Английская Википедия:Hypergeometric function

Plot of the hypergeometric function 2F1(a,b; c; z) with a=2 and b=3 and c=4 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

Шаблон:Use American English

In mathematics, the Gaussian or ordinary hypergeometric function ₂F₁(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation.

For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Шаблон:Harvtxt and Шаблон:Harvtxt. There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.

History

The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum.

Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Шаблон:Harvs.

Studies in the nineteenth century included those of Шаблон:Harvs, and the fundamental characterisation by Шаблон:Harvs of the hypergeometric function by means of the differential equation it satisfies.

Riemann showed that the second-order differential equation for ₂F₁(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities.

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

The hypergeometric series

The hypergeometric function is defined for Шаблон:Math by the power series

<math display=block>{}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} = 1 + \frac{ab}{c}\frac{z}{1!} + \frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!} + \cdots.</math>

It is undefined (or infinite) if Шаблон:Mvar equals a non-positive integer. Here Шаблон:Math is the (rising) Pochhammer symbol, which is defined by:

<math display=block>(q)_n = \begin{cases} 1 & n = 0 \\

 q(q+1) \cdots (q+n-1) & n > 0
\end{cases}</math>

The series terminates if either Шаблон:Mvar or Шаблон:Mvar is a nonpositive integer, in which case the function reduces to a polynomial:

<math display=block>{}_2F_1(-m,b;c;z) = \sum_{n=0}^m (-1)^n \binom{m}{n} \frac{(b)_n}{(c)_n} z^n.</math>

For complex arguments Шаблон:Mvar with Шаблон:Math it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line Шаблон:Math.

As Шаблон:Math, where Шаблон:Mvar is a non-negative integer, one has Шаблон:Math. Dividing by the value Шаблон:Math of the gamma function, we have the limit:

<math display=block>\lim_{c\to -m}\frac{{}_2F_1(a,b;c;z)}{\Gamma(c)}=\frac{(a)_{m+1}(b)_{m+1}}{(m+1)!}z^{m+1}{}_2F_1(a+m+1,b+m+1;m+2;z)</math>

Шаблон:Math is the most common type of generalized hypergeometric series Шаблон:Mvar, and is often designated simply Шаблон:Math.

Differentiation formulas

Using the identity <math> (a)_{n+1}=a (a+1)_n</math>, it is shown that

and more generally,

Special cases

Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are

<math display=block>\begin{align} _2F_1\left(1, 1; 2; -z\right) &= \frac{\ln(1+z)}{z} \\ _2F_1(a, b; b; z) &= (1-z)^{-a} \quad (b \text{ arbitrary}) \\ _2F_1\left(\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; z^2\right) &= \frac{\arcsin(z)}{z} \\ \,_2F_1\left(\frac{1}{3}, \frac{2}{3}; \frac{3}{2}; -\frac{27x^2}{4}\right) &= \frac{\sqrt[3]{\frac{3x\sqrt{3}+\sqrt{27x^2+4}}{2}}-\sqrt[3]{\frac{2}{3x\sqrt{3}+\sqrt{27x^2+4}}}}{x\sqrt{3}} \\ \end{align}</math>

When a=1 and b=c, the series reduces into a plain geometric series, i.e.

<math display=block>\begin{align} _2F_1\left(1, b; b; z\right) &= 1 + z + z^2 + z^3 + z^4 + \cdots \end{align}</math>

hence, the name hypergeometric. This function can be considered as a generalization of the geometric series.

The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function

<math display=block>M(a,c,z) = \lim_{b\to\infty}{}_2F_1(a,b;c;b^{-1}z)</math>

so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.

Legendre functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example

<math display=block>{}_2F_1(a,1-a;c;z) = \Gamma(c)z^{\tfrac{1-c}{2}}(1-z)^{\tfrac{c-1}{2}}P_{-a}^{1-c}(1-2z)</math>

Several orthogonal polynomials, including Jacobi polynomials PШаблон:Su and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using

<math display=block>{}_2F_1(-n,\alpha+1+\beta+n;\alpha+1;x) = \frac{n!}{(\alpha+1)_n}P^{(\alpha,\beta)}_n(1-2x)</math>

Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.

