Reflecting (on) the Kanade-Russell conjectures

I want to write a short note on the Kanade-Russell conjectures especially concerning their brief history and my involvement with them. This comes right after my new manuscript titled “Reflecting (on) the modulo 9 Kanade-Russell (conjectural) identities” with Wadim Zudilin. Long story way too short, I find these conjectures to be marvelous and equally frustrating. I hope the day will come that I will be able to either produce or follow a proof of them. Moreover, instead of solving the original conjectures, Wadim and I produced more sum-product type conjectures directly related to the modulo 9 Kanade-Russell conjectures in our work.

Creating clear targets in partition theory research is usually a hard task for me. I sometimes find myself like a car mechanic with no cars in his shop. It is a lazy existence. I have tools but no target to aim them at. It is boring, and even worrisome at times. You just don’t know when or where a new problem will arise or if the last paper you wrote is indeed your last paper (modulo all the projects you left behind for a future date). Kanade-Russell modulo 9 conjectures, as they stand, completely eliminates this problem for me. They are clear targets, I just don’t have the tools/don’t know how to use the tools to crack them open.

I should maybe start with a small recap. In their Ph.D. studies, Shashank Kanade and Matthew Russell put together a Maple program called IdentityFinder which does exact enumeration of partitions with some gap conditions in distance rules. They computed many truncated q-series generating functions for various gap conditions, and then checked to see if any one of these series’ conjecturally have a product representation (indicating the existence of equinumerous sets of partitions with some congruence conditions). In their first paper, published in 2015, they presented 4 modulo 9 and 2 modulo 12 conjectures. The 5th modulo 9 conjecture was noted in Matthew’s thesis the next year. In 2018, they wrote down another 9 modulo 12 conjectures. Around this time both Kanade-Russell, and Kağan Kurşungöz independently wrote down the analytic versions of these conjectures too. The one I stared at the most must be the first modulo 9 conjecture:

 \displaystyle \sum_{m,n\ge0} \frac{q^{m^2+3mn+3n^2}}{(q;q)_m(q^3;q^3)_n} \overset?= \frac{1}{(q,q^3,q^6,q^8;q^9)_\infty},

where we use the standard definitions (and abbreviations) of the q-Pochhammer symbols  (a;q)_n = (1-a)(1-aq)\dots(1-aq^{n-1}) for n=0,1,2,\dots  ,  (a;q)_\infty = \lim_{n\rightarrow\infty} (a;q)_n,  and  (a_1,a_2,\dots,a_k;q)_n = (a_1;q)_n(a_2;q)_n\dots(a_k;q)_n.

For anyone not-so-familiar with q-series, this is a good time to stop and ask yourself:

Isn’t it beautiful to see the full factorization of an infinite series?

If this conjecture is true, it will mean that this double series on the left has a reciprocal product representation. Not only that, this product representation has a clear structure where we know all the exponents of the variable q to appear. Looking on the right-hand side these exponents are coming from the residue classes 1, 3, 6, 8 modulo 9. Hence, the naming of this being a “modulo 9 conjecture“. Also we call this product symmetric since the residue classes can be grouped into additive inverse pairs, such as  \pm 1 and  \pm 3 modulo 9 in this example.

In 2018, Kathrin Bringmann, Chris Jennings-Shaffer, and Karl Mahlburg together were able to crack 7 of the 11 modulo 12 conjectures. They reduced some of the other modulo 12 conjectures. Their proofs were not out of this world either. In fact, from start to finish, it was just known techniques applied brilliantly. Their hard work payed off. This breakthrough, of course, raised my hopes towards all the other conjectures. In 2019, Hjalmar Rosengren re-proved some of the identities of Bringmann et al. and he also proved the remaining 4 modulo 12 conjectures. This part needs confirmation from Rosengren, but I believe he became aware of these conjectures for the first time in Chris’ talk in my OPSFA 2019 special session*. So I feel that by organizing a special session and inviting Chris to Linz, I played an infinitesimal (and clearly a non-mathematical) role in the proofs of the last 4 modulo 12 conjectures.

*Chris confirmed this on June 15, 2021.

We reached 2020 and there were only the modulo 9 conjectures left. I talked about them with many great researchers; Alexander Berkovich, Jehanne Dousse, Chris Jennings-Shaffer, Carsten Schneider to name just a few. I also wrote many papers about Capparelli’s identities (joint with Alexander Berkovich) which seems to be the closest cousin to these conjectures. In fact, take a look at the first Capparelli’s identity in its analytic form

 \displaystyle \sum_{m,n\ge0} \frac{q^{2(m^2+3mn+3n^2)}}{(q;q)_m(q^3;q^3)_n} = (-q^2,-q^4;q^6)_\infty(-q^3;q^3)_\infty.

You can mistake the sum side of this identity with the first modulo 9 Kanade-Russell conjecture above any day. The difference between the sum sides is only a factor of two in the exponent of q. On the other hand, the difficulty level is completely opposite. Capparelli’s identities are well studied. There are many proofs of Capparelli’s identities starting from George E. Andrews‘ 1994 proof. Not only q-series proofs either. For example, Stefano Capparelli himself gave a representation theoretic proof to this identity in 1995. With Alexander Berkovich, I proved this Capparelli’s identity 3 times already, each time by discovering a new finite analogue!

