A legendary mathematician, amazing visionary, fun loving easy to chat overall nice person.

Bruno changes the game in everything he does. For example, we couldn’t stop at the 15 minute mark. It was an enjoyable conversation. I am glad we found the time.

## I received the ACA ERA 2022

It was a great honor to be given the Applications of Computer Algebra 2022 Conference’s Early Career Researcher Award. I want to thank the deciding committee.

This raises my motivation while also puts more responsibility on me to do the best I can in the advancement of mathematics and its interactions with computer algebra.

More on the award and the recipients are here.

Miklós is a known name for anyone interested in combinatorics. It is either due to his great introductory books that covers the fundamentals in such clear terms or it is due to his phenomenal research. Regardless of how you know the name, the person behind it is just as great.

I had the great pleasure of sit down at Istanbul and hear his opinions on some topics. I hope you’ll also enjoy it as much as I did.

## Ramanujan and Euler: Online Conference and Summer School

I had the utmost pleasure of being invited to give some lectures at the Euler Institute’s summer school on Partitions, Mock Theta Functions and q-series. I want to thank the main organizer Eric Mortenson once again here for inviting me to this event. The conference portion was scientifically fulfilling and the summer school went really smoothly. Moreover, although it was an online event, I felt that the audience was engaging, which made the teaching experience ten-fold more enjoyable.

#### Short History Leading Up to the Event

This event wasn’t organized this way. Merely a year back, I was hoping to go and see the International Congress of Mathematics (ICM) at St. Petersburg. I was so happy when I got the invitation to the Euler Institute to give a scientific talk. This was supposed to take place in a satellite event of ICM 2022. Well… then things happened, foreseen and unforeseen… sad things… things that are not in the hands of mathematicians. This led to the eventual cancellation of the in-person ICM event.

Thankfully, this satellite event decided to become a stand-alone affair. They not only kept the conference (not in-person anymore) but to expand it with a summer school. I was just done with the Second Workshop on Integer Partitions at the Nesin Mathematics Village, when Eric asked me if I would be interested in giving three lectures in the summer school.

Knowing the dividing nature of the country name where Euler Institute resides. They said that they will keep a neutral website that mentions minimum of such ties… look at all the weird things conference organizers needs to deal with, I thought to myself.

#### How could I say no?

Teaching and sharing new research is a part of the job that I cherish. So when then invitation came from Eric, it was a no-brainer. I just wanted to teach something new. My lectures in the Workshop on Partitions were fresh. They were all recorded anyway. (I talked about Schmidt type partitions there.) I didn’t want to repeat myself. I picked one of my papers that I didn’t/couldn’t promote in the pandemic times. The ideas were fundamental and applicable and versatile.

In fact, I decided to build up these techniques in a way people fresh to the topic can understand and guide them to the last place these ideas were applied in my work and in others’.

#### Conference Portion

The scientific talks from mathematics legends like George Andrews and Bruce Berndt are never to be missed. There were world leading scientists like Jeremy Lovejoy and Atul Dixit as well. I had the opportunity to talk about my recent work on cylindric partitions, and my collaborator and friend Walter Bridges followed my talk with our spin on these problems and the results we recently found.

We had some great questions from the audience. It is clear some people got interested in the topics. Let me try to add some encouragement to that interest: I am here to collaborate if you want! Just send me an email.

#### Summer School

I learned that Walter was going to be one of the other lecturers in the summer school. We also understood that we were the ones that will stick with partition theory topics. We discussed what we wanted to teach and made sure that we don’t overlap. I was going to teach the basics of partitions and move on with combinatorial ideas to write generating functions. Walter was going to stay within bijective combinatorics to prove partition inequalities, etc.

The first lecture was mine, so I better gave a good impression and clear definitions of how we treat the objects.