Given <math>z\in\mathbb{C}\setminus\{0,1\}</math>, let

Then

<math display=block>\lambda (\tau) = \frac{\theta_2(\tau)^4}{\theta_3(\tau)^4}=z</math>

is the modular lambda function, where

<math display=block>\theta_2(\tau)=\sum_{n\in\mathbb{Z}}e^{\pi i\tau (n+1/2)^2},\quad \theta_3(\tau)=\sum_{n\in\mathbb{Z}}e^{\pi i\tau n^2}.</math>

The j-invariant, a modular function, is a rational function in <math>\lambda (\tau)</math>.

Incomplete beta functions B_x(p,q) are related by

<math display=block> B_x(p,q) = \tfrac{x^p}{p}{}_2F_1(p,1-q;p+1;x).</math>

The complete elliptic integrals K and E are given by

<math display=block>\begin{align} K(k) &= \tfrac{\pi}{2}\, _2F_1\left(\tfrac{1}{2},\tfrac{1}{2};1;k^2\right), \\ E(k) &= \tfrac{\pi}{2}\, _2F_1\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^2\right). \end{align}</math>

The hypergeometric differential equation

The hypergeometric function is a solution of Euler's hypergeometric differential equation

<math display=block>z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0.</math>

which has three regular singular points: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order linear differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.

Solutions at the singular points

Solutions to the hypergeometric differential equation are built out of the hypergeometric series ₂F₁(a,b;c;z). The equation has two linearly independent solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form x^s times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at a regular singular point. This gives 3 × 2 = 6 special solutions, as follows.

Around the point z = 0, two independent solutions are, if c is not a non-positive integer,

and, on condition that c is not an integer,

If c is a non-positive integer 1−m, then the first of these solutions does not exist and must be replaced by <math>z^mF(a+m,b+m;1+m;z).</math> The second solution does not exist when c is an integer greater than 1, and is equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a more complicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plus another series in powers of z, involving the digamma function. See Шаблон:Harvtxt for details.

Around z = 1, if c − a − b is not an integer, one has two independent solutions

and

Around z = ∞, if a − b is not an integer, one has two independent solutions

<math display=block> z^{-a}\, _2F_1 \left (a,1+a-c;1+a-b; z^{-1} \right)</math>

and

<math display=block> z^{-b}\, _2F_1 \left (b,1+b-c;1+b-a; z^{-1} \right ).</math>

Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated.

Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (Шаблон:Su) = 20 linear relations between them called connection formulas.

Kummer's 24 solutions

A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group W(D_n) of order 2ⁿ⁻¹n!. The hypergeometric equation is the case n = 3, with group of order 24 isomorphic to the symmetric group on 4 points, as first described by Kummer. The appearance of the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a Klein 4-group (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a,b;c;z) to one of

<math display=block>\begin{align} (1-z)^{-a} F \left (a,c-b;c; \tfrac{z}{z-1} \right ) \\ F(a,b;1+a+b-c;1-z) \\ (1-z)^{-b} F \left(c-a,b;c; \tfrac{z}{z-1} \right ) \end{align}</math>

which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independent solution to the differential equation.)

Applying Kummer's 24 = 6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities

<math display=block>\begin{align} {}_2F_1(a,b;c;z) &= (1-z)^{c-a-b} \, {}_2F_1(c-a,c-b;c;z) && \text{Euler transformation} \\ {}_2F_1(a,b;c;z) &= (1-z)^{-a} \, {}_2F_1(a,c-b;c; \tfrac{z}{z-1}) && \text{Pfaff transformation} \\ {}_2F_1(a,b;c;z) &= (1-z)^{-b} \, {}_2F_1(c-a,b;c; \tfrac{z}{z-1}) && \text{Pfaff transformation} \end{align}</math>

Q-form

The hypergeometric differential equation may be brought into the Q-form

by making the substitution u = wv and eliminating the first-derivative term. One finds that

and v is given by the solution to

which is

The Q-form is significant in its relation to the Schwarzian derivative Шаблон:Harv.

Schwarz triangle maps

Шаблон:Main The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.

where k is one of the points 0, 1, ∞. The notation

<math display=block>D_k(\lambda,\mu,\nu;z)=s_k(z)</math>

is also sometimes used. Note that the connection coefficients become Möbius transformations on the triangle maps.

