Hypergeometric series: Definition from Answers.com

In mathematics, a hypergeometric series, in the most general sense, is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, will define a hypergeometric function, which may then turn out to be defined over a wider domain of the argument by analytic continuation. Hypergeometric functions have many particular special functions as special cases, including many elementary functions, the Bessel functions, the incomplete gamma function, the error function, the elliptic integrals and the classical orthogonal polynomials. This phenomenon is in part because the hypergeometric functions are solutions to the hypergeometric differential equation, which is a fairly general second-order ordinary differential equation. The term hypergeometric series also refers to a specific type of these series, also known as Gauss’s series (Carl Friedrich Gauss), which were the object of a great deal of interest in the 19th century. One application of hypergeometric series is the inversion of elliptic integrals; these are constructed by taking the ratio of the two linearly independent solutions of the hypergeometric differential equation to form Schwarz–Christoffel maps of the fundamental domain to the complex projective line or Riemann sphere. Another application is Gauss's continued fraction, which can be used to produce continued fraction expansions of many elementary and special functions.

History

Gauss’s series were studied by Euler, but the first full systematic treatment is found in Gauss' seminal paper of 1812, Disquisitiones Generales Circa Seriem Infinitam $1+\frac{\alpha\beta}{\gamma.1}+\mbox{etc.}$ The term "hypergeometric series" is due to Pfaff^[1].

Studies in the nineteenth century included those of Ernst Kummer, and the fundamental characterisation by Bernhard Riemann of the F-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: that effectively the entire algorithmic side of the theory was a consequence of basic facts and the use of Möbius transformations as a symmetry group.

The cases where the solutions are algebraic functions were found by H. A. Schwarz (Schwarz's list).

Notation

A hypergeometric series is formally defined as a power series

$\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_n \beta_n z^n$

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

$\frac{\beta_{n+1}}{\beta_n} = \frac{A(n)}{B(n)}$

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

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

$1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\dots$ ,

$\beta_n = \frac{1}{n!}$ and $\frac{\beta_{n+1}}{\beta_n} = \frac{1}{n+1}$ . So this satisfies the definition with A(n) = 1 and B(n) = n+1.

It is customary to factor out the leading term, so $β 0$ is assumed to be 1. The polynomials can be factored into linear factors of the form $(a j + n)$ and $(b k + n)$ respectively, where the $a j$ and $b k$ are complex numbers.

For historical and practical 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

$\frac{c(a_1+n)\dots(a_p+n)}{d(b_1+n)\dots(b_q+n)(1+n)}$ ,

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

$1 + \frac{a_1\dots a_p}{b_1\dots b_q.1}\frac{cz}{d} + \frac{a_1\dots a_p}{b_1\dots b_q.1} \frac{(a_1+1)\dots(a_p+1)}{(b_1+1)\dots (b_q+1).2}\left(\frac{cz}{d}\right)^2+\dots$ ,

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

$1 + \frac{a_1\dots a_p}{b_1\dots b_q}\frac{z}{1!} + \frac{a_1(a_1+1)\dots a_p(a_p+1)}{b_1(b_1+1)\dots b_q(b_q+1)}\frac{z^2}{2!}+\dots$ .

This has the form of an exponential generating function. The standard notation for this series is

$\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)$ ,

although variations are sometimes used^[2].

Using the rising factorial or Pochhammer symbol:

$(a)_n=a(a+1)(a+2)...(a+n-1),\,(a)_0 = 1$ ,

this can be written

$\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n} \, \frac {z^n} {n!}$

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

Examples

The simplest example is the series for the exponential function: $\,_0F_0(;;z) = e^z$ .

As another example, start with the series for $log(1 + z)$ :

$z-\frac{z^2}{2}+\frac{z^3}{3}\dots$ .

Factoring out the first term, the series becomes

$1-\frac{z}{2}+\frac{z^2}{3}\dots$ ,

in which the nth coefficient is $\beta_n=(-1)^n\frac{1}{n+1}$ . The ratio of consecutive coefficients is then

$\frac{\beta_{n+1}}{\beta_n}=-\frac{n+1}{n+2}$ .

Since (n+1) is not a factor of the denominator, multiply both the numerator and the denominator by this factor to get

$-\frac{(n+1)(n+1)}{(n+2)(n+1)}$ .

This produces the expression

$\log(1+z)=z\,_2F_1(1,1;2;-z)$ .