Three lectures is a lot of time on paper but they go by really quickly. I was able to cover maybe 80% of what I intended on the first lecture. I couldn’t define what a partition identity is. No problem though, this made me time myself better. The second lecture was building up on how we write generating functions using combinatorial constructions and writing double sum generating functions for partitions with uniform gap conditions.

The questions after each lecture also helped me understand where the audience was, what they wanted, etc. You gotta give the audience what they want, right? At the end of my second lecture I was asked how or if the way I was writing generating functions had anything to do with my papers (joint with Alexander Berkovich) in Capparelli’s identities. The answer is, of course, or course! I added a mention of Capparelli’s finitizations to the last lecture as well.

The last lecture was there to tie things together. We wrote generating functions in different ways, wrote their polynomial refinements and mentioned how we can reflect them. In my recent work with Wadim Zudilin that’s exactly what we did anyway. This also led us to new conjectures. I wanted these new conjectures to be the highlight and a good ending of the lecture series. I wanted to pay homage to the St. Petersburg university and the Euler Institute. That was easy in this case, because a recent student of this university, Stepan Konenkov, worked on my paper with Wadim and wrote down 4 more conjectures (and 2 more theorems) using the very technique I was teaching. Turns out there were 2 more modulo 9 conjectures discovered by D. Hickerson. You can look into those even newer conjectures here.

It was a great conference and a great summer school to follow that. I was a bit disappointing at first to have this event online, but the engagement from the students, both in lecture and after the lectures via e-mail gave me a feeling of fulfillment. I was happy to be a part of this wonderful event and it was a great honor to be a summer school lecturer at the Euler Institute. Hope to visit the location, the Sketlov Institute, Chebyshev lab, and so on in a peaceful future.

Come on! I don’t need to introduce George E. Andrews, do I? I also frankly don’t have the space to write down half the things he accomplished.

Instead, I will say that I am so lucky to have met him. He is a regular visitor of University of Florida and the number theory seminars that my former advisor Alex Berkovich ran was always better with George’s presence. I was fortunate to have him in many of my academic milestones; I am honored that he was the one who announced I passed my PhD defense.

## Second Workshop on Integer Partitions

##### May 30 – June 5, 2022, @ Nesin Mathematics Village

It finally happened! Since 2018, I have been trying to find a time to organize this event. The first Workshop on Integer Partitions at the Nesin Mathematics Village (NMK, their Turkish abbreviation) was great fun and a major success. I wanted to keep the ball rolling and introduce these research topics (that I love) together with more people in a friendly environment.

As always, I felt the welcome of NMK from the start of the organizations. This time we got a new ally. Turkish Math Society (TMD, their Turkish abbreviation) also decided to support our event.

I would like to thank the Village and its wonderful team –special thanks to Asli, Aycan, and Tugce– and the TMD MAD fund for their support. I appreciate the recognition and this encourages me to do more.

### How did it start?

It was early 2020, and we didn’t know what was there to come, I first contacted NMK. Their program was full. We aimed this workshop to take place right after the Antalya Algebra Days (the major algebra conference in Turkey, which used to be in Antalya but now takes place at NMK). We couldn’t get the time we wanted. The village was busy for the whole summer. Even then they were able to carve us a small space to have this event. We started the organizations slowly only to cancel later due to the lock-downs.

In the beginning of 2022, I started by contacting my previous co-organizers Zafeirakis Zafeirakopoulos (Zaf) and Kağan Kurşungöz to see if they were up for another round of our workshop. The responses were encouraging. I knew I wanted Jehanne Dousse to be a part of this event too. I approached her. Voila! A positive response came and it got things officially moving.

It was time to contact, the Queen-Bee of the Village, Asli. I contacted her and requested some time and accommodations for this event. She was friendly as ever. It has been 12 years since we first met each other and to this day our conversion has been friendly, polite and professional. I find myself impressed by her professional conduct. We started with the basics, decided on the number of people, the level of participants we expect, the dates, etc. After some emails back-and-forth we decided to have a week-long workshop followed by couple more days for the organizers to stay and collaborate on their research projects. We aimed for the week after Antalya Algebra Days once again (although that event later got cancelled). Once our plans were solidified, Aycan took over and handled our organizational needs. She was just as great as Asli and helped me along the way leading up to the event itself.

