Pochhammer

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed a

For fixed n

Values at fixed points

General characteristics

Domain and analyticity

(Subscript[(a), n]) is an analytical function of a and n which is defined over (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to n

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

With respect to a

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

Branch points

With respect to n

The function (Subscript[(a), n]) does not have branch points with respect to n.

With respect to a

The function (Subscript[(a), n]) does not have branch points with respect to a.

Branch cuts

With respect to n

The function (Subscript[(a), n]) does not have branch cuts with respect to n.

With respect to a

The function (Subscript[(a), n]) does not have branch cuts with respect to a.

Series representations

Generalized power series

Expansions at a==0

Expansions at a==b

Expansions of Pochhammer[a+ϵ,n] at ϵ==0/;a≠-m

Expansions of Pochhammer[-m+ϵ,n] at ϵ==0/;m-n∉ℕ∧m∈ℕ

Expansions of Pochhammer[-m+ϵ,n] at ϵ==0/;n∈ℤ∧n≤m∧m∈ℕ

Asymptotic series expansions

Expansions at a==∞

Expansions at n==∞

Limit representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Differentiation

Low-order differentiation

With respect to a

With respect to n

Symbolic differentiation

With respect to a

Fractional integro-differentiation

With respect to a

With respect to n

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Zeros

History

– A. L. Crelle (1831) used a similar symbol
– L. A. Pochhammer (1890)
– P. E. Appell (1880) used the name "Pochhammer symbol"

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