NorlundB3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n,α

For fixed n,z

For fixed α,z

General characteristics

Domain and analyticity

The function (Subscript[B, n]^((α)))(z) is defined over ℕ⊗ℂ⊗ℂ. For fixed n, the function (Subscript[B, n]^((α)))(z) is a polynomial in α and z of degree n.

Symmetries and periodicities

Parity

No parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to α

The function (Subscript[B, n]^((α)))(z) is polynomial and has pole of order n at α =(OverTilde[∞]).

With respect to z

The function (Subscript[B, n]^((α)))(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

Branch points

With respect to α

The function (Subscript[B, n]^((α)))(z) does not have branch points.

With respect to z

The function (Subscript[B, n]^((α)))(z) does not have branch points.

Branch cuts

With respect to α

The function (Subscript[B, n]^((α)))(z) does not have branch cuts.

With respect to z

The function (Subscript[B, n]^((α)))(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==∞

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

Expansions at α==0

Expansions at α==∞

Generating functions

Identities

Functional identities

Relations between contiguous functions

Differentiation

Low-order differentiation

With respect to α

With respect to z

Symbolic differentiation

With respect to α

With respect to z

Fractional integro-differentiation

With respect to α

With respect to z

Integration

Indefinite integration

For the direct function with respect to α

For the direct function with respect to z

Summation

Infinite summation

Theorems

Asymptotical expansions of ratios of the gamma functions

Norlund polynomials are used in the asymptotical expansions of ratios of the gamma functions:

((Γ(a+z))/(Γ(b+z)))∝(z^(a-b)) (∑_(k=0)^∞)(((((-1))^k) (Subscript[(b-a), k]) )/(k!))(Subscript[B, k]^((a-b+1)))(a) (z^(-k))/;|arg(a+z)|<π∧(|z|->∞)

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