After the organizations were done, Walter Bridges said he wanted to join. Back then, we were expecting a mixed group of students from upper level undergrads to early PhD candidates. So, I suggested that he joins us as a lecturer. He accepted and that finalized the list of people to give lectures in this event

### The Workshop

Let’s start with the timetable and the videos. You can watch all the lectures on YouTube on NMK’s channel, or directly from the links below:

Day 3

Day 4

Day 5

Honestly, we didn’t put much thought on how the program should be after the teaching group was formed. I knew what I wanted to teach, I knew it wouldn’t interfere with the others’ lectures and that was good enough for me. One thing I know for sure was that, in the last workshop we had four 90-minute lectures everyday and that was way too much for the students to endure.

Maybe a month before the workshop, I had separate conversations with Zaf and Jehanne. Hour long lectures seemed to make sense the most. That would also leave some time for us to chat and possibly collaborate. With this vague understanding we all made our way to the Village on 29th of May. We finalized the program to have 5 lectures every day (except for the research talks and the social event on Thursday) and a later 1 hour obligatory meeting time with the participants to exchange ideas and answer any and all questions.

The lecture videos above would speak for themselves. The quality of the lectures (excluding a judgment on my own lectures) were quite high and they were all at an accessible level to a wide audience. We required that accessibility because we ended up with a group of participants all the way from freshmen to postdocs. It needed to be both interesting to everyone and to be understandable. I was impressed with how the instructors (again excluding a judgment on my lectures) understood this and kept the level so right. I am so glad that the Village asked us to record the event and upload it all on their webpage. I hope these lectures will be watched and more importantly bring the theory of partitions to the attention of math-loving audience.

### The Participants

We had a mixed bag. It was quite diverse. Including the lecturers, we were 18 people from 8 nationalities coming from institutes that lie in 5 different countries. We had 3 participants that were in the in the first workshop as well. One of them, Halime, finished her PhD studies in the meantime under Kağan’s supervision. It was a really proud moment to have a former participant to join us again, this time as a colleague. The other two were at the end of their doctoral studies too. Clearly, this bunch stayed busy in between these two workshops and it was pleasant to see that mathematics were gaining some new blood, especially in this topic of study.

As mentioned earlier, we had undergrads, Masters students, PhD candidates, and postdoctoral researchers. This gave me a good opportunity to see what was being lost on different levels of participants. For example, due to the fast start, some undergrads got slightly overwhelmed on the first day. I and these couple of undergrads sat down together after dinner, while other people were having drinks and getting to know each other better. We had a chat about what are some common goals of the theory of partitions research, how things connected, what the new and weird notations meant, etc. It was good to see that they became more comfortable. It would have been an utter shame for them to come all the way to the village and not get anything out of it. I didn’t want to let that happen. These two undergrads participated more and stayed interested in the lectures, so I think it worked.

### The Mini-Symposium and The Social Event

We wanted to have a break day after the 3rd day of lectures and see the ancient city of Ephesus. It might have been a one time opportunity for international participants to be this close to that site (although I hope they will all come to NMK many more times). We decided to have our social event after lunch and this allowed us to organize a quick symposium in the workshop. After all, we had 8 participants that do research in discrete mathematics; some doing research directly in the theory of partitions.

The Thursday morning session was filled with really nice talks. I am thankful to all our participants that volunteered to give a talk. Their talks sparked many new ideas in the group and, for example, I already started to collaborate with one of them. This is what we wanted for the workshop. It created new connections.

