FW: Truncated q-exponentials

["Dan Lozier" <lozier@nist.gov>]

Sender: Christian Krattenthaler <kratt@igd.univ-lyon1.fr>
Subject: Truncated q-exponentials


Here are several questions by Greg Kuperberg.
He posted them at another mailing list, and, with his permission,
I post them on this list (which seems much more appropriate).

   With best wishes,

      Christian Krattenthaler

---------- Forwarded message ----------
Date: Thu, 22 Apr 2004 17:25:20 -0700
From: Greg Kuperberg <greg at math dot ucdavis dot edu>
Subject: Truncated q-exponentials
Resent-Date: Thu, 22 Apr 2004 19:31:05 -0500 (CDT)

Dear dominoers,

This isn't really a domino tiling question, but it involves
q-hypergeometric series, so it is at least distantly related.  I am
studying the roots of the polynomial

P_n(x) = x^n - x^(n-1)/3 + x^(n-2)/3/15
   - x^(n-3)/3/15/63 + ... +- 1/3/15/.../(4^n-1).

As Eric Rains pointed out to me, this polynomial is equivalent to
a truncated q-exponential series with q=4, which I'll write as
exp_q^(n)(x).  We have:

    exp_q^(n)(x) = 1 + x + x^2/(1+q) + x^3/3!_q + ... + x^n/n!_q,

so

    P_n(x) = x^n * exp_q^(n)(1/x/(1-q)).   (q=4)

I need to know, and can prove, that P_n(x) has n distinct real roots
r_1 < ... < r_n (note that it is not true when, say, q=2).  I also need
to
know, and can prove (I think), that

    r_1^k + r_2^k + ... + r_n^k = (1-q)/(1-q^k) for k <= n.

Another interesting set of quantities is the complete symmetric
polynomials in r_1,...,r_n.  These also seem to be round rational
functions of q.  Although I'm satisfied enough with my arguments for
these facts, I would be interested for references to equivalent things
that are already in the literature.

For my purpose I also need to know that

    s_n = sqrt(r_1) + sqrt(r_2) + ... + sqrt(r_n) <= 1 (*)

but I am having some trouble proving this.   Experiments with Maple
show (modulo floating-point uncertainties) that it is true for n <= 50;
indeed they indicate that

    2^n*(1-s_n) -> .8141277036597754973514955...

In fact I conjecture more than this one inequality that I need.
I conjecture that

    sqrt(r_1) + sqrt(r_2) + ... + sqrt(r_{k-1}) <= r_k for k <= n, (**)

that 4^n*r_k approaches a limit t_k as n -> infty, that t_k/4^k -> 1/4
as
k -> infty, and that the two limits converge rapidly enough to establish
(*) and (**).  Part of the basis of this belief is the relation

    P_n(x) = 4^{-n}*P_{n-1}(4*x)*(4*x-1) +- 1/3/15/63/.../(4^n-1)/4^n.

The idea is that the correction term is very small, so to first
approximation the roots shrink by 4 and 1/4 is added as a new root.

Maybe it's not really beyond me to prove enough of this, but with the
ideas that came to me so far, it's at best inelegant.

Here is my application:  I'm trying to match the first n moments of the
uniform distribution on [-1,1] by taking a linear combination X of n
i.i.d., unbiased, centered Bernoulli random variables.  I can show that
there is a unique solution; the coefficients are sqrt(r_1),...,sqrt(r_n)
above.  I want to show that the range of X is a subset of [-1,1], and I
also believe that the values of X appears in the natural dyadic order.
It looks like something coming from a horseshoe map, although that
interpretation hasn't helped me prove anything.

--
  /\  Greg Kuperberg (UC Davis)
 /  \
 \  / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
  \/  * All the math that's fit to e-print *