IdentityFinder and Some New Identities of Rogers–Ramanujan Type

Abstract

The Rogers-Ramanujan identities and various analogous identities (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with integer partitions. These identities (written in generating function form) are typically of the form "product side" equals "sum side," with the product side enumerating partitions obeying certain congruence conditions and the sum side obeying certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic computation to generate various such sum sides and then use Euler's algorithm to see which of them actually do produce elegant conjectured product sides. We not only rediscover many of the known identities but also discover some apparently new ones, as conjectures supported by strong mathematical evidence.

Proofs and reductions of various conjectured partition identities of Kanade and Russell

“…The second being a difference at a distance condition, meaning a requirement that the difference between the parts π i and π i+k is at least d, for fixed k and d. The third being a congruence at a distance condition, meaning that if the difference between parts π i and π i+k is at most d, then the sum of successive parts π i +π i+1 +· · ·+π i+k is congruent to r modulo m, for fixed k, d, r, and m. Kanade and Russell then calculated all such partitions over a wide range of possible search parameters, and used Euler's algorithm to determine when the resulting series is equivalent to a simple infinite product. In the end, Kanade and Russell found a total of six conjectural identities in [14]; below we state the two conjectures with restrictions modulo 12 (the remaining four conjectures involved the modulus 9).…”

“…Inspired by such results, as well as similar identities mentioned below (notably Capparelli's work in [5,6]), Kanade and Russell conducted an extensive search for new gap-product identities in [14]. Their method was to explicitly construct partitions satisfying three types of conditions.…”

A Variant of IdentityFinder and Some New Identities of Rogers–Ramanujan–MacMahon Type

“…To verify a given potential candidate to a high degree of certainty, we proceed as in [18]. We first find recursions satisfied by P j .…”

“…Six new conjectural identities were found in [18]. It can be checked that the initial conditions in the identities I 2 -I 6 in [18] are all given by one or more fictitious zeros. I 1 does not have an initial condition.…”

“…This paper should be viewed largely as a continuation of [18] with a search space that is broader and differently parametrized than the one in [18]. For the general background on experimentally finding new partition identities and the importance thereof we refer the reader to a small but by no means exhaustive selection of articles: [3,4,18,23,28]. We ask the reader to recall the relevant terminology (sum-sides, product-sides, difference conditions, initial conditions etc.)…”