### Post-Workshop

Jehanne, Zaf and I stayed for another three days to collaborate after the workshop. I need to admit that I was drained with all the organizational responsibilities and especially the other work that I needed to address during the event. Zaf had a lot on his plate as well. Either way, I didn’t want to slack. This was a great opportunity to work with Jehanne on our common goals. I had a wonderful time working with her and I wish we could stay there longer.

After our workshop, CIMPA summer-school came to the NMK and filled it with a new batch of young mathematicians that were hungry for research. We met the organizers of this event back in 2018 when the first workshop on partitions was taking place. It was a great coincidence that they came back to the Village 4 years later at the same time as us.

Their event was a much larger than ours and the idea was to have small groups of 4, 5 students with a researcher to tackle some new research problems that the group leader chose. I should say that it gave me some ideas for the future of the workshop on integer partitions. We already had a strong group with many researchers in it. We could have also approached the partitions workshop in the CIMPA summer-school way.

On our last night, CIMPA event had their warming up and welcome party on the Village deck. We crashed their party an hour after it started and had a blast. We let loose and danced with the Village employees, the other event’s organizers  and participants into the night (dramatization). It was a superb ending to our event and our time at NMK.

Once more I left the Village longing for more; wanting more for the Workshop; wanting more for the future; wanting more from the times to come… I believe, I wasn’t the only one that felt this way either, and –in my book– that means there will be more to come.

##### Some trivia:

• Elaine and Vishnu were the pioneers of the WhatsApp group. They fostered such a friendly environment from the get go. I couldn’t have thought of doing such a thing. Thanks to them we had great transfer of information and a fun group chat too.
• The group photo was taken by a middle-schooler named Mert that was at the Village for a couple of days on a school trip.
• Caption of the group photo: Front-row left to right: Nicolas Smoot, Vishnupriya Anupindi, Ali Kemal Uncu, Kağan Kurşungöz, Walter Bridges, Shamus Albion. Second-row left-to-right: Kevin Allen, Yalçın Can Kılıç, Alp Eren Yılmaz, Jehanne Dousse, Halime Ömrüuzun Seyrek, Elaine Wong. Third Row left to right: Ömer Selçuk, Zafeirakiz Zafeirakopoulos, Mohammad Zadeh Dabbagh, Zohreh Aliabadi, Murat Ertan. Top row: Hasan Bilgili.
• NMK approached us and requested to record our lectures right after Jehanne’s first lecture. We needed to record her lecture again at the end of the second day. It was incredible to watch Jehanne recreate the same board and teach the class almost verbatim. She should be awarded an academy award for the best reproduction of a lecture.
• I decided to organize the next event not to be on the week after Antalya Algebra Days but to organize it to be on the week before a CIMPA school. They sure know how to have fun. 🙂

Matthew England is a great collaborator. I am glad to be in the same project as him. Coventry University is also a great host. I enjoy my visits to this group a lot. After working for 3 days non-stop, we had a short time to have this chat. I am so happy that we get to talk about important questions like what motivates him to do research. I learned a lot, I think you can too.

Matthew also has great humor and such a quick wit. It is quite fun to chat with him. That is why, this time, I am not editing the early minutes of our conversation out of this video. I think you’d like it. I am so dissapointed I don’t seem like I am enjoying the jokes but I love how he “threatened me with a pen”. 😛

I was just too preoccupied, thinking how I am supposed to start the video, what to ask, and in which order, etc. These early jokes deserve much more laughter that I didn’t bring to the table… I hope you will enjoy them more than I couldn’t back then.

Hebron & Medlock Professor of Information Technology James H. Davenport needs no introduction. I have been meaning to have this interview for a while now, but 15 minutes is a long time to sit and chat when you are as busy as James. He is so courteous to spare me this time, so that I can share some of his knowledge, and some of his thought on academia with you.

I hope you will enjoy this conversation as much as I did. I should schedule the next one with James really soon.

The thumbnail is from the Oberwolfach photo collection and it is used under the creative commons license.

