PolyGamma2

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Above formula presents (not unique) continuation of the classical definition of (ψ^((ν)))(z) from positive integer values of ν to its arbitrary complex values. Below four formulas are the particular cases of the above general
definition.

The classical definition of (ψ^((n)))(z) for positive integer n is the following:

Below formula accumulates above definitions for different values of parameter ν in one expression.

Above formula reflects the main functional property of classical polygamma function. In classical interpretation it is valid only for ν==0,1,2,3,… But it is very convenient to use it as basic definition of polygamma function for any arbitrary complex values ν, where symbol (((∂^ν)ψ(z))/(∂(z^ν))) denotes the νth fractional integro-derivative of ψ(z) with respect to z  (which provides the Riemann-Liouville-Hadamard fractional left-sided integro-differentiation with beginning at the  point 0). Such approach was realized in Mathematica.

Specific values

Specialized values

For fixed ν

For fixed z

Values at fixed points

General characteristics

Domain and analyticity

(ψ^((ν)))(z) is an analytical function of ν and z which is defined in (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed nonnegative integer ν, the function (ψ^((ν)))(z) has an infinite set of singular points:
a) z==-m/;ν==0∧m∈ℕ, are the simple poles with residues -1;
b) z==-m/;ν>0∧m∈ℕ, are the poles of order ν+1 with residues 0;
c) z==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to ν

For fixed z, the function (ψ^((ν)))(z) does not have poles.

Branch points

With respect to z

With respect to ν

For fixed z, the function  (ψ^((ν)))(z) does not have branch points.

Branch cuts

With respect to z

For fixed noninteger ν or fixed negative integer ν, the function  (ψ^((ν)))(z) is a single-valued function on the z-plane cut along the interval (-∞, 0), where  (ψ^((ν)))(z)  is continuous from above. This interval includes an infinite set of branch cut lines of combined power-logarithmic type along (-∞, -n)/;n∈ℕ.

For nonnegative integer ν, the function  (ψ^((ν)))(z) does not have branch cuts.

With respect to ν

For fixed z, the function (ψ^((ν)))(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==-m

Expansions of PolyGamma[ν,-m+ϵ] at ϵ==0

Asymptotic series expansions

Residue representations

Other series representations

Integral representations

On the real axis

Of the direct function

Contour integral representations

Continued fraction representations

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

Differentiation

Low-order differentiation

With respect to ν

With respect to z

Symbolic differentiation

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

Integral transforms

Laplace transforms

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Through Meijer G

Classical cases for the direct function itself

Representations through equivalent functions

With related functions

History

– B. Ross (1974)
– N. Grossman (1976)
– R.W. Gosper (1997) defined and studed cases ν==-2, -3 of function  (ψ^((ν)))(z)
– V. S. Adamchik (1998) suggested the idea to define  (ψ^((ν)))(z) for complex ν via Liouville's fractional integration operator
-O. I. Marichev (1998) derived representation of  (ψ^((ν)))(z) for complex ν through sums with hypergeometric or zeta functions
          using fractional integro differentiation

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