0
$\begingroup$

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.

  1. 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 $\frac{1}{2} j \left((a+b)^n-(a-b)^n\right)+\frac{1}{2} \left((a-b)^n+(a+b)^n\right)$

  1. 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?

  1. 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?

$\endgroup$

1 Answer 1

3
$\begingroup$

Not sure I understand all aspects of your question, but at least a few of those are related to analytic solutions for recursive sequences of order 2: https://en.wikipedia.org/wiki/Constant-recursive_sequence

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.