Existing Definitions: $$\zeta(n)=\sum_{k=1}^\infty \frac{ 1 }{k^n}$$ $$\lambda(n)=\sum_{k=1}^\infty \frac{ 1 }{(2k-1)^n}=\frac{\left(2^n-1\right)}{2^n}\zeta (n)$$ $$\eta(n)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^n}=\left(1 -2^{1-n} \right) \zeta (n)$$ $$\beta(n)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{(2k-1)^n}$$
The existing well known trigonometric functions $\csc(x)$, $\sec(x)$, $\tan(x)$ and $\cot(x)$ in infinite series form are:
$$\csc(x)=\frac{1}{x}+2 \sum _{k=1}^{\infty } \frac{\eta(2 k) }{\pi ^{2 k}}\;x^{2 k-1}$$
$$\sec(x)=2\sum _{k=1}^{\infty } \frac{\ 2^{2 k-1} \beta(2 k-1) }{\pi ^{2 k-1}}\,x^{2 k-2}$$
$$\tan(x)=2 \sum _{k=1}^{\infty } \frac{2^{2 k} \lambda(2 k) }{\pi ^{2 k}}\,x^{2 k-1}$$
$$\cot(x)=\frac{1}{x}-2 \sum _{k=1}^{\infty } \frac{ \zeta (2 k)}{\pi ^{2 k}}\,x^{2 k-1}$$
I am now going to define some analogous new definitions, using the odd Zeta constants and the even Beta constants, and postfix them with an "i":
$$\text{csci}(x) =2 \sum _{k=1}^{\infty } \frac{\eta(2 k+1) }{\pi ^{2 k+1}}\,x^{2 k}-\frac{1}{x}+\frac{2 \log (2)}{\pi }$$
$$\text{seci}(x)=2\sum _{k=1}^{\infty } \frac{\ 2^{2 k}\; \beta(2 k) }{\pi ^{2 k}}\,x^{2 k-1}$$
$$\text{tani}(x)=2 \sum _{k=1}^{\infty } \frac{2^{2 k+1} \;\lambda(2 k+1) }{\pi ^{2 k+1}}\,x^{2 k}+\frac{2 \log (2)}{\pi }$$
$$\text{coti}(x)=-2 \sum _{k=1}^{\infty } \frac{ \zeta (2 k+1)}{\pi ^{2 k+1}}\,x^{2 k}-\frac{1}{x}+\frac{2 \log (2)}{\pi }$$
You could try the same analogy with "dark sector" hyperbolic functions. e.g. $$\text{sechi}(x)=2\sum _{k=1}^{\infty } \frac{(-1)^{k-1} 2^{2 k}\; \beta(2 k) }{\pi ^{2 k}}\,x^{2 k-1}$$
It is immediately apparent that there are certain similarities between normal trigonometry and "dark sector" trigonometry
For example $\text{seci}(x)=\text{csci}(x+\frac{\pi}{2})$, and $\text{csci}(x)=\text{coti}(x/2)-\text{coti}(x)$.
But also differences for example $\text{tani}(x)$ is not equal to the inverse of $\text{coti}(x)$. However both the inverses appear to lead to the same new function, that differs only by an inversion and a phase shift of $\pi/2$.
Graph for $1/\text{tani}(x)$
Graph for $1/\text{coti}(x)$
Similar thing happens with the inverse of $1/\text{csci}(x)$ and $1/\text{seci}(x)$ to give $\text{sini}(x)$ and $\text{cosi}(x)$
Graph for $\text{sini}(x)$
Graph for $\text{cosi}(x)$
Examples of combining "dark sector" functions with normal trigonometric functions look quite interesting:
Graph for $\text{cosi}(x)+\cos(x)$
Graph for $\text{cosi}(x)-\cos(x)$
I've just sketched this structure using Mathematica before I waste too much time on it. There may be a better way of defining the four starting analogous functions: $\text{csci}(x)$,$\text{seci}(x)$,$\text{tani}(x)$ and $\text{coti}(x)$.
Does anyone know of attempts to develop what I call here "dark sector" trig or hyperbolic functions?
Does anyone recognise any of these functions and where they might have an application?