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Function Repository Resource:

KurepaK

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Evaluate the Kurepa function

Contributed by: Jan Mangaldan

ResourceFunction["KurepaK"][z]

gives the Kurepa function K(z).

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Kurepa function is also known as the left factorial.
The Kurepa function is defined as for positive integers n and is defined through analytic continuation elsewhere.
For integer arguments, ResourceFunction["KurepaK"] automatically evaluates to exact values.
ResourceFunction["KurepaK"] can be evaluated to arbitrary numerical precision.
ResourceFunction["KurepaK"] automatically threads over lists.

Examples

Basic Examples (3) 

Evaluate numerically:

In[1]:=
ResourceFunction["KurepaK"][10]
Out[1]=

Plot over a subset of the reals:

In[2]:=
Plot[ResourceFunction["KurepaK"][z], {z, -1, 2}]
Out[2]=

Series expansion at the origin:

In[3]:=
Series[ResourceFunction["KurepaK"][x], {x, 0, 2}] // FullSimplify // Quiet
Out[3]=

Scope (3) 

Evaluate for complex arguments:

In[4]:=
ResourceFunction["KurepaK"][1.43 + I]
Out[4]=

Evaluate to high precision:

In[5]:=
N[ResourceFunction["KurepaK"][1/3], 50]
Out[5]=

The precision of the output tracks the precision of the input:

In[6]:=
ResourceFunction["KurepaK"][0.333333333333333333333333333333333333]
Out[6]=

KurepaK threads elementwise over lists:

In[7]:=
ResourceFunction["KurepaK"][{1, 2, 3, 4, 5}]
Out[7]=

Properties and Relations (2) 

A functional equation satisfied by KurepaK:

In[8]:=
ResourceFunction["KurepaK"][z] - ResourceFunction["KurepaK"][z - 1] ==
   Gamma[z] // FullSimplify
Out[8]=

Compare KurepaK with its integral representation:

In[9]:=
With[{z = 4/3},
 {N[ResourceFunction["KurepaK"][z], 25], NIntegrate[(t^z - 1)/(t - 1) E^-t, {t, 0, \[Infinity]}, WorkingPrecision -> 25]}]
Out[9]=

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