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 specicic 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 generalisations 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: $$ \forall_{(x\ y)\in D}\quad (s\circ f)(x) + s\circ g(y)\ =\ f(x)\cdot g(x) $$ Is it possible to have two different functions $s$ for the same maximal rhythmic domain? (I doubt it).

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).