Hypergeometric2F1

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Basic definition

For |z|<1 and generic parameters a, b, c, the hypergeometric function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) is defined by the above infinite sum (that is convergent). Outside of the unit circle |z|<1the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) is defined as the analytic continuation with respect to z of this sum, with the parameters a, b, c held fixed.

Complete definition

Outside of the unit circle |z|<1the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) can be defined by the above formula. Under the stated restrictions, all occurring sums are convergent.

Specific values

Values at z==0

Values at z==1

Values at z==-1

For fixed a, b

For fixed b, c

For fixed b

For other parameters

Values at z==(1/2)

For fixed a, b

For fixed a, c

For fixed a

For fixed b

For fixed c

Values at other z

Values at z==-(1/2)

Values at z==-(1/3)

Values at z==-(1/8)

Values at z==(1/9)

Values at z==(1/4)

Values at z==(1/3)

Values at z==(8/9)

Values at z==2

Values at z==((4-3(Sqrt[2]))/8)

Values at z==((1-(Sqrt[2]))/2)

Values at z==((2-(Sqrt[2]))/4)

Values at z==2(Sqrt[2])-2

Values at z==12(Sqrt[2])-16

Values at z==((2-(Sqrt[3]))/4)

Values at z==(e^(±((π i)/3)))

Bill Gosper

Bill Gosper

Values at z including GoldenRatio

Bill Gosper

Bill Gosper

Bill Gosper

Bill  Gosper

Bill Gosper

Specialized values

For fixed a, b, z

For fixed a, c, z

For fixed b, c, z

For fixed a, z

For fixed b, z

For fixed c, z

For fixed z and with symbolical integers in parameters

For integer and half-integer parameters and fixed z

For rational parameters with denominators 3 and fixed z

For rational parameters with denominators 4 and fixed z

For rational parameters with denominators 5 and fixed z

For rational parameters with denominators 6 and fixed z

For rational parameters with denominators 8 and fixed z

For rational parameters with denominators 8 and fixed z and a>0

Values at fixed points

For z==-1 and integer parameters

Values at z==-1

Values at z==(1/2)

For z==(1/9) and integer parameters

For z==(1/4) and integer parameters

For z==(8/9) and integer parameters

For z==(4/3) and integer parameters

For z==2 and integer parameters

For z==4 and integer parameters

For z==-8 and integer parameters

General characteristics

Some abbreviations

Domain and analyticity

(Subscript[, 2])(Subscript[F, 1])(a,b;c;z) is an analytical function of a, b, c and z which is defined in (ℂ^4). For fixed b, c, z, it is an entire function of a. For fixed a, c, z, it is an entire function of b. For negative integer aor b, (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) degenerates to a polynomial in z of order -aor -b.

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to c

For fixed a, b, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) has an infinite set of singular points:
a) c==- k/;k∈ℕ, are the simple poles with residues ((((-1))^k)/(k!)) (Subscript[, 2])(Subscript[(OverTilde[F]), 1])(a,b;-k;z);
b) c==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to b

For fixed a, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) has only one singular point at b= (OverTilde[∞]).  It is an essential singular point.

With respect to a

For fixed b, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) has only one singular point at a= (OverTilde[∞]).  It is an essential singular point.

Branch points

With respect to z

With respect to c

For fixed a, b, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch points.

With respect to b

For fixed a, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch points.

With respect to a

For fixed b, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch points.

Branch cuts

With respect to z

With respect to c

For fixed a, b, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch cuts.

With respect to b

For fixed a, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch cuts.

With respect to a

For fixed b, c, z, the function (Subscript[, 2])(Subscript[F, 1])(a,b;c;z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==0

Expansions at z==1

Expansions at z==∞

Expansions at z==∞ for polynomial cases

Residue representations

General case

Logarithmic cases

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

where (Subscript[u, j,k])(z) are arbitrary functions from the following list of 24 functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Consecutive neighbors (nine basic relations)

Distant neighbors

Functional identities

Relations between contiguous functions

Additional relations between contiguous functions

Relations for fixed z

Relations of special kind

Reduction to polynomial

Division on even and odd parts and generalization

Generic general cases

Confluent general cases

Relations including three Kummer's solutions

Below relations hold for z∈ℂ\ℝ and all values of parameters a,  b,  c,  for which the gamma factors are finite.

Quadratic transformations with fixed a, b, z

Quadratic transformations with fixed a, c, z

Cubic transformations

Differentiation

Low-order differentiation

With respect to a

With respect to b

Brychkov Yu.A. (2006)

With respect to c

With respect to element of parameters |||  With respect to element of parameters

With respect to z

Symbolic differentiation

With respect to a

With respect to b

With respect to c

With respect to element of parameters |||  With respect to element of parameters

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

For the direct function itself

Involving the direct function

Integral transforms

Laplace transforms

Summation

Finite summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving Hypergeometric2F1Regularized

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions with linear arguments

Classical cases involving algebraic functions with rational arguments

Classical cases involving algebraic functions with quadratic arguments

Classical cases involving algebraic functions with squares in arguments

Classical cases involving algebraic functions with cubics in arguments

Classical cases with rational arguments and unit step UnitStep

Classical cases involving algebraic functions with linear arguments and unit step UnitStep

Classical cases involving algebraic functions with rational arguments and unit step UnitStep

