Hypergeometric2F1Regularized

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Basic definition

For |z|<1 and generic parameters a, b, c, the regularized hypergeometric function (Subscript[, 2])(Subscript[(OverTilde[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[(OverTilde[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[(OverTilde[F]), 1])(a,b;c;z) can be defined by the above formula. Under the stated restrictions, all occurring sums are convergent.

General characteristics

Some abbreviations

Domain and analyticity

(Subscript[, 2])(Subscript[(OverTilde[F]), 1])(a,b;c;z) is an analytical entire function of  a, b, c and z which is defined in (ℂ^4).  For negative integer aor b, (Subscript[, 2])(Subscript[(OverTilde[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[(OverTilde[F]), 1])(a,b;c;z) has only one singular point at c= (OverTilde[∞]).  It is an essential singular point.

With respect to b

For fixed a, c, z, the function (Subscript[, 2])(Subscript[(OverTilde[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[(OverTilde[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[(OverTilde[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[(OverTilde[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[(OverTilde[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[(OverTilde[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[(OverTilde[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[(OverTilde[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

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

Major 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

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 HypergeometricPFQRegularized

Involving HypergeometricPFQ

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 Hypergeometric2F1Regularized

Classical cases for products of  Hypergeometric2F1Regularized with linear arguments

Classical cases involving products of  Hypergeometric2F1Regularized with linear arguments

Classical cases for products of  Hypergeometric2F1Regularized with algebraic arguments

Classical cases involving products of  Hypergeometric2F1Regularized with algebraic arguments

Classical cases involving Hypergeometric2F1 with linear arguments

Classical cases involving algebraic functions and Hypergeometric2F1 with linear arguments

Classical cases involving Hypergeometric2F1 with algebraic arguments

Classical cases involving algebraic functions and Hypergeometric2F1 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 Hypergeometric2F1Regularized

Generalized cases for products of  Hypergeometric2F1Regularized with algebraic arguments

Generalized cases involving products of  Hypergeometric2F1Regularized with algebraic arguments

Generalized cases involving Hypergeometric2F1 with algebraic arguments

Generalized cases involving algebraic functions and Hypergeometric2F1 with algebraic arguments

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Copyright