Vox Meditantis

Sister Mary Celine Fasenmyer (1906-1996) was an American mathematician and Catholic nun whose 1945 doctoral thesis introduced a purely algorithmic method for finding recurrence relations satisfied by hypergeometric polynomials. Her groundbreaking approach transformed what had been an art requiring mathematical intuition into a mechanical procedure that could be automated – decades before the computers that would make her method truly powerful existed. When mathematicians Doron Zeilberger and Herbert Wilf rediscovered her work in 1978, they developed it into the celebrated “WZ theory,” winning the prestigious Leroy P. Steele Prize in 1998. Today, Sister Celine’s Method underpins algorithms in Mathematica, Maple, and other computer algebra systems used daily by mathematicians worldwide, yet her foundational contribution remains largely invisible to those who benefit from it.

Her story illuminates the profound challenges faced by brilliant women whose innovations arrived before their technological moment – and who lacked the institutional power to ensure their ideas survived the decades-long journey from obscurity to recognition.

Sister Celine, thank you for joining me today from what I understand is your retirement home in Erie, Pennsylvania. When Herbert Wilf tracked you down here in 1993, you’d been away from mathematics for nearly half a century. What was it like to discover that your work had become the foundation for an entire field?

Well, it was quite extraordinary, I must say. Professor Wilf arrived here at the motherhouse asking about work I’d done as a young woman – work I’d rather assumed had been forgotten entirely. When he explained what he and Dr. Zeilberger had built upon my thesis, I felt rather like someone who’d planted a seed in 1945 and returned to find it had grown into a mighty oak. Though I must confess, I didn’t entirely understand why it had taken thirty-three years for anyone to notice that particular seed.

You grew up in Pennsylvania’s oil country during the early 1900s. What led a young woman from Crown, Pennsylvania, to pursue advanced mathematics?

Mathematics was simply the subject that made sense to me. At St. Joseph’s Academy in Titusville, the sisters would set problems that confounded my classmates, but I found myself seeing patterns others missed. Numbers behaved in orderly ways – much more orderly than people, I thought then. My father worked his own oil lease, you understand. He was a practical man who believed in education for his children, which wasn’t universal in those days. When I showed aptitude, he supported my studies, though I suspect he never imagined his daughter would spend years working with something called hypergeometric polynomials.

After high school in 1923, you spent ten years teaching before earning your bachelor’s degree. That’s an unusual path for someone who would later complete a doctorate at the University of Michigan.

Those ten years were essential, though I didn’t realise it at the time. I taught in various schools whilst studying at Mercyhurst College after it opened in 1926. Teaching forces you to understand mathematics differently – you must grasp not merely what works, but why it works, and how to explain it to minds that think differently than your own. When I eventually reached Michigan in 1942, I had spent years thinking about how mathematical concepts unfold step by step. That experience proved invaluable when I began developing what would become my algorithm.

Let’s talk about that algorithm. Can you walk us through what you discovered in your thesis about finding recurrence relations for hypergeometric polynomials?

Ah, yes. Before my work, finding pure recurrence relations for these polynomial families was rather like solving a puzzle without knowing if all the pieces were in the box. Mathematicians would try various algebraic tricks, hoping to stumble upon the right combination. It was quite inefficient and required considerable intuition – or luck.

My insight was that this process could be made entirely mechanical. If you have a hypergeometric polynomial sequence, you can write down a recurrence relation of a particular form and determine its coefficients by solving a linear system of equations. No guesswork required. The method works the same way every time, regardless of which polynomial family you’re studying – Legendre, Jacobi, Bateman’s polynomials, it matters not.

Can you give us the technical details of how this actually works?

Certainly. Suppose we have a hypergeometric polynomial sequence P_n(x) and we wish to find a pure recurrence relation of the form:

a_0(n)P_n(x) + a_1(n)P_{n-1}(x) + … + a_r(n)P_{n-r}(x) = 0

The traditional approach was to guess the form of the coefficients a_i(n) and check whether they worked. My method proceeds differently.

First, I express each polynomial in the sequence using its hypergeometric series representation. Then I assume the recurrence relation exists with unknown polynomial coefficients a_i(n). Substituting the series representations and collecting terms according to powers of x gives us a system of linear equations in the unknown coefficients.

The crucial insight is that this system always has a solution when the coefficients are polynomials of sufficiently high degree. Moreover, the degree required can be determined in advance from the parameters of the hypergeometric series. Once you solve this linear system – which involves only rational arithmetic – you have your recurrence relation.

What made this approach revolutionary?

The algorithm requires no intuition whatsoever. A person could follow the steps mechanically, without understanding the underlying mathematics, and still obtain the correct recurrence relation. This was quite different from the artistic approach that had prevailed. Professor Rainville, my thesis supervisor, found it fascinating precisely because it removed the guesswork.