Similarly, several other familiar elementary functions can be expressed as hypergeometric series. These include^[3]

$\begin{align} (1-z)^k & = \,_1F_0\left(-k;;z\right)\\[3pt] \rm{arctanh}\,z = \tfrac{1}{2}\log\left((1+z)/(1-z)\right) & = z\,_2F_1\left({\scriptstyle\frac{1}{2}},1; {\scriptstyle\frac{3}{2}};z^2\right)\\[3pt] \arcsin z & = z \,_2F_1\left({\scriptstyle\frac{1}{2}},{\scriptstyle \frac{1}{2}}; {\scriptstyle\frac{3}{2}};z^2\right)\\[3pt] \arctan z & = z \,_2F_1\left({\scriptstyle\frac{1}{2}},1; {\scriptstyle\frac{3}{2}};-z^2\right)\, \end{align}$

Many other special cases are listed in the Category:Special hypergeometric functions.

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

$\Gamma(a,z) \sim z^{a-1}e^{-z}\left(1+\frac{a-1}{z}+\frac{(a-1)(a-2)}{z^2}\dots\right)$

which could be written $z^{a-1}e^{-z}\,_2F_0(1-a,1;;-1/z)$ . 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 $a j$ and $b k$ for which the numerator or the denominator of the coefficients is 0.

If any $a j$ is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree $- a j$ .
If any $b k$ is a non-positive integer (excepting the previous case with $- b k < - a j$ ) then the denominators become 0 and the series is undefined.

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

If p=q+1 then the ratio of the coefficients approaches 1. This implies that the radius of convergence is 1.
If p≤q then the ratio of the coefficients approaches 0. This implies that the radius of convergence is infinity.
If p>q+1 then the ratio of the coefficients approaches infinity. This implies that the radius of convergence is 0 and the series does not define an analytic function.

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

$\Re\left(\sum b_k - \sum a_j\right)>0$ .

Basic properties

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

$\,_2F_1(3,1;1;z) = \,_2F_1(1,3;1;z) = \,_1F_0(3;;z)$ .

Differentiation

Term by term differentiation gives

$\frac{ {\rm{d}} }{{\rm{d}}z}\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \frac{ \prod_{n=1}^{p}a_n }{\prod_{n=1}^{q}b_n}\,_pF_q(a_1+1,\dots,a_p+1;b_1+1,\dots,b_q+1;z)$ ,

and repeating this gives

$\frac{{\rm{d}}^n}{{\rm{d}}z^n}\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \frac{ \prod_{k=1}^{p}(a_k)_n }{\prod_{k=1}^{q}(b_k)_n}\,_pF_q(a_1+n,\dots,a_p+n;b_1+n,\dots,b_q+n;z)$ .

A slightly different term by term differentiation gives

$\frac{{\rm{d}}}{{\rm{d}}z}z^\alpha\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \alpha z^{\alpha-1}\,_{p+1}F_{q+1}(a_1,\dots,a_p,\alpha+1;b_1,\dots,b_q,\alpha;z)\quad(\alpha \ne 0, -1, -2 \dots)$ .

Applying the product rule and simplifying produces

$(z\frac{{\rm{d}}}{{\rm{d}}z} + \alpha)\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \alpha\,_{p+1}F_{q+1}(a_1,\dots,a_p,\alpha+1;b_1,\dots,b_q,\alpha;z)\quad(\alpha \ne 0, -1, -2 \dots)$

Putting $\alpha = a_j\,$ and using the cancellation property,

$(z\frac{{\rm{d}}}{{\rm{d}}z} + a_j)\,_pF_q(a_1,\dots,a_j,\dots,a_p;b_1,\dots,b_q;z) = a_j\,_pF_q(a_1,\dots,a_j+1,\dots,a_p;b_1,\dots,b_q;z)$ .

Applying this p times gives

$\prod_{n=1}^{p}(z\frac{{\rm{d}}}{{\rm{d}}z} + a_n)\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) =\prod_{n=1}^{p}a_n\,_pF_q(a_1+1,\dots,a_p+1;b_1,\dots,b_q;z)$ .

Similarly,

$(z\frac{{\rm{d}}}{{\rm{d}}z} + b_k - 1)\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_k,\dots,b_q;z) = (b_k - 1)\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_k-1,\dots,b_q;z)$ .

Applying this q times to the derivative gives

$\prod_{n=1}^{q} (z\frac{{\rm{d}}}{{\rm{d}}z} + b_n)\frac{{{\rm{d}}}}{ {\rm{d}}z}\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) = \prod_{n=1}^{p}a_n\,_pF_q(a_1,\dots,a_p;b_1+1,\dots,b_q+1;z)$ .

