PolyGamma

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

ψ(z) is an analytical function of z which is defined over the whole complex z-plane with the exception of countably many points z=-k/;k∈ℕ.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

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

Branch points

The function ψ(z) does not have branch points.

Branch cuts

The function ψ(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

Expansions at z==(Subscript[z, 0])/; (Subscript[z, 0])≠-n

Expansions at z==-n

Expansions of PolyGamma[z+ϵ] at ϵ==0/;z≠-n

Expansions of PolyGamma[-n+ϵ] at ϵ==0

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

A. Radović

Contour integral representations

Limit representations

Generating functions

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Products, sums, and powers of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations of special kind

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

Involving the direct function

Summation

Finite summation

Infinite summation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Zeros

History

– J. Stirling (1730)
– A.-M. Legendre (1809)
– S. Poisson (1811)
– C. F. Gauss (1810)
 – M.A. Stern (1847) proved convergence of the Stirling series for digamma function

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