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A rhythmic identity:

Let $\ \mathbb K\ $ be a field of characteristics $\ne 2\ \ (i.e.\ 1+1\ne 0).\ $ Then

$$ \forall_{x\in\mathbb K}\quad \binom{\binom x2}2+\binom{\binom {x+1}2}2\ \ =\ \ \binom x2\cdot\binom{x+1}2 $$

First, let me ask specific questions (while some of them a bit vague):

Q1:   Was this result published in the past? Could you provide or think of an entire class of similar identities?

Eventual generalizations may gain another dimension due to an equivalent formulation:

$$ \forall_{x\in\mathbb K}\quad \binom{\binom x2}2+\binom{\binom {-x}2}2\ \ =\ \ \binom x2\cdot\binom{-x}2 $$

A diophantine equation:

The following equation (non-trivial when the field's characteristic is $\ne 2)$

$$ (x-y)^2 = x+y $$

admits a full solution in rational integers:

$$ \{(x\ y)\in\mathbb Z : (x-y)^2 = x+y\}\ \ =\ \ \left\{ \left(\binom t2\,\ \binom{t+1}2\right) : t\in\mathbb Z\right\} $$

Remark:  It's nicer to consider here the integers rather than natural numbers or non-negative integers.

The solution feels natural since the above equation is equivalent (in characteristic $\ne 2)$ to a rhythmic equation:

$$ \binom x2 + \binom y2\ \ =\ \ x\cdot y $$

Let me ask routinely again:

Q2:   Was this result published in the past? Could you provide or think of an entire class of similar equations.

The rhythmic domains

Let's restrict ourselves to universe $\ \mathbb C\times\mathbb C,\ $ to discuss the functional rhythmic equation:

$$ s(x) + s(y) = x\cdot y $$

The main question is this one, about the rhytmic domains:

Q3: What are the maximal subsets $\ D\subseteq\mathbb C^2\ $ for which there exists a function $\ s:D\rightarrow\mathbb C\ $ such that $\ s\ $ satisfies the above rhythmic equation? Still better, what is the totality of such (rhythmic) domains?

REMARK   Define $\ D^T:= \{(y\ x): (x\ y)\in D\}.\ $ If $D$ is a rhythmic domain then so is $D\cup D^T$. Thus, every maximal rhythmic domain $D$ is symmetric, i.e. such that $\ D^T=D.\ $ Also, $\ D\cup D^-\ $ is a domain (the complex conjugates of both coordinates are considered for $D^-).\ $ Furthermore, every rhythmic domain is contained in a maximal one (Kuratowski-Zorn theorem).

To prepare the final question (Q4), let's look at two examples of rhythmic domains (I haven't quite tackled their maximality but I swear that they are--I have some properties of domains).

EXAMPLE 1   Let $\ D:=\left\{\left(\binom t2\ \ \binom{-t}2\right): t\in\mathbb C\right\}\ $ and $\ \forall_{z\in\mathbb C}\ s(z):=\binom z2.$ Observe that $D$ is symmetric.

EXAMPLE 2   Let $\ D\:=\left\{\left(t\,\ \frac t{t-1}\right): t\in\mathbb C\setminus\{0\}\right\}\cup\{(1\ 1)\}\ $ and $\ \forall_{z\in\mathbb C\setminus\{0\}}\ s(z):=z\ $ (the identity function so far) and $\ s(1)=\frac 12.\ $ Observe that $D$ is symmetric.

Let me finish with:

Q4.   Supply new various (maximal or interesting) domains $D$, a related function $s$, and complex-valued functions $f$ and $g$ such that: $$ \forall_{(x\ y)\in D}\quad (s\circ f)(x) + (s\circ g)(x)\ =\ f(x)\cdot g(x) $$ Is it possible to have two different functions $s$ for the same maximal rhythmic domain? (I doubt it).

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    $\begingroup$ artofproblemsolving.com/community/c3046h1049224__1 $\endgroup$ – individ Oct 29 '16 at 7:16
  • $\begingroup$ @individ, very nice. It follows from your blog and from the above together that the same integer solutions can be obtained in more than one way, which should lead to respective identities. $\endgroup$ – Włodzimierz Holsztyński Oct 29 '16 at 8:27
  • $\begingroup$ Israel Gelfand said that the formulas are more important than spaces. Henryk Toruńczyk most likely never heard Gelfand's expression but Toruńczyk has illustrated Gelfand's principle in a huge way early in the Infinite-Dimensional Topology. In a modest way, I am turning the attention to the Gelfand's view, where one provides an equation first, and only then one looks for the domains (there can be more than one, as my Question shows). $\endgroup$ – Włodzimierz Holsztyński Oct 29 '16 at 18:34

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