Comparing these gives a differential equation for the function $w = \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ defined by the series,

$\prod_{n=1}^{p}(z\frac{{\rm{d}}}{{\rm{d}}z} + a_n)w = \prod_{n=1}^{q}(z\frac{{\rm{d}}}{{\rm{d}}z} + b_n)w$ .

Integration

Reversing the differentiation formulas above gives integration formulas

$\int\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) {\rm{d}} z=\frac{\Pi_{n=1}^q(b_n-1)}{{\Pi_{n=1}^p(a_n-1)}}\,_pF_q(a_1-1,\dots,a_p-1;b_1-1,\dots,b_q-1;z)+C\quad(a_j, b_k \ne 1)$ .

$\int z^\alpha\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) {\rm{d}}z= \frac{z^{\alpha+1}}{\alpha+1}\,_{p+1}F_{q+1}(a_1,\dots,a_p,\alpha+1;b_1,\dots,b_q,\alpha+2;z)+C\quad(\alpha \ne -1).$

Using integration by substitution, this implies

$\int \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z^\alpha) {\rm{d}}z= z\,_{p+1}F_{q+1}(a_1,\dots,a_p,\tfrac{1}{\alpha};b_1,\dots,b_q,\tfrac{1}{\alpha}+1;z^\alpha)+C\quad(\alpha > 0).$

For example

$\log(1-x) = -\int_0^x \frac{{\rm{d}}z}{1-z} = -\int_0^x{}_1F_0(1;;z)dz = -\frac{z}{1}{}_2F_1(1,1;2;z) \Big|_0^x = -x\,_2F_1(1,1;2;x)$ .

$\arcsin x = \int_0^x \frac{1}{\sqrt{1-z^2}} {\rm{d}}z = \int_0^x{} {}_1F_0(\tfrac{1}{2};;z^2)dz = z {}_2F_1(\tfrac{1}{2}, \tfrac{1}{2};\tfrac{3}{2};z^2)\Big|_0^x = x {}_2F_1(\tfrac{1}{2}, \tfrac{1}{2};\tfrac{3}{2};x^2).$

Even and odd components

The ratio of coefficients of a series obtained by taking every other term of a hypergeometric series is also rational. Expanding these out according to the process given above produces for the even terms

$\frac{1}{2}\left[\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) + \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;-z)\right]$

$= \,_{2p}F_{2q+1}\left(\frac{a_1}{2},\frac{a_1+1}{2},\dots,\frac{a_p}{2},\frac{a_p+1}{2}; \frac{b_1}{2},\frac{b_1+1}{2},\dots,\frac{b_q}{2},\frac{b_q+1}{2},\frac{1}{2};\frac{z^2}{4^{q+1-p}}\right)$

and for odd terms

$\frac{1}{2}\left[\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z) - \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;-z)\right]$

$= z\frac{\Pi_{n=1}^pa_n}{\Pi_{n=1}^qb_n} \,_{2p}F_{2q+1}\left(\frac{a_1+1}{2},\frac{a_1+2}{2},\dots,\frac{a_p+1}{2},\frac{a_p+2}{2}; \frac{b_1+1}{2},\frac{b_1+2}{2},\dots,\frac{b_q+1}{2},\frac{b_q+2}{2},\frac{3}{2};\frac{z^2}{4^{q+1-p}}\right).$

For example

$\cosh z = \frac{e^z+e^{-z}}{2} = {}_0F_1(;\frac{1}{2};\tfrac{z^2}{4})$ ,

$\sinh z = \frac{e^z-e^{-z}}{2} = z\,_0F_1(;\frac{3}{2};\tfrac{z^2}{4})$ ,

$\frac{1}{2(1-z)^a}+\frac{1}{2(1+z)^a}= {}_2F_1\left(\frac{a}{2},\frac{a+1}{2};\frac{1}{2};z^2\right)$

$\frac{1}{2(1-z)^a}-\frac{1}{2(1+z)^a}= az\,_2F_1\left(\frac{a+1}{2},\frac{a+2}{2};\frac{3}{2};z^2\right)$

${\rm{arctanh}} z = \frac{1}{2}\left[\log(1+z)-\log(1-z)\right] = \frac{1}{2}z\left[\,_2F_1(1,1;2;z)+\,_2F_1(1,1;2;-z)\right]$