Note that each triangle map is regular at z ∈ {0, 1, ∞} respectively, with

<math display=block>\begin{align} s_0(z) &= z^\lambda (1+\mathcal{O}(z)) \\ s_1(z) &= (1-z)^\mu (1+\mathcal{O}(1-z)) \end{align}</math> and <math display=block>s_\infty(z)=z^\nu (1+\mathcal{O}(\tfrac{1}{z})).</math>

In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are conformal maps of the upper half-plane H to triangles on the Riemann sphere, bounded by circular arcs. This mapping is a generalization of the Schwarz–Christoffel mapping to triangles with circular arcs. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.

Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether λ + μ + ν – 1 is positive, zero or negative; and the s-maps are inverse functions of automorphic functions for the triangle group 〈p, q, r〉 = Δ(p, q, r).

Monodromy group

The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point. That is, when the path winds around a singularity of ₂F₁, the value of the solutions at the endpoint will differ from the starting point.

Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):

<math display=block>\pi_1(\mathbf{C}\setminus\{0,1\},z_0) \to \text{GL}(2,\mathbf{C})</math>

where π₁ is the fundamental group. In other words, the monodromy is a two dimensional linear representation of the fundamental group. The monodromy group of the equation is the image of this map, i.e. the group generated by the monodromy matrices. The monodromy representation of the fundamental group can be computed explicitly in terms of the exponents at the singular points.^[1] If (α, α'), (β, β') and (γ,γ') are the exponents at 0, 1 and ∞, then, taking z₀ near 0, the loops around 0 and 1 have monodromy matrices

<math display=block>\begin{align} g_0 &= \begin{pmatrix} e^{2\pi i\alpha} & 0\\ 0 & e^{2\pi i\alpha^\prime}\end{pmatrix} \\ g_1 &= \begin{pmatrix} {\mu e^{2\pi i \beta} -e^{2\pi i\beta^\prime}\over \mu -1} & {\mu (e^{2\pi i \beta} -e^{2\pi i\beta^\prime})\over (\mu -1)^2}\\e^{2\pi i\beta^\prime} - e^{2\pi i\beta} & {\mu e^{2\pi i \beta^\prime} -e^{2\pi i\beta}\over \mu -1}\end{pmatrix}, \end{align}</math>

where

<math display=block>\mu = {\sin \pi(\alpha +\beta^\prime +\gamma^\prime) \sin \pi(\alpha^\prime + \beta+\gamma^\prime)\over \sin \pi(\alpha^\prime + \beta^\prime +\gamma^\prime) \sin \pi(\alpha + \beta +\gamma^\prime)}.</math>

If 1−a, c−a−b, a−b are non-integer rational numbers with denominators k,l,m then the monodromy group is finite if and only if <math>1/k + 1/l + 1/m > 1</math>, see Schwarz's list or Kovacic's algorithm.

Integral formulas

Euler type

If B is the beta function then

<math display=block>\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 \qquad \real(c) > \real(b) > 0, </math>

provided that z is not a real number such that it is greater than or equal to 1. This can be proved by expanding (1 − zx)^−a using the binomial theorem and then integrating term by term for z with absolute value smaller than 1, and by analytic continuation elsewhere. When z is a real number greater than or equal to 1, analytic continuation must be used, because (1 − zx) is zero at some point in the support of the integral, so the value of the integral may be ill-defined. This was given by Euler in 1748 and implies Euler's and Pfaff's hypergeometric transformations.

Other representations, corresponding to other branches, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy action.

Barnes integral

Barnes used the theory of residues to evaluate the Barnes integral

<math display=block>\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</math>

as

<math display=block>\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)}\,_2F_1(a,b;c;z),</math>

where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a − 1, ..., −b, −b − 1, ... . This is valid as long as z is not a nonnegative real number.

John transform

The Gauss hypergeometric function can be written as a John transform Шаблон:Harv.

Gauss' contiguous relations

The six functions

are called contiguous to Шаблон:Math. Gauss showed that Шаблон:Math can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of Шаблон:Math, and Шаблон:Mvar. This gives

<math display=block> \begin{pmatrix} 6 \\ 2 \end{pmatrix} = 15</math>

relations, given by identifying any two lines on the right hand side of

<math display=block>\begin{align} z\frac{dF}{dz} &= z\frac{ab}{c}F(a+,b+,c+) \\ &=a(F(a+)-F) \\ &=b(F(b+)-F) \\ &=(c-1)(F(c-)-F) \\ &=\frac{(c-a)F(a-)+(a-c+bz)F}{1-z} \\ &=\frac{(c-b)F(b-)+(b-c+az)F}{1-z} \\ &=z\frac{(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)} \end{align}</math>

where Шаблон:Math, and so on. Repeatedly applying these relations gives a linear relation over Шаблон:Math between any three functions of the form

where m, n, and l are integers.

