrence (1994). "The quadratic formula made hard: A less radical approach to solving equations". arXiv: math.CA/9411224 .

Related Research Articles

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

<i>p</i>-adic number Number system extending the rational numbers

In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.

<span class="mw-page-title-main">Fourier series</span> Decomposition of periodic functions into sums of simpler sinusoidal forms

A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.

<span class="mw-page-title-main">Error function</span> Sigmoid shape special function

In mathematics, the error function, often denoted by erf, is a function defined as:

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.

<span class="mw-page-title-main">Hamiltonian mechanics</span> Formulation of classical mechanics using momenta

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

<span class="mw-page-title-main">Differential operator</span> Typically linear operator defined in terms of differentiation of functions

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz.

In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

<span class="mw-page-title-main">Confluent hypergeometric function</span> Solution of a confluent hypergeometric equation

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. The term confluent refers to the merging of singular points of families of differential equations; confluere is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions:

<span class="mw-page-title-main">Hypergeometric function</span> Function defined by a hypergeometric series

In mathematics, the Gaussian or ordinary hypergeometric function2F1(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.

In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.

<span class="mw-page-title-main">Bring radical</span> Real root of the polynomial x^5+x+a

In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial

In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are :

<span class="mw-page-title-main">Meijer G-function</span> Generalization of the hypergeometric function

In mathematics, the G-function was introduced by Cornelis Simon Meijer 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 line integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953.

In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction , where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction.

<span class="mw-page-title-main">Fox H-function</span> Generalization of the Meijer G-function and the Fox–Wright function

In mathematics, the Fox H-functionH(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

<span class="mw-page-title-main">Jacobi polynomials</span> Polynomial sequence

In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.

In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):

References