:: generalised trigonometry : product and sum formuli ::

[galathaea]

i wanted to demonstrate the generalised trigonometry in more depth
as i throw together the paper i am working on
to illustrate the algebra in more concrete terms

my hope is that a reader could point me
to work that precedes this
or possibly to stimulate their developments

first lets start with a classic
(hyperbolic) trigonometric derivation of the product rule
in terms of multisections

simpson's multisection formula gives

|0 x 1 x -x
| e = - ( e + e )
|2 2

so

|0 (a + b) 1 a b -a -b
| e = - (e e + e e )
|2 2

|0 (a - b) 1 a -b -a b
| e = - (e e + e e )
|2 2

and

/ |0 a \/ |0 b \
| | e || | e | =
\ |2 /\ |2 /

1 a b a -b -a b -a -b
- (e e + e e + e e + e e ) =
4

1 / |0 (a+b) |0 (a-b) \
- | | e + | e |
2 \ |2 |2 /

and we have the product rule for cosh

note
the basic steps for derivation were
a) expand out the product
b) collect terms as
multisections of sums

now
lets apply these steps to the
(0, 3)-multisection of the exponential

define w as

2 pi / 3
w := e

ie. it is a 3-root of unity
so that

2
|0 x 1 x wx w x
| e = - (e + e + e )
|3 3

and so the derivation goes as

/ |0 a \/ |0 b \
| | e || | e | =
\ |3 /\ |3 /

2
1 a b a wb a w b
- (e e + e e + e e +
9
2
wa b wa wb wa w b
e e + e e + e e +

2 2 2 2
w a b w a wb w a w b
e e + e e + e e )

now
collect by the diagonals

\\\_wrap around
-\\\_wrap around
-\\\

distribute a third
and you get the expressions

2
/ |0 (a+b) |0 (a+wb) |0 (a+w b) \
= | | e + | e + | e |
\ |3 |3 |3 /

a product rule for the (0,3)-exponential projection!

as with the classic trigonometric case
inversion gives the sum rule

of course at this point
the generalisation is obvious

take
2 pi / n
w := e
n

then

/ |m a \/ |m' b \
| | e || | e | =
\ |n /\ |n /

n-1 n-1
1 --- --- i j
-- \ \ -im -jm' w_n a w_n b
2 / / w w e e =
n --- --- n n
i=0 j=0

and summing across the appropriate subterms gives

n-1
1 --- l
(above) = - \ -m'l |(m+m') (a + w b)
n / w | e n
--- n |n
l=0

isn't that a pretty formula?

doing this for q-exponentials
gives you generalised q-trigonometric
product and sum formula

you might also notice that the matrix to invert
in order to get a sum formula from the product
is somewhat famous in combinatorics

so it seems strange that
i cannot find these results in the literature

if i can get no references
i am going to call the theory
"galathaea's universal trigonometry"
or g. u. t. for short

just to drive the physicists crazy with acronyms =)

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar