The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out.

Instead of $\exp(x)$ and $\ln(x)$ functions as the building blocks, let us take the following ones:

$l(x)=\ln \Gamma(x)$ and its derivative, digamma function:


We apply these functions, as well as field operations, starting with number $1$.

So, what do we get? $$\pi=\psi(3/4)-\psi(1/4)$$ $$\gamma=-\psi(1)$$ $$i=\frac{l(2)-l(3)+l\left(-\frac{1}{2}\right)-l\left(\frac{1}{2}\right)}{\psi\left(\frac{3}{4}\right)-\psi\left(\frac{1}{4}\right)}$$ $$e=\frac{i \pi +\left(\psi \left(\frac{i}{2 \pi }+\frac{1}{2}\right)-\psi \left(\frac{1}{2}-\frac{i }{2 \pi }\right)\right)}{i \pi -\left(\psi \left(\frac{i }{2 \pi }+\frac{1}{2}\right)-\psi \left(\frac{1}{2}-\frac{i }{2 \pi }\right)\right)}$$

Moreover, we can express $e^x$ and $\ln x$ via these two functions:

$$e^x=\frac{i \pi +\left(\psi \left(\frac{i x}{2 \pi }+\frac{1}{2}\right)-\psi \left(\frac{1}{2}-\frac{i x}{2 \pi }\right)\right)}{i \pi -\left(\psi \left(\frac{i x}{2 \pi }+\frac{1}{2}\right)-\psi \left(\frac{1}{2}-\frac{i x}{2 \pi }\right)\right)}$$

$$\ln x=l(x+1)-l(x)$$

As such, it turns out that the EL-numbers are a subset of this system.

That said, I wonder whether one of the functions I took as the elements is excessive? Can we keep the same system, using only one of them? Or, maybe, $\psi^{(1/2)}(x)$? What would be the qualities of the respective systems if we included only one?

What if, instead, we include all integer derivatives of digamma $\psi^{(n)}(x)$? It seems, in the later case, Catalan constant also can be expressed (as $\frac1{16}(\psi'(3/4)-\psi'(1/4))$).

P.S. It seems, if we take $i$ as the basic number, we can express $\pi, \gamma, e$ using only digamma function, without need for the first function.

  • 1
    $\begingroup$ This question has been posted on Math.SE as well. -- When cross-posting, please give links between both instances of your question. $\endgroup$
    – Stefan Kohl
    May 7, 2021 at 14:58


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