Hypergeometric1F2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Values at z==0

Specialized values

For fixed (Subscript[a, 1]), (Subscript[b, 2]), z

For fixed (Subscript[b, 1]), (Subscript[b, 2]), z

For fixed (Subscript[a, 1]), z

For fixed (Subscript[b, 1]), z

For fixed (Subscript[b, 2]), z

For fixed z and with symbolical integers in parameters

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

For integer and half-integer parameters and fixed z

For fixed z and (Subscript[a, 1])=-(11/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(9/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(7/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(5/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(3/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(1/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(1/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=1, (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(3/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=2, (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(5/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=3, (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(7/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=4, (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(9/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=5, (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(11/2), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=6, (Subscript[b, 1])≥-(11/2)

For rational parameters with larger denominators and fixed z

For fixed z and (Subscript[a, 1])=1, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=2, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=3, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=4, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=5, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=6, (Subscript[b, 1])≥-(23/4)

For fixed z and (Subscript[a, 1])=-((23)/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(21/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(19/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(17/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(15/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(13/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(11/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(9/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(7/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(5/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(3/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=-(1/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(1/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(3/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(5/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(7/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(9/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(11/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(13/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(15/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(17/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(19/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(21/4), (Subscript[b, 1])≥-(11/2)

For fixed z and (Subscript[a, 1])=(23/4), (Subscript[b, 1])≥-(11/2)

General characteristics

Domain and analyticity

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

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to (Subscript[a, 1])

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

With respect to (Subscript[b, j])

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

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

Branch points

With respect to z

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch points with respect to z.

With respect to (Subscript[a, 1])

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch points with respect to (Subscript[a, 1]).

With respect to (Subscript[b, k])

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch points with respect to (Subscript[b, k]).

Branch cuts

With respect to z

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch cuts with respect to z.

With respect to (Subscript[a, 1])

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch cuts with respect to (Subscript[a, 1]).

With respect to (Subscript[b, k])

The function (Subscript[, 1])(Subscript[F, 2])((Subscript[a, 1]);(Subscript[b, 1]),(Subscript[b, 2]);z) does not have branch cuts with respect to (Subscript[b, k]).

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==∞ for polynomial cases

Asymptotic series expansions

Expansions for Abs[Arg[z]]<π

The general formulas

Expansions for any z  in exponential form

Expansions for any z  in trigonometric form

Residue representations

Limit representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Functional identities

Relations between contiguous functions

Relations of special kind

Division on even and odd parts and generalization

Differentiation

Low-order differentiation

With respect to (Subscript[a, 1])

With respect to (Subscript[b, 1])

With respect to (Subscript[b, 2])

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

With respect to z

Symbolic differentiation

With respect to (Subscript[a, 1])

With respect to (Subscript[b, 1])

With respect to (Subscript[b, 2])

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

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

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