1.2.2 The Cyclic Trigonometric Functions and their Derivatives

We define the cyclic trigono­met­ric func­tions as the n-way splits of the ex­po­nen­tial series with $n\in \mathbb{Z}$, oc­cur­ring when the Lie al­ge­bra of a cyclic group el­e­ment is ex­po­nen­ti­at­ed. For the el­e­ments $Y_{j}^{\mathbb{R}} \subset \left( \mathfrak{g}^{\mathbb{R}} \right)^{'} = \{{Y_{1}^{\mathbb{R}}}^{'},{Y_{2}^{\mathbb{R}}}^{'},{Y_{3}^{\mathbb{R}}}^{'}\}$, the iden­ti­ty el­e­ment $\left( Y_{1}^{\mathbb{R}} \right)^{'}$ is un­changed com­pared to $Y_{1}^{\mathbb{R}}$. Look­ing at the re­sults for the de­riv­a­tive of $G\left( {y_{2}Y}_{2}^{\mathbb{R}} \right)$, we find there is a simple cyclic shift­ing of its trigono­met­ric series el­e­ments. Taking the $G\left( {y_{2}Y}_{2}^{\mathbb{R}} \right)$ gen­er­a­tor ex­am­ple

its de­riv­a­tive $G^{'}\left( Y_{2}^{\mathbb{R}} \right)$ is a right­ward cyclic shift of the $G\left( Y_{2}^{\mathbb{R}} \right)$ el­e­ments

To better un­der­stand the n-way trigono­met­ric func­tions, we first define them by name and de­scribe how their de­riv­a­tives relate to one an­oth­er.

De­f­i­n­i­tion (1.2.20) – For the three-way trigono­met­ric func­tions, we re­de­fine them as the Cyclic Trigono­met­ric Func­tions, de­not­ing these func­tions with the frak­tur symbol for $O,$as $\mathfrak{O}_{k}^{m}$, where $m$ is the degree of split­ting, as the number of in­di­vid­ual split trigono­met­ric func­tions, and $k = (0,1,2,\ \ldots,m - 2,m - 1)$ is the index of the in­di­vid­ual split func­tions in terms of their ex­po­nen­tial order

For the three-way cyclic trigono­met­ric func­tions $\in U(3)$, we can rep­re­sent them in the in­dexed frak­tur no­ta­tion as

The­o­rem (1.2.20) – For all Cyclic Trigono­met­ric func­tions $\mathfrak{O}_{k}^{m}(X)$ of the same degree $m$, the func­tion set de­fined by $\mathfrak{O}^{m} = \{\mathfrak{O}_{0},\mathfrak{O}_{1},\ldots,\mathfrak{O}_{m - 1}\}$ con­tains all of the de­riv­a­tives of that same func­tion set $\mathfrak{O}^{m}$, for a square matrix $X \in \mathbb{R}^{+}$, and the de­riv­a­tive of any func­tion $\mathfrak{O}_{j} \in \ \mathfrak{O}^{m}$ is equal to $\mathfrak{O}_{j - 1}$ where the index $j$ is con­gru­ent modulo $m$.

Proof – Ver­i­fi­ca­tion that $\frac{d^{n}}{dX^{n}}\left( \mathfrak{O}_{k}^{m} \right) = \mathfrak{O}_{k - n}^{m}$. As the left­most el­e­ment of the in­fi­nite set $\mathfrak{O}_{0}^{m}$ is equal to $\frac{X^{0}}{0!},$ when taking the de­riv­a­tive of $\mathfrak{O}_{0}^{m}$, we find

and the next el­e­ment in the in­fi­nite series $\frac{d^{n}}{dX^{n}}\left( \mathfrak{O}_{0}^{m} \right)$ will be equal to the max­i­mum index value of the set

As the de­riv­a­tives are cyclic and con­gru­ent modulo $m$, for any Cyclic Trigono­met­ric func­tion $\mathfrak{O}_{j} \in \ \mathfrak{O}^{m}$, we have