Gauss' continued fraction

Шаблон:Main

Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:

<math display=block>\frac{{}_2F_1(a+1,b;c+1;z)}{{}_2F_1(a,b;c;z)} = \cfrac{1}{1 + \cfrac{\frac{(a-c)b}{c(c+1)} z}{1 + \cfrac{\frac{(b-c-1)(a+1)}{(c+1)(c+2)} z}{1 + \cfrac{\frac{(a-c-1)(b+1)}{(c+2)(c+3)} z}{1 + \cfrac{\frac{(b-c-2)(a+2)}{(c+3)(c+4)} z}{1 + {}\ddots}}}}}</math>

Transformation formulas

Transformation formulas relate two hypergeometric functions at different values of the argument z.

Fractional linear transformations

Euler's transformation is <math display=block>{}_2F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2F_1 (c-a, c-b;c ; z).</math> It follows by combining the two Pfaff transformations <math display=block>\begin{align} {}_2F_1 (a,b;c;z) &= (1-z)^{-b} {}_2F_1 \left (b,c-a;c;\tfrac{z}{z-1} \right ) \\ {}_2F_1 (a,b;c;z) &= (1-z)^{-a} {}_2F_1 \left (a, c-b;c ; \tfrac{z}{z-1} \right ) \\ \end{align}</math> which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see Шаблон:Harvtxt and Шаблон:Harvtxt. It can also be written as linear combination <math display=block> \begin{align} {}_2F_1(a,b;c,z) = {} & \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_2F_1(a,b;a+b+1-c;1-z) \\[6pt] & {} + \frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b} {}_2F_1(c-a,c-b;1+c-a-b;1-z). \end{align} </math>

Quadratic transformations

If two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by Шаблон:Harvtxt, and a complete list was given by Шаблон:Harvtxt. A typical example is

<math display=block>{}_2F_1(a,b;2b;z) = (1-z)^{-\frac{a}{2}} {}_2F_1 \left (\tfrac{1}{2}a, b-\tfrac{1}{2}a; b+\tfrac{1}{2}; \frac{z^2}{4z-4} \right)</math>

Higher order transformations

If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by Шаблон:Harvtxt. A typical example is

<math display=block>{}_2F_1 \left (\tfrac{3}{2}a,\tfrac{1}{2}(3a-1);a+\tfrac{1}{2};-\tfrac{z^2}{3} \right) = (1+z)^{1-3a} \, {}_2F_1 \left (a-\tfrac{1}{3}, a; 2a; 2z(3+z^2)(1+z)^{-3} \right )</math>

There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers Шаблон:Harv. For example, <math display=block>{}_2F_1 \left (\tfrac{1}{4},\tfrac{3}{8};\tfrac{7}{8}; z \right) (z^4-60z^3+134z^2-60z+1)^{1/16} =

 {}_2F_1 \left (\tfrac{1}{48}, \tfrac{17}{48}; \tfrac{7}{8}; \tfrac{-432 z (z-1)^2 (z+1)^8}{(z^4-60z^3+134z^2-60z+1)^3} \right ).</math>

Values at special points z

See Шаблон:Harvtxt for a list of summation formulas at special points, most of which also appear in Шаблон:Harvtxt. Шаблон:Harvtxt gives further evaluations at more points. Шаблон:Harvtxt shows how most of these identities can be verified by computer algorithms.

Special values at z = 1

Gauss's summation theorem, named for Carl Friedrich Gauss, is the identity

<math display=block>{}_2F_1 (a,b;c;1)= \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}, \qquad \Re(c)>\Re(a+b) </math>

which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity as a special case.

For the special case where <math> a=-m </math>, <math display=block>{}_2F_1 (-m,b;c;1)=\frac{ (c-b)_{m} }{(c)_{m} } </math>

Dougall's formula generalizes this to the bilateral hypergeometric series at z = 1.