## Some Recent Advances in Cylindric Partitions

Studying theory of partitions, I come across many natural and exotic definitions. Cylindric partitions is one of those that is both. I don’t know how but it somehow is. Informally cylindric partitions encodes information about non-intersecting lattice walks and that is clearly natural. However, it is also a finite vector of partitions that can be put together in such a way that you can wrap this shifted arrangement around a cylinder. I don’t think that as something I would be able to come up, so I find it quite exotic in that sense. Thankfully we have great visionaries around such as Ira Gessel and Christian Krattenthaler in this case.

Cylindric partitions are actually really nice. An integer partition is a finite list of non-increasing natural numbers. So given a partition $\lambda = (\lambda_1,\lambda_2,\dots)$, we have an order relation that the parts satisfy: $\lambda_i \geq \lambda_{i+1}$. In the cylindric partitions, we start with a finite list of partitions $(\lambda^{(1)}, \dots, \lambda^{(k)})$ and a list of non-negative integers called a profile $c = (c_1,c_2,\dots,c_k)$ where the parts $\lambda^{(i)}_j$ of the partitions $\lambda^{(i)}$ satisfy an interlacing order relation $\lambda^{(i)}_{j} \geq \lambda^{(i+1)}_{j+c_i},$ where $\lambda^{(k+1)} = \lambda^{(1)}$.

Why do I/we care? Most classical partition identities gets realized as analytic identities where one side is an infinite sum and the other side is an infinite product. Number theoretically, to me, this is a bone chilling structural finding. We see that an infinite series can be completely factored. In the classical results, the sums are easier to find and easier to associate with the partition theory. However, it is quite hard to prove (and unclear to do so) that this sum is equal to a product even when the numerics provides indubitable evidence.

In cylindric partitions, all thanks to Alexei Borodin, we already know that the generating function for the number of such objects is an infinite product. To be precise, generating function for the number of cylindric partitions with profile $c = (c_1,c_2,\dots,c_k)$ (and $t=c_1+c_2+\dots+c_k +k$) is

$\displaystyle\frac{1}{(q^t;q^t)_\infty} \prod_{i=1}^k \prod_{j=i}^k \prod_{m=1}^{c_i} \frac{1}{(q^{m+j-i+s(i+1,j)};q^t)_\infty} \prod_{i=2}^k \prod_{j=2}^i \prod_{m=1}^{c_i} \frac{1}{(q^{t-m+j-i-s(j,i-1)};q^t)_\infty}$

*This product of products is written using q-Pochhammer symbols. Later we will also use q-Binomial coefficients.

This goes against the grain of classical results. This time we have the product representations for these generating functions for free but we have no idea what the infinite series are supposed to look like. For the profile sizes $k=2$, we know the cylindric partitions’ related products coincide with the Andrews-Gordon and Bressoud products, so the sum sides are known through these powerful theorems. However, the $k=3$ cases are still filled with unknowns and conjectures for the sum representation of the generating functions, most advancements are at this level now. $k\geq 4$ is a mystery. That is the product generating functions are there, but we don’t know the closed formula sum representations.

The recent $k=3$ journey starts with a paper by Sylvie Corteel and Trevor Welsh. Their idea is simple and that is what makes it extra brilliant. In simple words, we know the interlacing order relations the cylindric partitions have to satisfy. Therefore, we know at what location or locations the largest part of a cylindric partition can be; that is once given the profile $c$. If those largest parts are removed from the object, a new cylindric partition with a new profile $c^*$ emerges. So we can derive recurrences for the generating functions and associate them with generating functions for different profiles. The number of generating functions involved in such a construction is finite so we are not churning water.

Sylvie and Trevor used this recurrence relations to find the sum generating functions for $k=3$ cylindric partitions with $c_1+c_2+c_3=4$. They coincided with Andrews-Schilling-Warnaar‘s $A2$ Rogers-Ranaujan identities plus found an identity that was not identified before. Later, Sylvie, Jehanne Dousse and I come together and identified the $k=3$ case with $c_1+c_2+c_3 = 5$ in this paper.