$=z\,_4F_3\left(\frac{1}{2},1,\frac{1}{2},1;1,\frac{3}{2},\frac{1}{2};z^2\right)=z\,_2F_1\left(\frac{1}{2},1;\frac{3}{2};z^2\right).$

Contiguous function and related identities

Let $\vartheta$ be the operator $z\frac{d}{dz}$ . From the differentiation formulas given above, the linear space spanned by $\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ and $\vartheta\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ contains each of

$\,_pF_q(a_1,\dots,a_j+1,\dots,a_p;b_1,\dots,b_q;z), \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_k-1,\dots,b_q;z)$ ,

$z\,_pF_q(a_1+1,\dots,a_p+1;b_1+1,\dots,b_q+1;z), \,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ .

Since the space has dimension 2, any three of these $p + q + 2$ functions are linearly dependent. These dependencies can be written out to generate a large number of identities involving $p F q$ .

For example, in the simplest non-trivial case,

$\,_0F_1(;a;z) = (1)\,_0F_1(;a;z)$ ,

$\,_0F_1(;a-1;z) = (\frac{\vartheta}{a-1}+1)\,_0F_1(;a;z)$ ,

$z\,_0F_1(;a+1;z) = (a\vartheta)\,_0F_1(;a;z)$ ,

$\,_0F_1(;a-1;z)-\,_0F_1(;a;z) = \frac{z}{a(a-1)}\,_0F_1(;a+1;z)$ .

This, and other important examples,

$\,_1F_1(a+1;b;z)-\,_1F_1(a;b;z) = \frac{z}{b}\,_1F_1(a+1;b+1;z)$ ,

$\,_1F_1(a;b-1;z)-\,_1F_1(a;b;z) = \frac{az}{b(b-1)}\,_1F_1(a+1;b+1;z)$ ,

$\,_1F_1(a;b-1;z)-\,_1F_1(a+1;b;z) = \frac{(a-b+1)z}{b(b-1)}\,_1F_1(a+1;b+1;z)$

$\,_2F_1(a+1,b;c;z)-\,_2F_1(a,b;c;z) = \frac{bz}{c}\,_2F_1(a+1,b+1;c+1;z)$ ,

$\,_2F_1(a+1,b;c;z)-\,_2F_1(a,b+1;c;z) = \frac{(b-a)z}{c}\,_2F_1(a+1,b+1;c+1;z)$ ,

$\,_2F_1(a,b;c-1;z)-\,_2F_1(a+1,b;c;z) = \frac{(a-c+1)bz}{c(c-1)}\,_2F_1(a+1,b+1;c+1;z)$ ,

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

Similarly, by applying the differentiation formulas twice, there are $\binom{p+q+3}{2}$ such functions contained in $\{1, \vartheta, \vartheta^2\}\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ , 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 $\pm 1$ to exactly one of the parameters $a_j,\ b_k$ in $\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ is called contiguous to $\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;z)$ . Using the technique outlined above, an identity relating $\,_0F_1(;a;z)$ and its two contiguous functions can be given, six identities relating $\,_1F_1(a;b;z)$ and any two of its four contiguous functions, and fifteen identities relating $\,_2F_1(a,b;c;z)$ 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.)

Derived is the identity

$_2F_2(a,b;c,d;x)= \sum_{i=0} \frac{{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}} \, _1F_1(a+i;c+i;x)\frac{x^i}{i!},$

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

Integral formulas

Euler type

Expanding

$I = \int_0^1 x^{\alpha-1} (1-x)^{\beta-1}\,_pF_q(a_1,\dots,a_p;b_1,\dots,b_q;zx)dx \quad \real(\alpha), \real(\beta) > 0$

produces a series where the coefficient of $z n$ is

$\beta_n \int_0^1 x^{\alpha-1+n} (1-x)^{\beta-1} dx = \beta_n \Beta(\alpha+n,\beta)$

Where $Β$ is the Beta function. It can be shown that

$\frac{\Beta(\alpha+n+1,\beta)}{\Beta(\alpha+n,\beta)} = \frac{\alpha+n}{\alpha+\beta+n}$ .

$I = \Beta(\alpha,\beta)\,_{p+1}F_{q+1}(\alpha, a_1,\dots,a_p;\alpha+\beta, b_1,\dots,b_q;z)$ .