Speaking of Professor Rainville, his 1960 textbook “Special Functions” included your results in two chapters. How do you feel about the way your work was presented there?

Professor Rainville was a wonderful supervisor and a gentleman. He certainly credited my work appropriately within academic circles. However, when one’s thesis results appear as chapters in another person’s textbook, the broader mathematical community often associates those techniques with the textbook’s author rather than the original researcher. This is particularly true when the author is well-established and the original work was done by a student – especially a female student who then disappeared from the research community.

I don’t believe this was intentional on Professor Rainville’s part. He was quite proud of what he called the “very pretty mathematics” in my thesis. But institutional memory works in peculiar ways, and attribution can become muddled over time.

You published just two papers after your thesis – in 1947 and 1949 – and then returned to teaching at Mercyhurst College for the rest of your career. Was this a difficult decision?

Not difficult at all. My vocation was to serve God through education and community service. The Sisters of Mercy had invested considerably in my education, and I felt called to return that investment by teaching young people. Research mathematics is a solitary pursuit, whereas teaching is fundamentally about human connection and service.

I loved working with undergraduates – watching them discover that mathematics could be beautiful, not merely difficult. Many of my students went on to successful careers in mathematics, engineering, and science. That seems to me a worthy contribution, even if it doesn’t lead to publications in prestigious journals.

Do you ever regret not pursuing research mathematics further?

Regret implies I made the wrong choice, which I did not. However, I do sometimes wonder what might have happened if I had continued. The work I’d done was quite novel, and there were certainly directions it could have been taken. But that path would have required a different sort of life – likely at a major research university, focused on individual achievement rather than community service. That simply wasn’t the life I was called to live.

When Doron Zeilberger rediscovered your thesis in 1978, thirty-three years had passed. Why do you think your work was overlooked for so long?

Several factors, I think. First, the work was ahead of its time. My algorithm was perfectly suited to computer automation, but in 1945, electronic computers barely existed. ENIAC had just been completed, but it was hardly available for mathematical research. The method’s true power could only be demonstrated once computers became common tools for mathematicians, which didn’t occur until the 1970s and 1980s.

Second, I was at Mercyhurst College, not a major research university. We had no doctoral students to carry forward the work, no research seminars where it might be discussed regularly. Small teaching colleges, however excellent their undergraduate programmes, simply don’t have the research infrastructure to keep specialised methods alive in the mathematical community’s memory.

Third, I suspect the algorithmic nature of the work counted against it initially. In 1945, mathematicians prized elegant theorems and deep theoretical insights. An algorithm – a mechanical procedure – was often viewed as mere technique rather than mathematics proper. This hierarchy began changing only with the computer revolution.

There’s also the matter of your identity as a Catholic nun. How did that affect your place in the mathematical community?

Being “Sister Celine” certainly marked me as unusual. Most mathematicians were either men or women who had chosen traditional academic careers. I was neither. My primary identity was religious, not mathematical, which meant I was somewhat outside the normal professional networks.

That said, the mathematical community was generally quite welcoming. My gender and religious vocation were far less important than whether my mathematics was sound. The difficulty was not discrimination, but rather that my life priorities and institutional commitments were different from those of career mathematicians.

Let’s talk about the moment when your work was rediscovered. What was your reaction when Zeilberger’s paper appeared?

I nearly dropped my tea when Professor Wilf first explained it to me. Here was this young Israeli mathematician – well, young to me at eighty-seven – who had taken my little algorithm and turned it into something that could prove vast classes of combinatorial identities automatically. He and Professor Wilf had developed what they called “WZ theory,” and it was winning prizes and revolutionising how mathematicians approached certain problems.

What surprised me most was how the computer had finally caught up to what I’d envisioned in 1945. My method had always been designed for mechanical execution, but it took thirty years for the machines to become sophisticated enough to demonstrate its full potential.

How do you feel about the fact that this breakthrough theory was named “WZ theory” after Wilf and Zeilberger, rather than something that honoured your original contribution?

Professors Wilf and Zeilberger were scrupulous about acknowledging my foundational role. They consistently called my original method “Sister Celine’s Technique” and made clear that their work built upon mine. I have no complaints about their conduct as scholars or as gentlemen.

However, the naming of mathematical theories follows certain conventions, and those conventions favour the individuals who develop fully mature theories rather than those who provide foundational insights. “WZ theory” reflects the men who formalised, generalised, and popularised the approach. My work provided the seed, but they grew it into something much larger.

From a practical standpoint, most mathematicians today encounter these methods through WZ theory, not through my original papers. They associate the techniques with Wilf and Zeilberger because that’s how they learned them. This is how mathematical knowledge typically develops and spreads.