As a side note, it is not easy to guess these sum representations at all. They are infinite multisums with 3 and 4 variables in the cases mentioned above, respectively. Once guessed/identified, it is not easy to prove that these objects satisfy the recurrences the generating functions for the number of cylindric partitions with certain products is not easy either due to the high number of variables.

Sylvie and Jehanne were the ones who were able to identify the 4-fold sum generating functions. I cannot take any credit in that miraculous find. I was working on the qFunctions package and as a part of that I was implementing the (coupled) recurrence system of Corteel-Welsh. My objective was to carry this system to a computer algebra system so that we can use other this in conjunction with other symbolic computation tools to prove (or disprove) guessed identities in the spirit of Corteel-Welsh paper.

This was a perfect match. Once we combined our powers, we proved 7 sum-product formulas such as the following.

$\displaystyle\sum_{n_1,n_2,n_3,n_4\geq 0 } \frac{q^{n_1^2+n_2^2+n_3^2+n_4^2+n_1+n_2+n_3+n_4-n_1n_2 + n_2n_4}}{(q;q)_{n_1}} {n_1\brack n_2}_q{n_1\brack n_4}_q{n_2 \brack n_3}_q=\frac{1}{(q^2,q^3,q^3,q^4,q^4,q^5,q^5,q^6;q^8)_\infty}$

Not long after we submitted our paper, Ole Warnaar was able to guess the generic sum representiton for the $k=3$ profile cylindric partitions with $c_1+c_2+c_3\not\equiv 0$ mod $3$. For example, the claim is that for every $j$ the following sum is the sum representation of the generating function for some particular cylindric partitions.

$\displaystyle\sum_{\substack{n_1,\dots,n_j\geq 0\\m_1,\dots,m_{j-1}\geq 0}} \frac{q^{n_j^2+\sum_{i=1}^j}}{(q;q)_{n_1}} \prod_{i=1}^{j-1} q^{n_i^2 - n_im_i + m_i^2 + m_i} {n_i\brack n_{i+1}}_q {n_i-n_{i+1}+m_{i+1}\brack m_i}_q$

If we can prove this, then sum (due to Borodin) factors completely and has a product represenatiton that we wrote above. More conjectures and many interesting results can be found in Ole’s paper.

The way Ole come up with these results is also commendable. He looked at the $k=3$ results that we mentioned above as well as two results from $k=2$ that can be viewed as initial cases of $k=3$ using some symmetries of cylindric partitions. There two steps were enough for him to guess the whole chain. This is an amazing advancement. I wonder if I could do this induction even if I had 20 steps on this chain… Ole only needed 2. Who knows maybe he would guess parts of the $k=4$, etc. chains in the future too.

Technically, we should be able to use the code in qFunctions with Christoph Koutschan‘s HolonomicFunctions package and prove these cases one by one. However, we do not have the computational power to prove that the multi-fold sums satisfy a given recurrence. This is not about physical limiting factor of a computer hardware either. We know that for these objects the theory says that Doron Zeilberger‘s creative telescoping algorithm would terminate, but we do not know when. My personal experience is that for the conjecture above with $j=3$, the code did not terminate in a month on a RISC server even thought the server was not having any memory shortage or anything of sorts.

There you have it. We have all these identities and conjectures, and that is just the beginning of it. We still haven’t touched anything with $k\geq 4$. So be my guest and/or be my collaborator. If you have a good idea on how one can guess these sums?, how we can prove Ole’s guessed sums?, etc. I’m all ears.