In the first proof we gave, we found a finite version of the left-side sum using the combinatorial techniques Kurşungöz laid out. I used this technique on Kanade-Russell conjectures too. I found combinatorially meaningful polynomial analogues of the sum sides for the modulo 9 Kanade-Russell conjectures. Not to drown you in details but the left-hand side sum of the first Kanade-Russell conjecture we presented above is counting some integer partitions with some gap conditions. I was able to show that if you want to count those partitions with the extra condition that the parts are all  \leq N , the left-hand side sum turns into

\displaystyle \sum_{m,n\geq 0} q^{m^2+3mn+3n^2}{N-m-3n+1 \brack m}_q {\lfloor\frac{2}{3}N\rfloor-m-n+1\brack n}_{q^3},

where we use the standard q-Binomial coefficients defined by

\displaystyle {m+n \brack m}_q := \left\lbrace \begin{array}{ll}\frac{(q;q)_{m+n}}{(q;q)_m(q;q)_{n}},&\text{for }m, n \geq 0,\\0,&\text{otherwise.}\end{array}\right.

Now coming back to my recent paper with Wadim. He visited the Research Institute for Symbolic Computation in February 2020 and we had some good time to talk about various possible projects that we can work on. These conjectures were already in both of our radars so we started sharing ideas and get to experimenting with things. Nothing really solved the conjecture (and I am not going to announce that we prove them at the end or anything) but we started to have a mathematical discussion together. We started sending TeX files back and forth. Unfortunately, and with sincere regrets on my part, with the events of 2020 plus me getting more and more occupied with the search of the next academic job, I started to slack and let our conversation to fizzle out.

Later came an unforeseen and golden opportunity. Wadim contacted me out of the blue and let me know, one month in advance, that he will be attending the 85th Séminaire Lotharingien de Combinatoire that was going to take place in September. This was supposed to take place in Bundesinstitut für Erwachsenenbildung, Strobl; only 2 hours away from my apartment in Linz. It was incredibly unexpected to hear about an in-person conference. It was the 40th anniversary of Séminaire Lotharingien de Combinatoire and Christian Krattenthaler was not letting it be an online event. I wanted to go but I was too anxious to join. There were 14 participants on the list but that was still a high number for me. I watched the conference participants list maybe every other day and kept on seeing it shrink; 12 people, 11 people, 9… Exactly 10 days before the conference’s start date the list shrunk to only 7 people. That was my cue to join. I send Christian an email and asked if I can join…

It was an enjoyable conference. It was nice to be back at Strobl. It was pleasant to be able to meet new people again. And most importantly, it was delightful to catch up with people that I already know. The change of place was a nice refresher and I was able to bounce some ideas. My mind was at the right place again. Wadim and I picked up where we left off. He had new ideas, we started to try them one by one. The modulo 9 conjectures were not unraveling but at least we were learning a little more as each attempt failed.

My TeX file traffic with Wadim picked itself up again after the conference. We kept on chatting on the conjectures. After a while, Wadim reminded me of some observations he shared with me, which I completely failed to address months prior. Ole Warnaar in a private conversation shared his observations about the generating function for the partitions with the gap conditions of the 4th modulo 9 conjecture where the parts are \leq N. He observed that if one reflects  q\mapsto 1/q and clears denominators, depending on the residue class of  N mod 3, these sub-sequences converge to some modulo 45 products or to a sum of two mod 45 products. Ole observed these new conjectures using the defining recurrences of these generating functions. I already had the exact expression for these generating functions. So the formulae for the reflected sum sides were already there for us.

The 4th and the, later found, 5th modulo 9 conjectures involve asymmetric modulo 9 products. After the reflections we look at sub-sequences with respect to  N modulo 3. We conjecturally see that the limit of two sub-sequences are modulo 45 products and for the last residue class modulo 3 the limit of the reflection is the sum of the two modulo 45 products that appear as the limit of the other cases. On the other hand, the reflection of the first three symmetric conjectures are not as easy to identify. None of the sub-sequences seemingly converge to a single product. In those cases, Wadim’s intuition was what took us to the finish line. He reduced the problem down to a finite search space for me. Only after that, we saw that although the first 3 mod 9 conjectures involve symmetric products, their reflections are nowhere symmetric. In fact, the limits of the reflected images are not even single products. For each case, these limits seem to be sums of three modulo 45 products.

Here is a small sample set of how these new conjectures look like:

\begin{aligned}\sum_{a,b\ge 0} \frac{q^{a^2 - 3ab + 3b^2 + b}}{(q^3;q^3)_b} {3b - a \brack a}_q&\overset?= \frac{1 }{(q^2;q^3)_\infty(q^3,q^9,q^{12},q^{21},q^{30},q^{36},q^{39};q^{45})_\infty},\\\sum_{a,b\ge 0} \frac{q^{a^2-3ab+3b^2+a-2}}{(q^3;q^3)_b}{3b-a-1 \brack a}_q &  \overset?=   \langle 2,10,16,19\rangle+q\langle 4,10,16,22\rangle-q^2\langle 8,10,14,19\rangle ,\end{aligned}

where

\displaystyle \langle c_1,c_2,c_3,c_4\rangle =\frac{(q^{45};q^{45})_\infty}{(q^3;q^3)_\infty\prod_{j=1}^4(q^{c_j},q^{45-c_j};q^{45})_\infty}.

These are only 2 out of the 10 main conjectures proposed in the paper. We also reflected Capparelli’s theorem in the same spirit and proved that its reflection yields mod 48 products. So, I highly suggest you look into that paper.

We dedicated this paper to our dear friend and legendary mathematician Doron Zeilberger on the occasion of his 20th prime birthday.

I do not know when the modulo 9 Kanade-Russell conjectures will be proven. I also don’t know when their reflections will be proven. I was trying to prove some conjectures and be a part of the solution. We ended up with more conjectures. I became a part of the problem so to say. Thankfully, it is the kind of the problem mathematicians like. I hope people will enjoy attempting these new conjectures as much as I enjoy trying to solve Kanade-Russell conjectures.

As my final word, I want to thank my new co-author Wadim wholeheartedly for his collaboration, patience, and leadership.