In the Elliptic Realm
Michael Somos
12 Jun 2016
The trigonometric functions have a natural generalization which is
discovered using only the simplest tools possible. These new functions
are part of what we have called "the Elliptic Realm" since they come
from elliptic functions. We construct sequences of values of circular
and hyperbolic trigonometric functions and then generalize to construct
elliptic sequences without requiring any knowledge of elliptic functions.
Section 1. Functions, Tables, and Sequence Construction.
A function will be specified by giving a table of values at equally
spaced points, usually starting at zero. The cube function is given by
n : 0 1 2 3 4 5 ...
n*n*n: 0 1 8 27 64 125 ...
which tabulates the function starting at zero. The table of sequence
values can be formed in more than one way using arithmetic operations.
We will be generalizing trigonometric functions so we construct
trigonometric sequence tables to illustrate our construction methods
and also to test potential constructions of generalized sequences.
One good example of a hyperbolic sequence is the following table
n : 0 1 2 3 4 5 6 7 8 9 10 ...
s(n): 0 1 3 8 21 55 144 377 987 2584 6765 ...
constructed starting with initial terms s(0) = 0 and s(1) = 1 . Here,
each term is a constant c times the previous term minus the term before
that. The recursion equation is s(n) = c*s(n-1) - s(n-2) for all n.
For example, n = 2 , gives s(2) = c*s(1) - s(0) = c, hence c must = 3.
Another example, n = 5 , gives s(5) = 3*21 - 8 = 55. Note the identity
s(n) = F(2*n) where F() denotes the Fibonacci sequence. We will
call this sequence "Fib2" for future reference. Any number could have
been used as the constant c to construct a sequence. If the constant
is greater than 2 , the sequence is hyperbolic, but if less than 2 in
absolute value then the sequence is circular. The linear case is if
the constant is 2 and hence s(n) = n by induction.
The trigonometric sequences can be constructed in other ways. For
example, we have the equation s(4)*s(1) = s(3)*s(2) - s(2)*s(1) and
for Fib2 this is 8*3 - 3*1 = 21 . We solve this equation to get s(4)
= 21 . We have also the equation s(5)*s(1) = s(3)*s(3) - s(2)*s(2)
and for Fib2 this is 8*8 - 3*3 = 55 which we solve to get s(5) = 55 .
These equations and related equations depend on triples of pairs of
positive integers which form a pair of patterns
[4,1] [3,2] [2,1] and [3,1] [2,2] [1,1]
[6,1] [4,3] [3,2] and [5,1] [3,3] [2,2]
... ... ... ... ... ...
[2n+2,1] [n+2,n+1] [n+1,n] and [2n+1,1] [n+1,n+1] [n,n]
In other words, the following equations
s(2*n+2)*s(1) = s(n+2)*s(n+1) - s(n+1)*s(n) , and
s(2*n+1)*s(1) = s(n+1)*s(n+1) - s(n)*s(n) ,
hold for all positive n . These equations can be solved for s(n)
where n is greater than 2 given any values for s(2) and s(1) but s(1)
must be non-zero. When s(1) = 1 and s(2) = x , these sequences are
related to Chebyshev polynomials of the second kind as follows
s(n) = U(n-1, x/2).
Now we seek a simple generalization of trigonometric sequences.
Section 2. Sums of Four Squares and Elliptic Sequences.
To discover generalized sequences we need a starting point. Among
many possible starting points, I have come across what I think is the
nicest so far. It begins with the sum of four positive squares. It
is known that every positive number is the sum of four squares, but
not always four positive squares. The smallest such number is 4 since
4 = 1+1+1+1 is a sum of four positive squares. Next on the list is
7 = 4+1+1+1 and then 10 = 4+4+1+1 and so on.
Suppose we now require the number to be a sum of four positive
squares in more than one way. In this case, the smallest such number
is 28 and we have the following three sums of squares
[5,1,1,1] [4,2,2,2] [3,3,3,1]
28 = 25+1+1+1 = 16+4+4+4 = 9+9+9+1 .
Above each square appears its square root. The three square root
quadruples have many arithmetic properties. For example, if we
take the product of the four numbers of each quadruple we get
[5,1,1,1] [4,2,2,2] [3,3,3,1]
5 = 5*1*1*1 32 = 4*2*2*2 27 = 3*3*3*1
and note that 5 = 32 - 27 . Now we use each number of each quadruple
to index into the Fib2 sequence mentioned earlier. Recall that
s(2) = 3 , s(3) = 8 , s(4) = 21 , s(5) = 55 , yielding
[5,1,1,1] [4,2,2,2] [3,3,3,1]
55 = 55*1*1*1 567 = 21*3*3*3 512 = 8*8*8*1
as the calculated product of corresponding terms and 55 = 567 - 512 .
Thus, these quadruples have nice unexpected arithmetic properties
when indexed into trigonometric sequences.
Let us now explore these properties further. First, we find that
the next smallest number which is the sum of four positive squares
also in three ways is 42. We have the following sums
[6,2,1,1] [5,3,2,2] [4,4,3,1]
42 = 36+4+1+1 = 25+9+4+4 = 16+16+9+1
and again the square root quadruples appear above. Note again that
we get the equality of difference of products as given by
[6,2,1,1] [5,3,2,2] [4,4,3,1]
12 = 6*2*1*1 60 = 5*3*2*2 48 = 4*4*3*1
with 12 = 60 - 48 . Also, taking the product of corresponding indexed
terms of the Fib2 sequence gives the expected result
[6,2,1,1] [5,3,2,2] [4,4,3,1]
432 = 144*3*1*1 3960 = 55*8*3*3 3528 = 21*21*8*1
which is 432 = 3960 - 3528 . This is an encouraging observation.
