Scientific Groups and Collaborations
We try to locate the
maximum sized coefficient of the special q-Pochhammer Symbol
(1-q)(1-q^2)...(1-q^n), which is sometimes also referred to as
finite Euler Products. This coefficient sizes, in combinatorics,
correspond to the differences between the number of distinct
partitions with even number and odd number of elements. These
products appear in various other locations such as high energy
physics, representation thery and Lie algebras, and computer
science and have their own respective interpretations in each of
these fields. Euler's Pentagonal Number Theorem proves that
asymptotically these differences would become less than or equal
to one in size, but in the finite level this separation grows
exponentially. Up till now, we only had information about the
growth constant of these sizes for these differences but the
location of these elusive peak values was a mystery.
Using the massive
capacity of the MACH2, we were able to carry out full precision
calculations for the first 75,000 products. This led to initial
conjectures about the location of these maximum separations.
Alexander Berkovich
& Ali Kemal Uncu (2020) Where Do the Maximum Absolute q-Series
Coefficients of (1 − q)(1 − q^2)(1 − q^3)…(1 − q^{n − 1})(1 − q^n)
Occur?,
Experimental Mathematics,
https://www.tandfonline.com/doi/full/10.1080/10586458.2020.1776177
Relevant Links
The On-line Encyclopedia of Integer Sequences - A160089: https://oeis.org/A160089
On some polynomials and series of Bloch–Pólya type. Berkovich, A, Uncu, A. K. Proc. Amer. Math. Soc. 146(7): (2018) 2827–2838. doi:10.1090/proc/13982
A personal account on the project by Ali Uncu:
https://akuncu.com/2020/12/19/where-do-the-maximum-absolute-q-series-coefficients-of-qq_n-occur/