As special cases:

$\int_0^1 x^{\alpha-1} (1-x)^{\beta-1}e^{zx} dx = \Beta(\alpha,\beta)\,_1F_1(\alpha;\alpha+\beta;z) \quad \real(\alpha), \real(\beta) > 0$ .

$\Beta(a,b-a)\,_1F_1(a;b;z) = \int_0^1 x^{a-1} (1-x)^{b-a-1}e^{zx} dx \quad \real(b) > \real(a) > 0$ .

$\int_0^1 x^{\alpha-1} (1-x)^{\beta-1}(1-zx)^{-a} dx = \Beta(\alpha,\beta)\,_2F_1(\alpha, a;\alpha+\beta;z) \quad \real(\alpha), \real(\beta) > 0$ ,

$\Beta(b,c-b)\,_2F_1(a,b;c;z) = \int_0^1 x^{b-1} (1-x)^{c-b-1}(1-zx)^{-a} dx \quad \real(c) > \real(b) > 0$ ,

provided $| z | < 1$ or $| z | = 1$ and both sides converge. This was given by Euler in 1748 and is the basis of Euler's Hypergeometric Transformations.

Putting z = 1 in the last equation gives

$\,_2F_1(a,b;c;1) = \frac{\Beta(b,c-b-a)}{\Beta(b,c-b)} = \frac{\Gamma(c)\Gamma(c-b-a)}{\Gamma(c-a)\Gamma(c-b)}$ ,

where $Γ$ is the Gamma function.

Barnes' type

The theory of residues can be used to evaluate the contour integral

$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)} (-z)^s ds$

$\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,_2F_1(a,b;c;z),$

where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a-1, ..., −b, −b−1, ... . There are several variations on this idea and they can be used to prove certain identities.

Identities

A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries.

Special cases

The series ₀F₀

As noted earlier, $\,_0F_0(;;z) = e^z$ . The differential equation for this function is $\frac{d}{dz}w = w$ , which has solutions $w = k e z$ where k is a constant.

The series ₁F₀

Also as noted earlier, $\,_1F_0(a;;z) = (1-z)^{-a}$ . The differential equation for this function is $\frac{d}{dz}w = (z\frac{d}{dz}+a)w$ , or $(1-z)\frac{dw}{dz} = aw$ , which has solutions $w = k (1 - z) - a$ where k is a constant.

The series ₀F₁

The functions of the form $\,_0F_1(;a;z)$ are called Confluent Hypergeometric Limit Functions and are closely related to Bessel functions. The differential equation for this function is $w = (z\frac{d}{dz}+a)\frac{dw}{dz}$ or $z\frac{d^2w}{dz^2}+a\frac{dw}{dz}-w = 0$ . When a is not a positive integer, the substitution $w = z 1 - a u$ , gives a linearly independent solution $z^{1-a}\,_0F_1(;2-a;z)$ , so the general solution is $k\,_0F_1(;a;z)+l z^{1-a}\,_0F_1(;2-a;z)$ where k, l are constants.

The series ₁F₁

Main article: Confluent hypergeometric function

The functions of the form $\,_1F_1(a;b;z)$ are called Confluent Hypergeometric Functions of the First Kind, also written $M (a; b; z)$ . The incomplete gamma function $γ(a, z)$ is a special case.

The differential equation for this function is $(z\frac{d}{dz}+a)w = (z\frac{d}{dz}+b)\frac{dw}{dz}$ or $z\frac{d^2w}{dz^2}+(b-z)\frac{dw}{dz}-aw = 0$ . When b is not a positive integer, the substitution $w = z 1 - b u$ , gives a linearly independent solution $z^{1-b}\,_1F_1(1+a-b;2-b;z)$ , so the general solution is $k\,_1F_1(a;b;z)+l z^{1-b}\,_1F_1(1+a-b;2-b;z)$ where k, l are constants.

When a is a non-positive integer, −n, $\,_1F_1(-n;b;z)$ is a polynomial. Up to constant factors, these are the Laguerre polynomials. This implies Hermite polynomials can be expressed in terms of $1 F 1$ as well.