Still, they won the Steele Prize – one of mathematics’ highest honours – for work that built directly on your algorithm. Does that feel unfair?

The Steele Prize recognised their achievement in creating a mature, powerful theory that has transformed how mathematicians work with combinatorial identities. Their contribution was substantial and deserving of recognition.

Do I wish there were better mechanisms for acknowledging foundational contributions, particularly when decades pass between the original insight and its full development? Certainly. But I chose a path that prioritised service over recognition. One cannot expect the rewards of academic careerism whilst living a life devoted to other principles.

Let’s talk about the technical impact of your work. Today, algorithms descended from your method are implemented in Mathematica, Maple, and other computer algebra systems. Millions of calculations rely on your approach. Did you anticipate this kind of practical impact?

Not the scale, certainly, but the principle, yes. I always knew the method was well-suited to mechanical computation. In my 1949 paper, I was quite explicit about this – I wrote that the algorithm could be “readily adapted to machine computation”. This was rather prescient, considering that electronic computers were still experimental curiosities.

What I envisioned was that once computing machines became common – and I was confident they would – mathematicians would no longer need to spend hours or days working out recurrence relations by hand. The computer could handle the tedious arithmetic while the mathematician focused on interpreting and applying the results.

Your algorithm works with hypergeometric functions, which appear throughout mathematical physics. Can you explain why these functions are so important?

Hypergeometric functions are remarkably versatile. They include as special cases many of the most important functions in mathematical analysis – Bessel functions, Legendre polynomials, Jacobi polynomials, and countless others. These functions appear naturally when solving differential equations that arise in physics, engineering, and applied mathematics.

For instance, when physicists study quantum mechanical systems, the solutions often involve hypergeometric functions. The same is true for problems in electromagnetism, fluid dynamics, and statistical mechanics. My algorithm provides a way to find systematic relationships among these functions, which can simplify calculations enormously.

Looking back on the 1940s, you developed your algorithm in a world without electronic computers, before the modern theory of algorithms existed, before computer science was even a discipline. What was it like to be working in such uncharted territory?

It was rather like working in a fog, mathematically speaking. I knew the method worked – I could execute it by hand and obtain correct results. But I had no framework for understanding why algorithmic approaches might be important or where they might lead.

The theoretical foundations for thinking about algorithms and computation were only beginning to be developed. Alan Turing’s work on computable functions was barely a decade old. The notion that mathematics itself might be mechanised was still quite radical.

What I had was a concrete technique that solved a specific class of problems efficiently. I couldn’t have anticipated that this represented an early example of what would become known as “computer algebra” or “symbolic computation.” Those concepts didn’t exist yet.

You mentioned executing the algorithm by hand. How long would it typically take you to find a recurrence relation for a polynomial family?

Oh, that depended entirely on the complexity of the polynomials involved. For something like the Legendre polynomials, perhaps a day or two of careful arithmetic. For more complicated families, it could take a week or more. The calculations were quite tedious – setting up the linear systems, solving them step by step, checking and rechecking the arithmetic.

This is precisely why the method was crying out for mechanical implementation. A modern computer can execute what took me days in a fraction of a second. The algorithm hasn’t changed fundamentally since 1945; it’s simply that the machines have finally caught up to the method.

Was there a particular moment when you realised you’d discovered something significant?

I remember working late one evening in the library at Michigan, probably in early 1945. I had been trying to find a recurrence relation for one of Bateman’s polynomial families using traditional methods and getting nowhere. Then I decided to try my systematic approach – setting up the equations mechanically rather than trying to guess the answer.

When the arithmetic worked out perfectly and I obtained a clean, simple recurrence relation, I felt a peculiar sort of satisfaction. It wasn’t the thrill of solving a difficult puzzle, but rather the quiet pleasure of discovering that a particular type of problem need never be difficult again. Any competent person could now solve such problems by following a fixed procedure.

I think I understood even then that this represented a different way of doing mathematics – less artistry, perhaps, but more reliability and much greater efficiency.

Your thesis advisor, Earl Rainville, was known for his work on differential equations and special functions. How did he react to your algorithmic approach?

Professor Rainville was wonderfully supportive, though I think he was somewhat puzzled by the implications of what I’d done. He appreciated the elegance of the method and encouraged me to develop it further. His mathematical tastes ran towards classical analysis, so an algorithmic approach was rather foreign to his usual way of thinking.

He often remarked that there was “very pretty mathematics” in my work, which I took as high praise coming from him. But I don’t think he fully grasped how the method might be generalised or automated. That required a different mathematical sensibility – one that wouldn’t emerge until computer science matured decades later.

