The title may sound a little bit odd, but I was wondering if anything has ever been documented about functions that aren't periodic in the sense \( f(x + \tau) = f(x) \), but rather (if {p} represents an operator of p magnitude and }p{ reps its root inverse) \( f(x\, \{p\}\, \tau) = f(x) \)
I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1, \( q:ln(x) = \exp^{[-q]}(x) \) ;
\( x \{p\} S(p) = x \) or \( S(p) \) is the identity function,:
\( sin_q(x) = q:ln(sin(-q:ln(x))) \)
\( cos_q(x) = q:ln(cos(-q:ln(x))) \)
they satisfy
\( sin_q(x \{-q\} q:ln(2\pi)) = sin_q(x) \)
\( cos_q(x \{-q\} q:ln(2\pi)) = cos_q(x) \)
they follow all the laws sin and cos follow only with lowered operators (using logarithmic semi operators); ie
\( e^{x\, \{1-q\}\, q:ln(i)}\, =\, cos_q(x)\, \{-q\}\, (q:ln(i)\, \{1-q\}\, sin_q(x)) \)
\( sin_q(x\, \{-q\}\, q:ln(\frac{\pi}{2}))\, =\, cos_q(x) \)
\( (sin_q(x)\,\{2-q\}\,2)\, \{-q\}\, (cos_q(x)\,\{2-q\}\,2)\,=\,S(1-q) \)
\( sin_q(x\, \{-q\}\, y)\, =\, (sin_q(x)\,\{1-q\}\,cos_q(y))\, \{-q\}\, (sin_q(y)\,\{1-q\}\,cos_q(x)) \)
Pretty much any trigonometric identity you can think of these lowered operator trigonometric functions obey.
They also have a logarithmic semi operator Taylor series very much the same as their sine and cosine counterparts.
if \( \{p\} \sum_{n=0}^{R}\, f(n)\, =\, f(0)\, \{p\}\, f(1)\, \{p\} ... f\(R\) \)
then
\( sin_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n+1!))\, \{1-q\}\, (x\,\{2-q\}\,2n+1) \)
\( cos_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n!))\, \{1-q\}\, (x\,\{2-q\}\,2n) \)
it can also be shown that if \( q:\frac{d}{dx}\, f(x)\, =\, \lim_{h\to\ S(-q)}\, [f(x\,\{-q\}\,h)\,\}-q\{\, f(x)]\,\}1-q\{\,h \)
\( q:\frac{d}{dx} sin_q(x)\, =\, cos_q(x) \)
I ask because I've come across a curious set of "lowered operator" trigonometric function; if 0 <= q < 1, \( q:ln(x) = \exp^{[-q]}(x) \) ;
\( x \{p\} S(p) = x \) or \( S(p) \) is the identity function,:
\( sin_q(x) = q:ln(sin(-q:ln(x))) \)
\( cos_q(x) = q:ln(cos(-q:ln(x))) \)
they satisfy
\( sin_q(x \{-q\} q:ln(2\pi)) = sin_q(x) \)
\( cos_q(x \{-q\} q:ln(2\pi)) = cos_q(x) \)
they follow all the laws sin and cos follow only with lowered operators (using logarithmic semi operators); ie
\( e^{x\, \{1-q\}\, q:ln(i)}\, =\, cos_q(x)\, \{-q\}\, (q:ln(i)\, \{1-q\}\, sin_q(x)) \)
\( sin_q(x\, \{-q\}\, q:ln(\frac{\pi}{2}))\, =\, cos_q(x) \)
\( (sin_q(x)\,\{2-q\}\,2)\, \{-q\}\, (cos_q(x)\,\{2-q\}\,2)\,=\,S(1-q) \)
\( sin_q(x\, \{-q\}\, y)\, =\, (sin_q(x)\,\{1-q\}\,cos_q(y))\, \{-q\}\, (sin_q(y)\,\{1-q\}\,cos_q(x)) \)
Pretty much any trigonometric identity you can think of these lowered operator trigonometric functions obey.
They also have a logarithmic semi operator Taylor series very much the same as their sine and cosine counterparts.
if \( \{p\} \sum_{n=0}^{R}\, f(n)\, =\, f(0)\, \{p\}\, f(1)\, \{p\} ... f\(R\) \)
then
\( sin_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n+1!))\, \{1-q\}\, (x\,\{2-q\}\,2n+1) \)
\( cos_q(x)\, =\, \{-q\} \sum_{n=0}^{\infty} ([q:ln(-1)\, \{2-q\}\, n] \,\}1-q\{\, q:ln(2n!))\, \{1-q\}\, (x\,\{2-q\}\,2n) \)
it can also be shown that if \( q:\frac{d}{dx}\, f(x)\, =\, \lim_{h\to\ S(-q)}\, [f(x\,\{-q\}\,h)\,\}-q\{\, f(x)]\,\}1-q\{\,h \)
\( q:\frac{d}{dx} sin_q(x)\, =\, cos_q(x) \)