Corol­lary (1.2.21) – For the Cyclic Trigono­met­ric Func­tions ${\mathfrak{O}_{k}^{m}}^{\pm}(X)$ with square matrix func­tion ar­gu­ments $X \in \mathbb{R}^{\pm}$, the func­tion set de­fined as $\mathfrak{O}^{m^{\pm}} = \{\pm \mathfrak{O}_{0}, \pm \mathfrak{O}_{1},\ldots, \pm \mathfrak{O}_{m - 1}\}$ con­tains all of the de­riv­a­tives of that same func­tion set $\mathfrak{O}^{m}$, for $X \in \mathbb{R}^{\pm}$, and the de­riv­a­tive of any func­tion $\mathfrak{O}_{j}^{\pm} \in {\mathfrak{O}^{m}}^{\pm}$, for $X \in \mathbb{R}^{+}$, is equal to $\mathfrak{O}_{j - 1}^{+}$, where the index $k$ is con­gru­ent modulo $m$. With $\mathfrak{O}_{j}^{\pm} \in {\mathfrak{O}^{m}}^{\pm}$, for $X \in \mathbb{R}^{-}$, we have al­ter­nat­ing $\pm m$-way trigono­met­ric func­tions with odd powers neg­a­tive, even powers pos­i­tive, and the de­riv­a­tive of any func­tion $\mathfrak{O}_{j}^{\pm} \in {\mathfrak{O}^{m}}^{\pm}$, for $X \in \mathbb{R}^{-}$, is equal to $\mathfrak{O}_{j - 1}^{-}.$

Proof – For $\frac{d^{n}}{dX^{n}}\left( \mathfrak{O}_{+ k}^{m} \right)$, we have the same result as The­o­rem (1.2.20) and verify that $\frac{d^{n}}{dX^{n}}\left( \mathfrak{O}_{- k}^{m} \right) = \mathfrak{O}_{k - n}^{m -}$.

We start by con­firm­ing we have al­ter­nat­ing $\pm m$-way trigono­met­ric func­tions with odd powers neg­a­tive, and even powers pos­i­tive, for the in­fi­nite series rep­re­sen­ta­tion of ${\mathfrak{O}_{j}^{m}}^{-}$

where it holds that the sign of the trigono­met­ric func­tion al­ter­nates by $( - 1)^{n}$. So, each el­e­ment of the in­fi­nite series is neg­a­tive for $n =$ odd and pos­i­tive for $n =$ even. The de­riv­a­tive of ${\mathfrak{O}_{j}^{m}}^{-}$ is

and this com­pletes the proof that $\frac{d}{dX}\mathfrak{O}_{- k}^{m} = \mathfrak{O}_{k - 1}^{m -}$.

To clar­i­fy the im­pli­ca­tions of the the­o­rem and corol­lary above, we first choose the fa­mil­iar cyclic group

This group above is abelian and cyclic, and $Y_{2}$ is the pri­ma­ry gen­er­a­tor of the group when $\alpha = 1$, given any prod­uct of $Y_{2}$ with itself will be $\in Y$. The $e^{\left( Y_{2} \right)}$ matrix ex­po­nen­tial will yield the hy­per­bol­ic group $SO(1,1)^{+}$

and in cyclic trigono­met­ric no­ta­tion, this is

In the in­fi­nite sum­ma­tion form, the ar­gu­ment is the gen­er­a­tor matrix $Y_{2}$, where the even func­tion is

and the odd func­tion is

For the case of

the group is also cyclic when $\alpha = 1$, given the pri­ma­ry gen­er­a­tor ${- Y}_{2}$, as any prod­uct of itself is in the group $Y$. The matrix of the ${- Y}_{2}$ gen­er­a­tor yields the hy­per­bol­ic group $SO(1,1)^{-}$

and in cyclic trigono­met­ric no­ta­tion, this is

In the in­fi­nite sum­ma­tion form, the ar­gu­ment is the gen­er­a­tor matrix $- Y_{2}$, where the even func­tion is

and the odd func­tion

So, for the simple case of $\mathfrak{O}^{2 -}$, when $m = 2$, there are two trigono­met­ric func­tions, $\cosh(\alpha)$ and ${- sinh}(\alpha)$, one pos­i­tive and the other neg­a­tive. The dif­fer­ing signs in the re­sults are due to the odd func­tion series el­e­ments having $\left( - Y_{2} \right)^{2n + 1}$ as nu­mer­a­tors, which is ${- Y}_{2}$ raised to odd powers for all $n \in \mathbb{Z}^{+}$, and the even func­tions having $\left( - Y_{2} \right)^{2n}$ as nu­mer­a­tors, and is $- Y_{2}$ raised to even powers for all $n \in \mathbb{Z}^{+}.$When $m > 2$ and odd, for $m \in \mathbb{Z}^{+}$, the pos­i­tive and neg­a­tive fac­tors will be spread across the in­fi­nite trigono­met­ric func­tion series. This is seen using the ex­am­ple $\mathfrak{O}^{3 -}\left( Y_{2}^{\mathbb{R}} \right)$, where ${- Y}_{2}^{\mathbb{R}}$ is the rep­re­sent­ed cyclic el­e­ment, which has the fol­low­ing matrix ex­po­nen­tial

in which, al­ter­nat­ing signs can be seen in each of the $\mathfrak{O}^{3 -}$ trigono­met­ric func­tion series.