From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: e and nth root of minus unity
Date: Wed, 7 Jul 1999 10:00:49 -0500 (CDT)
Newsgroups: [missing]
To: cnelson9@gte.net
Keywords: Generalizing trig identities to cover F(x)=exp(alpha*x)

OK, I think I see what you want. If I'm right, the right interpretation is
not that i and -i are pth roots of -1 (p=2) but rather that they are
the primitive n-th roots of unity (n=4); what you are proposing is the
special case of what I am about to say when  n = 2*p  (p=prime).

For any fixed positive integer  n  we can define functions
  C_k(x) = Sum(  zeta^(j*k) exp( zeta^j x ) , j in (Z/nZ)^* )
that is, the sum is taken over those integers  j  in the range  [1, n]  which
are relatively prime to  n.  Here  zeta=exp(2 pi i / n)  is a fixed primitive
n-th root of unity. Taking  n=4  gives  C_0 = -C_2 = 2*cos, C_1 = - C_3 = 
-2*sin, so these are generalizations of the usual sine and cosine; you might 
want, in the general case, to divide by an appropriate constant.  These
functions  C_k  can easily be checked to be real. You can recover the
exponential functions from them, that is, for any integer  r  coprime to  n,
  Sum( zeta^(r*k) C_k(x) , k=0..n ) = n * exp( zeta^(-r) x )  

Depending on your point of view, this is all either standard number theory
("Gauss sums") or (group) representation theory ("orthogonality relations").
These functions have probably never been named, because they are trivially
related to the exponential function (indeed, sine and cosine would probably
never have been named either if they hadn't been discovered first!); you
could look in the comprehensive formularies of special functions if you're
determined to make sure they aren't individually studied.

The sums of the squares of the  C_k  isn't constant, for most  n. If we simply
sum of the squares of the definitions of the  C_k  and interchange the
order of summation, we do get some simplification, but it settles down to
 2*n Sum( exp( (zeta^j + zeta^(-j)) x ), j in (Z/nZ)^* )
The case  n=4  is exceptional here because there is only one term, with
i^1 + i^3  in the exponent, and this happens to be zero !

You'll have to decide what you mean by "orthogonality", but the answer is
likely to fall out of the definitions, too. For example, if you want
to know whether or not  Integral( C_k(x)  C_l(x) dx, x=some interval )  is
zero, you need only simplify a sum of some integrals of the exponential
function. Since the functions  C_k  are not periodic for any n<>4, I can't
suggest an appropriate interval over which to integrate.

As far as
>What is the geometrical interpretation of the functions that result? 
I must say you've prejudiced the issue by assuming there is one! I don't see
that there's any particular geometrical interpretation at all, even to the
sine and cosine function -- or more precisely, if I advance one geometrical
interpretation it could easily be attacked as "not the right one" :-)
Likewise I suggest you steer clear of content-free phrases like
>the multidimensional meaning of the exponential

When you get to this part
>The nth root of minus unity for Bn numbers is
I think you're departing from convential mathematical notation. If you can
put the rest of your question into a standard framework I can try to help
you out, but I must say I think Conway spoke for the mathematical
community generally when he took a dim view of the fervor of synergistics in
  http://forum.swarthmore.edu/epigone/geom.research/tunquorblel
(Likewise would a post to sci.math.research have to be expressed in the
language of the math research community. You can attempt a translation
using this approach by Chapman:
  http://forum.swarthmore.edu/epigone/sci.math.research/querdphextwox
I take it from his comments that your questions about "B_n numbers" are
supposed to be questions about some particular associative (and commutative?)
algebras over the reals; a question about such rings may well be appropriate
for sci.math.research -- or at least easy to answer.)

I might take this opportunity to mention a website I maintain which attempts
to guide people to the appropriate parts of mathematics for your query:
  welcome.html
The topics mentioned in this letter can be found in various sections:
  11-number theory
  13-commutative rings and algebras
  16-associative rings and algebras
  20-group theory
  33-special functions
I suppose, depending on how serious you are in this query, that you might
also want to look at 42: Fourier analysis  or one of the allied areas, where
orthogonal bases of function spaces would arise.

dave