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Background #1

Here is a part of an answer of @Sankyu Kim in MathOverflow.

Consequently, we get the Euler-chi function $\chi(z):=\frac{\zeta(1-z)}{\zeta(z)}$.

And I want to know if Sankyu Kim's last expression is a closed form. (First assuming that its true though I failed to check the computation)

$$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\chi(z)$$

This question is in context of this comment.

@ Sangkyu Kim: But the gamma function has no closed form.

If it is a closed form then he may have found the closed form of Gamma function.

Background #2

I am gathering information about this question's(about gamma function) solution.

What I know

$$\xi(z)=\xi(1-z)$$, where $$\xi(z):=π^{-\frac z2}\Gamma(\frac z2)\zeta (z)$$.

What I've tried to derive the closed form of gamma function

I used the equation above and derived the following.

$$π^\frac{2z-1}{2}\frac{\Gamma(\frac{1-z}{2})}{\Gamma(\frac z2)}\chi(z)=1$$

Further I applied $$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\chi(z)$$ and it gave me function equation. It gave me:

$$\Gamma(z)=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\frac{\Gamma(\frac z2)}{\Gamma(\frac{1-z}2)}π^{-\frac{2z-1}2}$$

This is a functional equation about the gamma function.

I recently learned that Laplace transform could be used to solve some functional equations like fibonacci, so I tried to find the laplace transform of Gamma function.

I failed. (As it doesn't exist)

I just now googled to find the laplace transform of gamma function, but I can't find any. So I launched a new question.

If you know how to solve that functional equation, please let me know. Thanks!

Question:

  1. Does $\chi (z)$ have a closed form?

  2. How do I solve the functional equation above?

What I mean by closed form

a form that the value of the function can be evaluated with finite numbers of evaluation s of elementary functions

What I mean by elementary functions

Rational, exponential, logarithmic, trigonometric, etc.. (Normal(?) functions)

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  • You'll have to define "closed form". Anyway, what formula(s) do you know for chi? – Gerry Myerson Oct 01 '18 at 09:52
  • https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function –  Oct 01 '18 at 10:17
  • @GerryMyerson May I drfine "closed form" as "a form that the value of the function can be evaluated with finite numbers of evaluation s of elementary function s"? – KYHSGeekCode Oct 01 '18 at 10:19
  • OK, but then you have to say which are the elementary functions. Also, you haven't told me what formula(s) you know for chi. – Gerry Myerson Oct 01 '18 at 12:29
  • @GerryMyerson Edited my question. – KYHSGeekCode Oct 01 '18 at 16:40
  • For $\Re(s) >0$, $\Gamma(s)$ is the Laplace transform of $e^{-e^{-u}}$ and $e^{-e^{-u}}e^{-\sigma u}$ is the inverse Fourier transform of $\Gamma(\sigma+i\omega)$. Also looking at a proof of the reflection formula $\Gamma(s)\Gamma(s-1)\sin(\pi s) = \pi$ might be interesting – reuns Oct 01 '18 at 21:32
  • OK, so, it's clear that $\Gamma$ has a closed form if and only if $\chi$ does. Given that $\Gamma$ has been around for a long time, has been studied intensively, and no one has found a closed form for it, my money would be on the negative. – Gerry Myerson Oct 01 '18 at 23:06

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