In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{x}{n} \right ) \left ( 1+ \frac{x}{n}\omega \right )\left ( 1+ \frac{x}{n} \omega^2 \right )$$
which was shown to be equal to:
$$f(x) = \sum_{k=0}^\infty x^{3k}\zeta_k(3)$$
where we have defined:
$$\zeta_k(3) = \sum_{1\le n_1 <n_2 \cdots <n_k} \left(\frac{1}{n_1 n_2 \cdots n_k} \right) ^3$$
and which is also equal to:
$$f(x) = \frac{1}{\Gamma(1+x)\Gamma(1+\omega x) \Gamma(1+\omega^2 x)}$$
Let me define similar to $\sin,\cos$ functions $c_0,c_1,c_2$ where $c_0$ corresponds to $\sin$:
Let $c_0(x) = x f(\frac{x}{\pi})$. Then $c_0( \omega x ) = \omega x f(\frac{\omega x}{\pi}) = \omega x f(\frac{x}{\pi}) = \omega c_0(x)$ and similariliy $c_0(\omega^2x) = \omega^2 c_0(x)$.
From this it follows that $$c_0(x) + c_0(\omega x) + c_0(\omega^2 x ) = (1+\omega + \omega^2) c_0(x) = 0$$.
We have:
\begin{align}
c_0'(x) &= f(\frac{x}{\pi}) + \frac{x}{\pi}f'(\frac{x}{\pi})\\
&= \sum_{k=0}^\infty \frac{x^{3k}}{\pi^{3k}} \zeta_k(3) + \frac{x}{\pi}\sum_{k=0}^\infty \frac{3k x^{3k-1}}{\pi^{3k-1}} \zeta_k(3)\\
&= \sum_{k=0}^\infty \frac{(3k+1)x^{3k}}{\pi^{3k}} \zeta_k(3) \\
&= \sum_{k=0}^\infty \frac{(3k+1)\zeta_k(3)}{\pi^{3k}} x^{3k}
\end{align}
Let
$$c_1(x) := c_0(\frac{\pi}{3}+\omega x), c_2(x) := c_1(\frac{\pi}{3}+\omega x) = c_0 ( \frac{\pi}{3}+\omega \frac{\pi}{3} + \omega^2 x)$$
then we also have:
$$c_2(\frac{\pi}{3}+\omega x) = c_0(x)$$
Question:
- Do the $c_0,c_1,c_2$ satisfy an "addition formula" similar to $\sin,\cos$?
- Is there a relationship, similar to $\sin,\cos$, between the derivatives of these functions?
- Is there a geometric interpretation of these functions?
Edit:
I have plotted $c_0(x)$ ($c_1(x)$ and $c_2(x)$ look similar) and it shows a three-fold symmetry, like there are three waves which meet at a threefold star and do a physical refraction:
Second edit:
As a number theoretic application one can deduce some equalities concering Aperys constant $\zeta(3)$:
$$\zeta(3) = \frac{81}{\pi} c_2(\frac{2 \pi}{3}) \omega -27 – \sum_{n=1}^\infty \frac{\zeta_{n+1}(3)}{27^n}$$
Also:
$$\zeta_k(3) = \frac{c_2^{(3k+1)}(\frac{\pi}{3})\omega \pi^{3k}}{(3k+1)!}, k=0,1,2,\cdots$$
especially we get:
$$\zeta(3) = \frac{c_2^{(4)}(\frac{\pi}{3})\omega \pi^{3}}{24}$$
Best Answer
There is no addition formula: functions satisfying an algebraic addition formula have been completely characterized,
Painlevé, P. Sur les fonctions qui admettent un théorème d’addition, Acta Math. 27, 1-54 (1903).
P. J. Myrberg, Über Systeme analytischer Funktionen, welche ein Additionstheorem besitzen. Preisschriften gekrönt und herausgegebenen von der Fürstlich Jablonowskischen Gesellschaft zu Leipzig, Teubner, Leipzig, 1922.
Abe, Yukitaka, A statement of Weierstrass on meromorphic functions which admit an algebraic addition theorem, J. Math. Soc. Japan 57 (2005), no. 3, 709–723.