Deriving the closed form of Gamma function using Euler-Chi function

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

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)

• You'll have to define "closed form". Anyway, what formula(s) do you know for chi? – Gerry Myerson Oct 1 '18 at 9:52
• en.wikipedia.org/wiki/… – Yves Daoust Oct 1 '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 1 '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 1 '18 at 12:29
• @GerryMyerson Edited my question. – KYHSGeekCode Oct 1 '18 at 16:40