But wait there is more! Walter Bridges was looking at a paper about the asymptotics related to the number of cylindric partitions by Han-Xiong. He saw that they have presented Borodin like product representations for symmetric cylindric partitions and double skew shifted plane partitions. Not only that, he also observed that we can adapt Corteel-Welsh idea for all these objects. I was happy to jump in this project right away, and this led to our joint paper. We discovered many nice new sum-product identities such as

$\displaystyle\sum_{n,m \geq 0} (-1)^m q^{3{n+1 \choose 2}-3m(m+1)} \frac{(-q,-q^5;q^6)_m}{(q^6;q^6)_m(q^3;q^3)_{n-2m}}=\frac{(q^4,q^8;q^{12})_{\infty}}{(q^6;q^{12})_{\infty}}.\\$

Furthermore, the more that we looked at the Borodin and Han-Xiong product generating functions and how they are constructed, the more clear it became that we can weight the counts of different diagonals (partitions) in the cylindric partitions and the product generating functions would reflect this change without the underlying structure changing. This opened us up to infinitely many combinatorial connections that we don’t know what to do with (yet). There are really cute ones, and there are the ones that were discovered before outside of the scope of cylindric partitions/skew shifted plane partitions.

For example, in our paper, we prove some Schmidt-type partition results, my favorite being the Schmidt’s theorem itself. (A great read on this with many new results by George Andrews and Peter Paule here.) The theorem says that counting the partitions into distinct parts ($\lambda = (\lambda_1,\lambda_2,\dots)\in\mathcal{D}$ i.e. $\lambda_i>\lambda_{i+1}>\dots$) with the “every other” part counting statistic takes you to the count of all the partitions:

$\displaystyle\sum_{\lambda\in\mathcal{D}} q^{\lambda_1 + \lambda_3+\lambda_5+\dots} = \frac{1}{(q;q)_\infty}$

*For any confused audience members: $1/(q;q)_\infty$ is the generating function for the number of all partitions.

In one of my previous papers, I proved this result without knowing that it already known. I was using a completely different technique, I am happy that it was not a full rediscovery. With Walter, I also proved this theorem using the weighted skew shifted plane partitions (see the line above Prop 5.1 here). Basically, we put distinct parts in a staircase of size two and then weight the diagonal that has the even indexed parts with 0 so that their contribution vanishes and the weighted version of Han-Xiong’s products did the work for us.

Anyway, I will leave it here. There is a lot more to discover, and next time when you have an infinite product generating function in hand check if it is a weighted cylindric partition or a skew shifted plane partition, you might just see a new combinatorial connection to your problem. We have these functionalities build in qFunctions now…

## Prove One, Get Infinitely Many Free!

This is a better deal than any deal you can find on the market.

Bailey’s lemma is an amazing tool. At its core it is a really simple observation but in practice it is a beast. I can (and will) state the actual lemma and you will think to yourself “is this it?”. I am serious. It is as easy as teaching a kid that they can roll a snowball on snow to get it larger to make a snowman. Yet, in practice the same action of snowball rolling down a hill can lead to huge avalanches.

Here is the lemma: Wilfrid N. Bailey noticed that if you start two identities of the form

$\displaystyle \beta_L = \sum_{r=0}^L \alpha_r u_{L-r} v_{L+r}$ and $\displaystyle \gamma_L = \sum_{r=L}^\infty \delta_r u_{L-r} v_{L+r}$

for some sequences $\alpha_j,\ \beta_j,\ \gamma_j, \delta_j,\ u_j,$ and $v_j$ , then we have

$\displaystyle \sum_{L=0}^\infty \alpha_L \gamma_L = \sum_{L=0}^\infty \beta_L \delta_L.$

OK, so I can hear you say that “Ali, you lied, this doesn’t look all that easy.” Well, it is not so easy to roll a snowball for a kid at their first try. Yes, Bailey’s observation is sophisticated but let me give you the proof. It will make you agree that this implication is actually easy. Follow from left to right:

$\displaystyle \sum_{L=0}^\infty \alpha_L \gamma_L = \sum_{L=0}^\infty \alpha_L \sum_{r=L}^\infty \delta_r u_{L-r} v_{L+r} = \sum_{r=0}^\infty \delta_r \sum_{L=0}^r \alpha_L u_{r-L} v_{r+L} = \sum_{r=0}^\infty \beta_r \delta_r.$

Of course, the most rigorous among us must be telling me that we need many convergence conditions to justify the magic trick I did above. You would be right, but this is supposed to be a general writing that I am failing to keep light. Please excuse me. I will not justify them here.

