It attracted my attention that in many areas of mathematics we sometimes encounter expressions of the form $(c/2+1/2)^n \pm(c/2-1/2)^n$, where $c$ is some kind of a known constant.
- Split-complex numbers
The expression $(j/2+1/2)^n-(j/2-1/2)^n$ alternates between $j$ and $1$. Expression $(1/2+j/2)^n-(1/2-j/2)^n$ is always $j$.
Formula for $(a+bj)^n$ is $\frac{1}{2} j \left((a+b)^n-(a-b)^n\right)+\frac{1}{2} \left((a-b)^n+(a+b)^n\right)$
- Golden ratio. Binet formula for Fibonacci numbers
- $F_n = \frac{\left(1/2+\sqrt{5}/2\right)^n- \left(1/2-\sqrt{5}/2 \right)^n}{\sqrt{5}}$
I wonder, whether golden ratio and its reciprocal play roles, similar to zero divisors or null elements?
- In the research of divergent integrals that I work on now, I encountered similar formulas:
$\int_0^\infty x^n dx=\frac{\left(\tau +\frac{1}{2}\right)^{n+2}-\left(\tau -\frac{1}{2}\right)^{n+2}}{(n+1)(n+2)}=\frac{\omega _+^{n+2}-\omega _-^{n+2}}{(n+1)(n+2)}$
$\int_0^\infty \frac1{x^n} dx=\frac{\left(\tau +\frac{1}{2}\right)^{n}-\left(\tau -\frac{1}{2}\right)^{n}}{(n-1)n!}=\frac{\omega _+^{n}-\omega _-^{n}}{(n-1)n!}$
Here $\tau=\int_0^\infty dx$, a basic divergent integral, but the value $2\tau=\int_{-\infty}^\infty dx$ also makes sense, so we can consider $\tau$ to be "a half" of another constant. Also, formally (via Fourier transform, $\tau=\pi\delta(0)$, so taken twice, $\pi$ becomes $2\pi$, which also makes sense. At the same time $\omega_-=\tau-1/2$ and $\omega_+=\tau+1/2$ have some properties of the "null" or "singular" elements, even though they are not zero divisors. For instance, one cannot divide by or take logarithm of $\omega_-$ (although it is not a zero divisor).
I also want to point out that $(j/2+1/2)^{j/2+1/2}=(1/2-j/2)^{1/2-j/2}=1$ and in the theory of divergent integrals the expressions $\omega_+^{\omega_+}$ and $\omega_+^{\omega_-}$ also play a role.
Given such similar behavior of objects from different fields, I wonder whether the expression $(c/2+1/2)^n \pm(c/2-1/2)^n$ can signalize something other than the existence of null elements? Does it have some other algebraic significance?