Classical cases involving  Sign

Classical cases involving powers of Hypergeometric2F1

Classical cases for products of  Hypergeometric2F1 with linear arguments

Classical cases involving products of  Hypergeometric2F1 with linear arguments

Classical cases for products of  Hypergeometric2F1 with algebraic arguments

Classical cases involving products of  Hypergeometric2F1 with algebraic arguments

Classical cases involving Hypergeometric2F1Regularized with linear arguments

Classical cases involving algebraic functions and Hypergeometric2F1Regularized with linear arguments

Classical cases involving Hypergeometric2F1Regularized with algebraic arguments

Classical cases involving algebraic functions and Hypergeometric2F1Regularized with algebraic arguments

Generalized cases for the direct function itself

Generalized cases involving algebraic functions with quadratic arguments

Generalized cases involving algebraic functions with squares in arguments

Generalized cases involving  Sign

Generalized cases involving powers of Hypergeometric2F1

Generalized cases for products of  Hypergeometric2F1 with algebraic arguments

Generalized cases involving products of  Hypergeometric2F1 with algebraic arguments

Generalized cases involving Hypergeometric2F1Regularized with algebraic arguments

Generalized cases involving algebraic functions and Hypergeometric2F1Regularized with algebraic arguments

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

The Gauss and Riemann-Papperitz differential equations

The Gauss hypergeometric differential equation

z (1-z)w''(z)+(c-(a+b+1)z) w'(z)-a b w(z)==0

has three regular singular points  z=0,1,∞ with corresponding pairs of indicial exponents {0,1-c},{0,c-a-b},{a,b} . After fractional-linear transformations like z->((A z+B)/(C z+D)) these three regular points can be moved into arbitrary points z=(OverTilde[a]), (OverTilde[b]), (OverTilde[c]) with corresponding pairs of indicial exponents {α,α'}, {β,β'}, {γ,γ'} and the Gauss'  equation will be transformed into the Riemann–Papperitz differential equation

w''(z)+(((1-α-α')/(z-(OverTilde[a])))+((1-β -β')/(z- (OverTilde[b])))+((1-γ-γ')/(z-(OverTilde[c])))) w'(z)+(((α α' ((OverTilde[a])-(OverTilde[b]))((OverTilde[a])-(OverTilde[c])))/(z-(OverTilde[a])))+((β β'( (OverTilde[b])-(OverTilde[c]))  ( (OverTilde[b])-(OverTilde[a])))/(z- (OverTilde[b])))+((γ γ'((OverTilde[c])-(OverTilde[a])) ((OverTilde[c])- (OverTilde[b])))/(z-(OverTilde[c])))) ((w(z))/((z-(OverTilde[a])) (z- (OverTilde[b])) (z-(OverTilde[c]))))==0/;   α+α'+β +β'+γ+γ'==1.

A solution of this equation may be expressed in the form

(((((z-(OverTilde[a]))/(z- (OverTilde[b])))))^α) (((((z-(OverTilde[c]))/(z- (OverTilde[b])))))^γ) (Subscript[, 2])(Subscript[F, 1])(α+β+γ,α+β'+γ;1+α-α';(((z-(OverTilde[a]))( (OverTilde[c])-(OverTilde[b])))/((z-(OverTilde[b]))( (OverTilde[c])-(OverTilde[a]))))).

23 other similar solutions can be produced when the parameters are interchanged.

The conformal maps from triangles with circular edges onto the unit disk

Using rational functions drawn from among hypergeometric functions, one can construct conformal maps from triangles with circular edges onto the unit disk.

The eigenvalues and (non-normalized) eigenfunctions of the Schrödinger equation with the Pöschl-Teller potential

The eigenvalues and (non–normalized) eigenfunctions of the Schrödinger equation with the Pöschl–Teller potential (with singularities at x==0 and x==π/(2α))

-(((∂^2)(Subscript[ψ, n])(x))/(∂(x^2)))+(α^2)(((κ(κ+1))/((sin^2)(α x)))+((λ(λ+1))/((cos^2)(α x))))(Subscript[ψ, n])(x)==(α^2)(κ+λ+2n)(Subscript[ψ, n])(x)/;         κ,λ>1, n∈ℕ

are given by  (Subscript[ψ, n])(x)==(sin^κ)(α x)(cos^λ)(α x)  (Subscript[, 2])(Subscript[F, 1])(-n,κ+λ+n;κ+(1/2);(sin^2)(α x)).

Traffic flow modelling with cellular automaton rule 184

The average flow F(t) in the cellular automaton rule 184 based model, starting from random initial conditions is

F(t)==1-ρ-(((((1-ρ))^(1+v+v t))(ρ^t)(1+v+t+v t)!)/((1+v+v t)(1+t)!(v+v t)!))(Subscript[, 2])(Subscript[F, 1])(2,-t;2+v+v t;1-(1/ρ))

where ρ is the initial random density of cars and v their maximal velocity.

History

– Chu Shih–Chieh (1303)
– J. Wallis (1655) introduced the name "hypergeometric"
–  L. Euler (1748, 1769, 1778, 1794)
– A. T. Vandermonde (1772)
– J. Fr. Pfaff (1797)
– C. F. Gauss (1812)
– E. E. Kummer (1832, 1836)
– B. Riemann (1856)
– H. A. Schwarz (1873);
– E. Goursat (1881)
– E. W. Barnes (1908)

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