HypergeometricPFQ

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

General characteristics

Some abbreviations

Domain and analyticity

(Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) is an analytical function of (Subscript[a, 1]), …, (Subscript[a, p]), (Subscript[b, 1]), …, (Subscript[b, q]) and z which is defined in (ℂ^(p+q+1)). In the cases p≤q for fixed (Subscript[a, 1]), …, (Subscript[a, p]), (Subscript[b, 1]), …, (Subscript[b, q]), it is an entire function of z. If parameters (Subscript[a, k])include negative integers, the function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) degenerates to a polynomial in z.

Symmetries and periodicities

Mirror symmetry

Permutation symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to (Subscript[a, l])

The function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) as a function of  (Subscript[a, l]),1≤l≤p, has only one singular point at (Subscript[a, l])= (OverTilde[∞]).  It is an essential singular point.

With respect to (Subscript[b, j])

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

Branch points

With respect to z

With respect to (Subscript[a, l])

The function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) as a function of  (Subscript[a, l]),1≤l≤p, does not have branch points.

With respect to (Subscript[b, j])

The function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) as a function of  (Subscript[b, j]),1≤j≤q, does not have branch points.

Branch cuts

With respect to z

With respect to (Subscript[a, l])

The function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) as a function of  (Subscript[a, l]),1≤l≤p, does not have branch cuts.

With respect to (Subscript[b, j])

The function (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) as a function of  (Subscript[b, j]),1≤j≤q, does not have branch cuts.

Series representations

Generalized power series

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

Expansions on branch cuts for p==q+1

Expansions at z==0

Expansions at z==1 for p==q+1

The point z==1 is the end point of the branch cut for the function (Subscript[, q+1])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, q+1]);(Subscript[b, 1]),…,(Subscript[b, q]);z), where it has a rather complicated behavior. The corresponding general formula (for noninteger (Subscript[ψ, q])==(∑_(j=1)^q)(Subscript[b, j])-(∑_(j=1)^(q+1))(Subscript[a, j])) includes two major terms - regular and singular which are an analytical functions. Moreover, the singular term has representation of the form const (((1-z))^(Subscript[ψ, q])) (1+O(z-1)) and regular term is bounded near point z==1. A more detailed description of this behavior is presented below.

At the singular point z==1 the function (Subscript[, q+1])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, q+1]);(Subscript[b, 1]),…,(Subscript[b, q]);z) is continuous for Re((Subscript[ψ, q]))>0, bounded for Re((Subscript[ψ, q]))==0,(Subscript[ψ, q])≠0 and has, in general, a logarithmic singularity for (Subscript[ψ, q])==0 while for Re((Subscript[ψ, q]))<0 it has a power singularity of order -(Subscript[ψ, q]) to which for integer (Subscript[ψ, q])a logarithmic singularity can also occur.

Expansions at z==∞ for p==q+1

Expansions at z==∞ for polynomial cases

Asymptotic series expansions

Expansions for q==p

Expansions for q==p+1

Expansions for q≥p+2

Expansions for (Subscript[, 0])(Subscript[F, 2])

Expansions for (Subscript[, 0])(Subscript[F, 3])

General formulas of asymptotic series expansions

Main terms of asymptotic expansions

Residue representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

The differential equation for the function  (Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z) has the order max(p,q+1). It has  two (z==0,z==(OverTilde[∞]) for p≤q) or three (z==0,z==1, z==(OverTilde[∞]), for p==q+1) singular points. If  p≤q, then the point  z==0 is a regular singular point, while  z==(OverTilde[∞])is a nonregular (essential) singular point; if p==q+1, then all three singular points are regular.

Below representation includes functions of two kinds. The function (Subscript[G, q+1,q+1]^(q+1,0))(z|({{1-(Subscript[a, 1]),…,1-(Subscript[a, q+1])}, {0,1-(Subscript[b, 1]),…,1-(Subscript[b, q])}})) is the piecewise analytical function with a discontinuity on the unite circle |z|==1. It has singularity near point z==1of the form const (((1-z))^(Subscript[ψ, q])) (1+O(z-1)), when |z|<1. The functions (Subscript[G, q+3,q+3]^(2,q+3))(z|({{0,(Subscript[b, k]),1-(Subscript[a, 1]),…,1-(Subscript[a, q+1])}, {0,(Subscript[b, k]),0,1-(Subscript[b, 1]),…,1-(Subscript[b, q])}})) are the analytical functions and are bounded near point z==1.

The function (Subscript[G, q+1,q+1]^(q+1,0))(z|({{1-(Subscript[a, 1]),…,1-(Subscript[a, q+1])}, {0,1-(Subscript[b, 1]),…,1-(Subscript[b, q])}})) inside of |z|<1can be reprezented through hypergeometric functions defined for all complex z.

Transformations

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Distant neighbors with respect to q

Functional identities

Relations between contiguous functions

Relations of special kind

Division on even and odd parts and generalization

Case (Subscript[, q+1])(Subscript[F, q])

Differentiation

Low-order differentiation

With respect to (Subscript[a, 1])

With respect to (Subscript[b, 1])

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 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

Summation

Finite summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Through hypergeometric functions of two variables

Through Meijer G

Classical cases for the direct function itself

Theorems

Connections between series and continued fraction representations

Euler established that the convergent series  (∑_(k=1)^∞)(Subscript[a, k]) can be expressed in a continued fraction form

(∑_(k=1)^∞)(Subscript[a, k])==(Subscript[a, 1])  ((( 1+ContinueFraction[{-((Subscript[a, k+1])/(Subscript[a, k])),1+((Subscript[a, k+1])/(Subscript[a, k]))},{k,1,∞}]))^(-1)).

In particular the following representation takes place:

(Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),(Subscript[a, 2]),…,(Subscript[a, p]);(Subscript[b, 1]),(Subscript[b, 2]),…,(Subscript[b, q]);z)==1+((z (∏_(k=1)^p)(Subscript[a, k]))/((∏_(k=1)^q)(Subscript[b, k])))  ((( 1+ContinueFraction[{-((z (∏_(j=1)^p)(k+(Subscript[a, j])))/((k+1) (∏_(j=1)^q)(k+(Subscript[b, j])))),1+((z (∏_(j=1)^p)(k+(Subscript[a, j])))/((k+1) (∏_(j=1)^q)(k+(Subscript[b, j]))))},{k,1,∞}]))^(-1)).

History

– J.  F. Pfaff (1797)
– T. Clausen (1828); J. Thomae (1870, 1879) studied differential equation
– S. Pincherle (1886, 1888)
– L. Pochhammer (1888)
– E. W. Barnes (1906–1908)
– T. W.Chaundy (1943)
– N. E. Nörlund (1955)
– A. P. Prudnikov, Y.A. Brychkov and O.I. Marichev (1986)

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