The series ₂F₁

Historically, the most important are the functions of the form $\,_2F_1(a,b;c;z)$ . These are sometimes called Gauss' hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function is used for the functions $\,_pF_q$ 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 $(z\frac{d}{dz}+a)(z\frac{d}{dz}+b)w = (z\frac{d}{dz}+c)\frac{dw}{dz}$ or $z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0$ . This is known as the hypergeometric differential equation. When c is not a positive integer, the substitution $w = z 1 - c u$ , gives a linearly independent solution $z^{1-c}\,_2F_1(1+a-c,1+b-c;2-c;z)$ , so the general solution for $| z | < 1$ is $k\,_2F_1(a,b;c;z)+l z^{1-c}\,_2F_1(1+a-c,1+b-c;2-c;z)$ 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, $\,_2F_1(-n,b;c;z)$ 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 $2 F 1$ 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.:

$\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y\ =\ \frac{x\left(\alpha\,{}_2F_1\left(\frac{1}{\alpha},\frac{1}{2};1+\frac{1}{\alpha};-x^\alpha\right)+2\sqrt{x^\alpha+1}\,\right)}{2+\alpha}\ ,\ \ \ \alpha\neq0.$

Generalizations

Hypergeometric series were generalised to several variables, for example by Paul Emile Appell; 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 ratio of successive terms, instead of being a rational function of n, are considered to be a rational function of $q n$ . 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 hypergeometric series, 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.

Notes

^ D.E. Smith History of Mathematics Vol. II (Dover 1958) p. 507
^ Weisstein, Eric W., "Generalized Hypergeometric Function" from MathWorld.
^ Abramowitz and Stegun (1972); Wall (1948).

References

Gauss's original paper can be found in Carl Friedrich Gauss Werke, p.125
W. N. Bailey Generalized Hypergeometric Series Cambridge 1935
MathWorld
- Weisstein, Eric W., "Generalized Hypergeometric Function" from MathWorld.
- Weisstein, Eric W., "Hypergeometric Function" from MathWorld.
- Weisstein, Eric W., "Confluent Hypergeometric Limit Function" from MathWorld.
- Weisstein, Eric W., "Confluent Hypergeometric Function of the First Kind" from MathWorld.
- Weisstein, Eric W., "Hypergeometric Function" from MathWorld.

Milton Abramowitz and Irene A. Stegun, eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4.
- Chapter 13. - $\,_1F_1$
- Chapter 15. - $\,_2F_1$
Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications. 71. Cambridge University Press. MR 1688958. ISBN 978-0-521-62321-6; 978-0-521-78988-2.

Gerrit Heckman and Henrik Schlichtkrull (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. Academic Press, San Diego. ISBN 0-12-336170-2. (Part 1 treats hypergeometric functions on Lie groups.)
Slater, Lucy Joan (1960). Confluent hypergeometric functions. Cambridge University Press. MR 0107026.
Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge University Press. MR 0201688. ISBN 978-0-521-09061-2.
H. S. Wall (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc..
E. T. Whittaker and G. N. Watson (1927). A Course of Modern Analysis. Cambridge University Press.
Masaaki Yoshida (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Friedrick Vieweg & Son. ISBN 3-528-06925-2.

External links

The book "A = B", this book is freely downloadable from the internet.

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Hypergeometric series

Contents

History

Notation

Examples

Terminology

Convergence conditions

Basic properties

Differentiation

Integration

Even and odd components

Contiguous function and related identities

Integral formulas

Euler type

Barnes' type

Identities

Special cases

The series ₀F₀

The series ₁F₀

The series ₀F₁

The series ₁F₁

The series ₂F₁

Generalizations

See also

Notes

References

External links

	Weber differential equation (mathematics)
	Whittaker differential equation (mathematics)
	Srinivasa Ramanujan Aiyangar

	When I plot 5 data series say Series 1 Series 2....Series 5 in Excel and I want the legend to only show Series 1-3 how do I do it? Read answer...
	Where can you find the series? Read answer...
	What is the Italian serie A? Read answer...

	What are the properties of hypergeometric distribution?
	What is relation between binomial multinomial hypergeometric and poisson probability?
	Find the relation ship between confluent hypergeometric function and laguerre polynomials?

		Sci-Tech Dictionary. McGraw-Hill Dictionary of Scientific and Technical Terms. Copyright © 2003, 1994, 1989, 1984, 1978, 1976, 1974 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Hypergeometric series". Read more

Hypergeometric series

Contents

History

Notation

Examples

Terminology

Convergence conditions

Basic properties

Differentiation

Integration

Even and odd components

Contiguous function and related identities

Integral formulas

Euler type

Barnes' type

Identities

Special cases

The series 0F0

The series 1F0

The series 0F1

The series 1F1

The series 2F1

Generalizations

See also

Notes

References

External links

The series ₀F₀

The series ₁F₀

The series ₀F₁

The series ₁F₁

The series ₂F₁