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We know from Calculus what a series is, and you might have seen infinite products as well. But the Elementary Symmetric Polynomials give an entire spectrum of operators between a sum and product over a finite set.

Given a sequence $s_n$ and an $a \in \mathbb{N}$(representing which operator we're picking), define $S_n$ to be the set $\{s_k | 1 \leq k \leq n\}$. Then our "generalized series" is

$T_a(s) = \lim_{n \to \infty} e_a(S_n)$

If $a = 0$, you get $1$ no matter what the set is. If $a = 1$, then you get a standard Calculus series. If $a = 2$, then we get

$T_2(s) = \sum_{n = 1}^\infty s_n \left(\sum_{m = n+1}^\infty s_m\right)$

If $a=3$, then $T_3(s) = \sum_{n=1}^\infty s_n \left(\sum_{m = n+1}^\infty s_m \left(\sum_{k = m+1}^\infty s_k \right) \right)$

and so on. I can provide a Mathematica function I wrote to compute them, if you like. My questions are:

  1. Is there a standard name or paper for these?
  2. I can argue to myself that if $a < b$ and $T_b(s)$ exists, then $T_a(s)$ must exist as well. Does there exist a sequence $s_n$ and a $b > 1$ such that $T_a(s)$ exists for every $a < b$, but $T_b(s)$ doesn't?
  3. You can't recover an infinite product with this $T$ function. Short of providing a new function $R$ that swaps $\sum$ for $\prod$ and multiplication for addition in the $T_2$ and $T_3$ expansions above, is there an easy way to recover them?

Thank you for your time,

-- Michael Burge

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  • $\begingroup$ For the answer of question number 2, I would try $s_n= n^{-k}$ with $k$ suitable chosen... $\endgroup$
    – Fabian
    Mar 2, 2011 at 19:39

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There's a book on the subject: Symmetric Functions and Hall Polynomials, Macdonald. You can find all sorts of bases for the linear space of all symmetric functions, and they lead to the representation theory of $S_n$.

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  • $\begingroup$ Thank you for the reference. $\endgroup$ Mar 2, 2011 at 23:09

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