Kummer's theorem (z = −1)

There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformation to change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer:

<math display=block>{}_2F_1 (a,b;1+a-b;-1)= \frac{\Gamma(1+a-b)\Gamma(1+\tfrac12a)}{\Gamma(1+a)\Gamma(1+\tfrac12a-b)}</math>

which follows from Kummer's quadratic transformations

<math display=block>\begin{align} _2F_1(a,b;1+a-b;z)&= (1-z)^{-a} \;_2F_1 \left(\frac a 2, \frac{1+a}2-b; 1+a-b; -\frac{4z}{(1-z)^2}\right)\\ &=(1+z)^{-a} \, _2F_1\left(\frac a 2, \frac{a+1}2; 1+a-b; \frac{4z}{(1+z)^2}\right) \end{align}</math>

and Gauss's theorem by putting z = −1 in the first identity. For generalization of Kummer's summation, see Шаблон:Harvtxt.

Values at z = 1/2

Gauss's second summation theorem is

<math display=block>_2F_1 \left(a,b;\tfrac12\left(1+a+b\right);\tfrac12\right) = \frac{\Gamma(\tfrac12)\Gamma(\tfrac12\left(1+a+b\right))}{\Gamma(\tfrac12\left(1+a)\right)\Gamma(\tfrac12\left(1+b\right))}. </math>

Bailey's theorem is

<math display=block>_2F_1 \left(a,1-a;c;\tfrac12\right)= \frac{\Gamma(\tfrac12c)\Gamma(\tfrac12\left(1+c\right))}{\Gamma(\tfrac12\left(c+a\right))\Gamma(\tfrac12\left(1+c-a\right))}.</math>

For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see Шаблон:Harvtxt.

Other points

There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in Шаблон:Harvtxt and Шаблон:Harvtxt. Some typical examples are given by

<math display=block>{}_2F_1 \left(a,-a;\tfrac{1}{2};\tfrac{x^2}{4(x-1)} \right ) = \frac{(1-x)^a+(1-x)^{-a}}{2},</math>

which can be restated as

<math display=block>T_a(\cos x)={}_2F_1\left(a,-a;\tfrac{1}{2};\tfrac{1}{2}(1-\cos x)\right)=\cos(a x)</math>

whenever −π < x < π and T is the (generalized) Chebyshev polynomial.

References

Шаблон:Sfn whitelist Шаблон:Reflist

Шаблон:Cite book
Шаблон:Cite book
Beukers, Frits (2002), Gauss' hypergeometric function. (lecture notes reviewing basics, as well as triangle maps and monodromy)
Шаблон:Dlmf
Шаблон:Cite book
Gasper, George & Rahman, Mizan (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. Шаблон:ISBN.
Шаблон:Cite journal
Шаблон:Cite book
Шаблон:Cite journal
Шаблон:Cite journal
Шаблон:Cite book (part 1 treats hypergeometric functions on Lie groups)
Шаблон:Cite book
Шаблон:Cite book
Шаблон:Cite book
Шаблон:Cite journal
Шаблон:Cite journal
Шаблон:Cite journal
Шаблон:Cite book
Шаблон:Cite journal
Шаблон:Cite journal
Шаблон:Cite journal (a reprint of this paper can be found in Шаблон:Cite web)
Шаблон:Cite book
Шаблон:Cite book (there is a 2008 paperback with Шаблон:ISBN)
Шаблон:Cite journal
Шаблон:Cite book
Шаблон:Cite book
Шаблон:Cite book

External links

Шаблон:Springer
John Pearson, Computation of Hypergeometric Functions (University of Oxford, MSc Thesis)
Marko Petkovsek, Herbert Wilf and Doron Zeilberger, The book "A = B" (freely downloadable)
Шаблон:MathWorld

Шаблон:Series (mathematics)

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[1] Шаблон:Harvnb

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Партнерские ресурсы
Криптовалюты	Обмен криптовалют - www.bestchange.ru Криптовалютная биржа CoinEx Криптовалютная биржа Binance HIVE OS - операционная система для майнинга e4pool - Мультивалютный пул для майнинга.
Магазины	AliExpress — глобальная виртуальная (в Интернете) торговая площадка, предоставляющая возможность покупать товары производителей из КНР; computeruniverse.net - Интернет-магазин компьютеров(Промо код 5 Евро на первую покупку:FWWC3ZKQ);
Хостинг	DigitalOcean - американский провайдер облачных инфраструктур, с главным офисом в Нью-Йорке и с центрами обработки данных по всему миру;
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