Let us briefly summarize our observations. We have found triples
of quadruples of positive integers with the following properties :
1) (Lagrange) The sum of squares of one quadruple is equal to
the sum of squares of the each of the other two quadruples.
2) The product of the first quadruple is the product of the
second quadruple minus the product of the third quadruple.
3) (Weierstrass) Same as property 2 except we use the product of
the corresponding indexed terms of trigonometric sequences.
We will call these "Pfaffian triples" since they come from Pfaffians.
The Weierstrass property relates Pfaffian triples to numeric sequences.
We will call non-trigonometric sequences that satisfy the Weierstrass
property "elliptic sequences".
A reasonable goal is now to find all Pfaffian triples and all the
sequences of numbers satisfying the Weierstrass property. The next sum
of four positive squares in more than two ways is 52, where
[7,1,1,1] [5,5,1,1] [5,3,3,3] [4,4,4,2]
52 = 49+1+1+1 = 25+25+1+1 = 25+9+9+9 = 16+16+16+4
We seek to select a Pfaffian triple of quadruples. In this case it is
[7,1,1,1] [5,3,3,3] [4,4,4,2]
7 = 7*1*1*1 135 = 5*3*3*3 128 = 4*4*4*2
with 7 = 135 - 128 . Can we generalize this? There are a few clues
here. For example, the first and third Pfaffian triples compared
[5,1,1,1] [4,2,2,2] [3,3,3,1]
[7,1,1,1] [5,3,3,3] [4,4,4,2]
might suggest there is a simple arithmetic progression here. Does
[9,1,1,1] [6,4,4,4] [5,5,5,3]
give us another? Yes, it does, and the general pattern is given by
[2*n+1,1,1,1] [n+2,n,n,n] [n+1,n+1,n+1,n-1]
This gives us one infinite set of triples. Are there any others? Yes.
For example, starting from the second triple we get a pattern
[6,2,1,1] [5,3,2,2] [4,4,3,1]
[8,2,1,1] [6,4,3,3] [5,5,4,2]
[10,2,1,1] [7,5,4,4] [6,6,5,3]
... ... ...
[2*n+2,2,1,1] [n+3,n+1,n,n] [n+2,n+2,n+1,n-1]
which is similar to the first pattern. From just these two patterns,
we can construct numeric sequences in a very simple way.
Section 3. Constructing Elliptic Sequences.
Now that we have infinite sets of Pfaffian triples we can use
them to construct elliptic sequences {a(n)}. We start with the values
of a(1),a(2),a(3),a(4). The first triple equation for a sequence is
a(5)*a(1)*a(1)*a(1) = a(4)*a(2)*a(2)*a(2) - a(3)*a(3)*a(3)*a(1)
and if we know the values of a(1),a(2),a(3),a(4), then the equation is
linear in a(5) and we solve for it (assuming non-zero a(1)). Now that
we know a(5), the next equation is linear in a(6)
a(6)*a(2)*a(1)*a(1) = a(5)*a(3)*a(2)*a(2) - a(4)*a(4)*a(3)*a(1)
and we solve for it (assuming non-zero a(1) and a(2)). So, given all
the equations from the two patterns we can solve for all of the terms
of the sequence given a(1),a(2),a(3),a(4). This gives a construction
of an infinite sequence we will call an elliptic sequence.
What about other patterns? For example, startng from the first two
Pfaffian triples leads to this pattern
[5,1,1,1] [4,2,2,2] [3,3,3,1]
[6,2,1,1] [5,3,2,2] [4,4,3,1]
[7,3,1,1] [6,4,2,2] [5,5,3,1]
... ... ...
[n,n-4,1,1] [n-1,n-3,2,2] [n-2,n-2,3,1]
but does the sequence fit into the other patterns? Yes, but the proof
is not obvious and requires work. What do non-trivial examples of
elliptic sequences look like? A simple example is
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
a(n): 0 1 1 1 -1 -2 -3 -1 7 11 20 -19 -87 -191 -197 ...
which satisfies a(n)*a(n-4) = a(n-1)*a(n-3) - a(n-2)*a(n-2) and
many other equations. A closely similar example is
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
a(n): 0 1 1 -1 1 2 -1 -3 -5 7 -4 -23 29 59 129 ...
which satisfies a(n)*a(n-4) = a(n-1)*a(n-3) + a(n-2)*a(n-2) and
other equations. In fact, the odd indexed terms alternate in sign,
and with the sign removed the resulting positive sequence
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
a(n): 1 1 1 1 2 3 7 23 59 314 1529 8209 83313 ...
satisfies the same recursion equation. It is called the Somos-4
sequence -- the simplest member of a large family of sequences.
All this and more is part of "the Elliptic Realm". The elliptic
sequences are simply related to Jacobi theta functions and Weierstrass
sigma functions evaluated at muliples of a number. Some special cases
were called "elliptic divisibility sequences" by Morgan Ward in 1948.
I have demonstrated that they can easily be discovered and constructed
using only simple tools, and without knowledge of elliptic functions.
One starting point is Pfaffian triples and the Weierstrass property.