For a revised copy, go here (Feb. 2004)

Factorizations of Symmetric Function Relations and a Generalization of the Binomial Theorem

Gus Wiseman

September 2003

Consider the symmetric function operators:

• f = Underoverscript[∑, i = 1, arg3] x _ i f(i) (◊ f) (n) = x _ n f(n) (V f) (n) = Underoverscript[∑, i = n, arg3] x _ i f(i) (v f) (n) = Underoverscript[∑, i = n + 1, arg3] x _ i f(i)

Because this notation may be foreign, we provide a definition of the differential operator for comparison:

(D f) (x) = Underscript[lim, h -> 0] (f(x) - f(x + h))/h

Note that ◊ f, v f and V f are themselves functions (whose variable is n above). The quantity r is merely the number of variables in the symmetric function.

We can use these operators to express the elementary symmetric polynomials e _ n, the homogeneous symmetric polynomials h _ n, and the power sums p _ n. For example,

e _ 3 = Underscript[∑, 1 <= i < j < k <= r] x _ i x _ j x _ k = Underoverscript[∑, i = 1, arg3] x _ i Underoverscript[∑, j = i + 1, arg3] x _ j Underoverscript[∑, k = j + 1, arg3] x _ k = • v v

and in general

e _ n = • v^(n - 1) h _ n = • V^(n - 1) p _ n = • ◊^(n - 1)

Furthermore, the identities relating these symmetric functions have simple factorizations with these operators. For example, notice that

e _ 2 = • v = 1/2 ! • (• - ◊) = 1/2 ! (p _ 1^2 - p _ 2) e _ 3 = • v v = 1/3 ! • (• - ◊) (• - 2 ◊) = 1/3 ! (p _ 1^3 - 3 p _ 1 p _ 2 + 2 p _ 3) e _ 4 = • v v v = 1/4 ! • (• - ◊) (• - 2 ◊) (• - 3 ◊) = 1/4 ! (p _ 1^4 - 6 p _ 1^2 p _ 2 + 8 p _ 1 p _ 3 + 3 p _ 2^2 - 6 p _ 4) : e _ n = • v^(n - 1) = 1/n ! • (• - ◊) (• - 2 ◊) ··· (• - (n - 1) ◊)

It is helpful to notice that the behavior of combinations of • and ◊ (or • and v) is analogous to the reduction of operators based on precedence that is used in most computer languages. For example,

• ◊ ◊ ◊ • • ◊ • ◊ • ◊ = p _ 4 p _ 1 p _ 2 p _ 2 = p _ 1 p _ 2^3 p _ 4

which is reduced the same way as

0 + n × n × n × n + n + n × n + n × n + n × n = n^4 + n + n^2 + n^2 = n + 3 n^2 + n^4

Other factorizations are

h _ n = • V^(n - 1) = 1/n ! • (• + ◊) (• + 2 ◊) ··· (• + (n - 1) ◊) e _ (n + 1) = • v^n = • (• - V)^n h _ (n + 1) = • V^n = • (• - v)^n

There are other factorizations for the other relations but they appear to require more complicated operators.

Now, since the elementary symmetric polynomials generate a binomial product

Underoverscript[∏, n = 1, arg3] (1 + x _ n t) = 1 + • t + • v t^2 + • v v t^3 + • v v v t^4 + ···

we have a nice generalization of the Binomial Theorem

Underoverscript[∏, n = 1, arg3] (1 + x _ n t) = 1 + • t + 1/2 ! • (• - ◊) t^2 + 1/3 ! • (• - ◊) (• - 2 ◊) t^3 + 1/4 ! • (• - ◊) (• - 2 ◊) (• - 3 ◊) t^4 + ···

Also, since

Underoverscript[∏, n = 1, arg3] 1/(1 - x _ n t) = 1 + • t + • V t^2 + • V V t^3 + • V V V t^4 + ···

we have

Underoverscript[∏, n = 1, arg3] 1/(1 - x _ n t) = 1 + • t + 1/2 ! • (• + ◊) t^2 + 1/3 ! • (• + ◊) (• + 2 ◊) t^3 + 1/4 ! • (• + ◊) (• + 2 ◊) (• + 3 ◊) t^4 + ···

for -1 < x _ i < 1. This generalizes the Negative Binomial Theorem. We find in general that

Underoverscript[∏, n = 1, arg3] (1 + x _ n t)^k = 1 + k • t + 1/2 ! k • (k • - ◊) t^2 + 1/3 ! k • (k • - ◊) (k • - 2 ◊) t^3 + 1/4 ! k • (k • - ◊) (k • - 2 ◊) (k • - 3 ◊) t^4 + ···

which shows that the symmetric functions e _ n and (-1)^n h _ n are special cases of the symmetric functions

u _ (n, k) = 1/n ! k • (k • - ◊) (k • - 2 ◊) ··· (k • - (n - 1) ◊)

where

Underoverscript[∏, n = 1, arg3] (1 + x _ n t)^k = 1 + u _ (1, k) t + u _ (2, k) t^2 + u _ (3, k) t^3 + ···

Converted by Mathematica  (September 4, 2003)