The Super-Symmetric Mean

The famous Arithmetic-Geometric Mean (AGM) has found many uses
in analysis (e.g., elliptic integrals) and number theory (e.g.,
quadratically convergent algorithms for computing the digits
of PI), but there does not seem to have been much attention
devoted to higher-order versions of iterated means of more
than two arguments.  One obvious extension is to consider the 
iterated Arithmetic-Geometric-Harmonic mean of three numbers 
a[0], b[0], c[0], defined by the recurrence

         a[n+1] = A(a[n],b[n],c[n])
  
         b[n+1] = G(a[n],b[n],c[n])

         c[n+1] = H(a[n],b[n],c[n])

Similarly, making use of all ten of the "means" defined by the ancient
Greeks, we can have iterations of up to 10 elements.  However, this
seems somewhat arbitrary, and the restriction to just a small number
of means is rather artificial.

Another obvious approach would be to use the "Holder means", which
are essentially just isomorphic to the simple arithmetic mean carried
out in a transformed domain.  In other words, for any invertible
transformation function f we can define the mean of the values x1,
x2,..,xn by the formula

                      f(x1) + f(x2) + ... + f(xn)
        f(x_mean) =  -----------------------------
                                    n

which is just the arithmetic average in the f domain.  The Holder
mean M_k() is defined by the above formula with f(x)=x^k, from
which it follows that M_1() is the Arithmetic mean, M_-1() is
the Harmonic mean, M_2() is the root-sum-square, and the limit of 
M_k() as k goes to infinity is the Geometric mean.  Of course, we 
can also get the Geometric mean by simply defining f(x)=log(x).
However, these means are all just (in a sense) weighted version
of a single prototype, and don't seem to capture the essence of
the desired generalization of the AGM.

By the way, it's interesting to consider the class of means of
complex numbers given by the above equation with f() taken from 
the class of the most general holomorphic functions of the complex
plane, namely, the linear fractional transformations

                       ax + b
             f(x)  =  --------
                       cx + d

This has connection with modular functions, and is very interesting
in its own right, but it still doesn't seem to capture the desired
generalization.

I think the optimum approach to generalizing the AGM would be to 
devise an nth-order mean by observing how our two basic means arise 
from two of the elementary symmetric functions of n objects.  We could 
devise "means" based on all n of the symmetric functions.  Recall that
the first elementary symmetric function of x1,x2,..,xn is just the sum

               S_1 = x1 + x2 + ... + xn

and the "mean" based on this is the single number that, if we set all 
the x's to that value, would give the same S_1.  In other words, it 
is (S_1)/n.  This gives the usual arithmetic mean.  Similarly, the 
nth elementary symmetric function is just the product of the n values, 
i.e.,
               S_n  =  (x1)(x2)...(xn)

and again the mean based on this function is the single value that,
if we set all the x's to that value, would give the same S_n.  In 
other words, it is the nth root of S_n.

Now it's clear how we can proceed to the other symmetric functions.
For example, S_2 is the sum of all products of two of our x values.
To illustrate, with n=4 we would have

S_2 = (x1)(x2) + (x1)(x3) + (x1)(x4) + (x2)(x3) + (x2)(x4) + (x3)(x4)

Obviously the "mean" based on this function would be the square root
of (1/6)th  of S_2.  In general, we can define the kth "symmetric mean" 
of n objects as follows

                     /  S_k   \(1/k)
            M_k  =  ( -------- )
                     \ C(n,k) /

where C(n,k) is the binomial coefficient (n choose k).  M_1 is just
the arithmetic mean and M_n is the geometric mean.

Now, our generalization of the AGM to n elements is the "super-symmetric 
mean" of the n values x1[0], x2[0], ..., xn[0], defined as the convergent 
value given by the braid of iterations

      xj[k+1]  =  M_j(x1[k],x2[k]...,xn[k])     j=1,2,..,n

Of course our original values of xj[0] are the roots of the polynomial

  f(x) =  x^n - (S_1) x^(n-1) + (S_2) x^(n-2) - ... +- (S_n)

and the next set of values, xj[1], are the roots of another polynomial
whose coefficients are the symmetric functions used on the next
iteration, and so on.  Thus we have a sequence of polynomials
converging on one of the form  F(x) = (x - r)^n,  where r is the
super-symmetric mean of the original x's.  This implies that we can
define the polynomials Mj(x) = (x - M_j(x1,x2,..,xn))^n  where the
jth such polynomial matches the (n-j)th derivative of f(x) at x=0.

To illustrate on a simple example, Figure 1 shows how the arithmetic
and geometric means of the numbers 3 and 15 are related to the two
polynomials that match the value and the derivative of the function
f(x)=(x-3)(x-15) respectively at x=0.

Figure 1


For a less trivial example, Figure 2 shows the three symmetric mean
polynomials for the three values 3,7,15.  Again the mean polynomials
match the value and the first and second derivatives of the function
f(x)=(x-3)(x-7)(x-15) respectively at x=0.

Figure 2


It's also interesting to see how we can construct various other means 
from the elementary symmetric means.  For example, the harmonic mean 
of n values is given by nM_n/(M_{n-1}).  More generally, we can define
the family of means of n numbers as follows

                        / (M_k)/C(n,k) \ 1/(k-j)
           R_{k,j}  =  ( -------------  )              n >= k > j
                        \ (M_j)/C(n,j) /

In these terms the ordinary arithmetic mean is R_{1,0}, the geometric 
mean is R_{n,0}, and the harmonic mean is R_{n,n-1}.