Now back to the main point. There are standard choices for $u_j,\ v_j,\ \delta_j,$ and $\gamma_j$ which makes Bailey’s lemma work. With these standard choices, we are left with free choices for $\alpha_j$ and $\beta_j$. A pair $(\alpha_j,\beta_j)$ that satisfies Bailey’s lemma is conveniently called a Bailey pair.

Bailey and later his student Lucy Slater did amazing things with this result. I am talking about hundreds of novel sum-product identities by merely applying Bailey’s lemma. Skipping ahead, two big visionaries, Peter Paule and George E. Andrews independently observed something Bailey and Slater missed completely. They observed that if you start with a Bailey Pair $(\alpha_j,\beta_j)$ then you can find a new pair $(\alpha'_j,\beta'_j)$ that is also a Bailey pair. So you can apply the lemma again and generate a new-new Bailey pair $(\alpha''_j,\beta''_j)$… That is the avalanche. You start with a snowball -the initial seed identity that defines your Bailey pair- then you apply Bailey’s lemma as many times as you like. This creates an infinite hierarchy on top of your seed identity.

Andrews-Gordon identities is one of the most famous q-series identities that builds on the Rogers-Ramanujan identities with this very lemma. These identities are the main content of Andrews’ American Mathematical Society Presidential Signature:

I am sure you also noticed Srinivasa Ramanujan‘s silhouette watching over this infinite hierarchy. He sow the case $k=2$ seed that turned into this infinite family of sum-product identities by Bailey’s lemma in the hands of Andrews.

One remark I need to make is that at each step of the application of the lemma you are summing both sides of your seed identity in a special way. It is the reason that you see an implicit $\underline{n} = (n_1,n_2,\dots,n_{k-1})$ vector on the left-hand side of the above identity. That is actually a $k-1$-fold sum.

To sum it up, Bailey’s lemma is an incredible tool. Every day there are more and more being discovered using this technique. In my recent paper with Alexander Berkovich, we reaped the seeds we sowed. For example, with a special form of Bailey’s lemma, we saw that

$\displaystyle \sum_{m,n\geq 0} \frac{q^{2m^2+6m n+6n^2}(q^3;q^3)_M}{(q;q)_m (q^3;q^3)_n(q^3;q^3)_{M-2n-m}} = \sum_{j=-M}^M q^{3j^2+j} {2M\brack M-j}_{q^3}$

(which we proved in an earlier joint paper) implies

\begin{aligned} \sum_{m,n,n_1,n_2,\dots,n_f\geq 0 } &\frac{q^{2m^2+6mn+6n^2+3(N_1^2 + N_2^2+\dots+N_f^2)} (q^3;q^3)_{n_f}}{(q;q)_m (q^3;q^3)_n (q^3;q^3)_{n_f-2n-m }(q^3;q^3)_{n_1}(q^3;q^3)_{n_2}\dots(q^3;q^3)_{n_{f-1}}(q^3;q^3)_{2n_f}}\\&\hspace{9cm} = \frac{(q^{6f+6}, -q^{3f+2}, -q^{3f+4}; q^{6f+6})_\infty}{(q^3;q^3)_\infty},\end{aligned}

for any $f=1,2,3,\dots$, where $N_i:=n_i + n_{i+1}+\dots + n_f$ with the standard definitions of q-Pochhammer symbols and q-Binomial coefficients.