You completed your doctorate in 1946, just as the first electronic computers were being built. Did you have any contact with the early computing community?

Not directly, no. ENIAC was unveiled at the University of Pennsylvania in February 1946, just as I was finishing my dissertation at Michigan. But the early computers were enormous, expensive, and primarily devoted to military calculations. They weren’t accessible to ordinary mathematicians for research purposes.

I was aware of the developments, of course. The mathematical community was quite excited about the potential of electronic computation. But I don’t think anyone imagined how quickly computers would become practical tools for symbolic manipulation rather than just numerical calculation.

If you had to identify the biggest obstacle to your work being recognised earlier, what would it be?

The timing, I believe. The algorithm was conceptually complete by 1945, but the technological and intellectual infrastructure needed to appreciate its significance didn’t exist yet. Electronic computers were in their infancy. The field of computer algebra wouldn’t emerge for another two decades. The mathematical community still privileged theoretical insights over algorithmic methods.

By the time those circumstances changed – by the late 1970s – I had long since left active research. My work existed only in two papers and some scattered references. Without someone actively championing it, as Professor Zeilberger eventually did, it might have remained forgotten indefinitely.

Do you have any advice for young mathematicians, particularly women, who might face similar challenges in getting their work recognised?

First, do mathematics because you love it, not because you expect recognition. The joy should come from the work itself – from understanding, from solving problems, from discovering patterns that no one has seen before.

Second, don’t underestimate the importance of community. I was fortunate to have the Sisters of Mercy supporting my education and career. Find colleagues, mentors, and institutions that value your contributions. Isolation makes it much harder for good work to survive and flourish.

Third, be patient but persistent. If your work is truly valuable, it will eventually find the recognition it deserves, though perhaps not in the timeframe you expect. Mathematics has a long memory, even when individual mathematicians do not.

Finally, remember that there are many ways to contribute to mathematics beyond publishing papers. Teaching, mentoring, and service to the mathematical community are all valuable forms of scholarship.

Looking at the current state of mathematics and computer science, what aspects would surprise the Sister Celine of 1945 most?

The sheer computational power available to ordinary mathematicians would astonish me. In 1945, I could barely imagine a machine that would execute my algorithm reliably. Today, a graduate student can verify combinatorial identities on a desktop computer that would have required months of hand calculation in my era.

But I think what would surprise me most is how algorithmic thinking has transformed mathematics itself. In 1945, algorithms were tools for solving specific problems. Today, the design and analysis of algorithms is a central mathematical discipline. Computer science has become a branch of mathematics in its own right.

The integration is so complete that young mathematicians today think algorithmically in ways that would have been quite foreign to my generation. They expect to be able to automate routine calculations and focus their human intelligence on genuinely creative aspects of problem-solving.

Any regrets about the path your life took?

None whatsoever. I lived according to my principles, served my community, and contributed what I could to mathematics and education. That my early work eventually found recognition and contributed to important developments in the field is simply a bonus – a gift I never expected to receive.

Life rarely unfolds as we anticipate. The young woman who developed that algorithm in 1945 could never have imagined she would be sitting here in 1993, discussing her work with mathematicians from around the world at the age of eighty-seven. Sometimes the most important seeds we plant are the ones we never expect to see grow.

Sister Celine, thank you for sharing your story with us today.

The pleasure has been entirely mine. And please, give my regards to all the mathematicians who are still discovering what algorithms can accomplish. They’re living in the future I helped to imagine, even if I never quite got to see it fully realised myself.

Letters and emails

We’ve selected five insightful letters and emails from our growing community, each eager to explore further with Sister Mary Celine Fasenmyer about her journey, mathematical insight, and lived experience. These thoughtful questions span continents and disciplines, reflecting sincere curiosity about her life, her contributions, and her advice for those taking on similar challenges today.

Alina Popescu, 34, computational mathematician, Cluj-Napoca, Romania
Sister Celine, I’m fascinated by the pure mechanics of your algorithm. When you were hand-calculating those linear systems in the 1940s, what was your error-checking process? Did you develop any shortcuts or verification techniques that might not have made it into your published papers? I’m curious whether those practical workarounds from the pre-computer era might still offer insights for debugging modern implementations when they produce unexpected results.

Ah, Miss Popescu, you’ve asked about the practical realities of hand computation – the part that never makes it into published papers but occupies most of one’s actual working time. I’m delighted someone wants to know about this, because error-checking was absolutely crucial when executing these calculations manually.

My primary verification method was what I called “parallel reconstruction.” Once I’d solved a linear system and obtained coefficients for a recurrence relation, I would use those coefficients to generate several terms of the polynomial sequence and compare them against the known values. If P_5(x) computed from my recurrence matched the standard tabulated value of P_5(x), that was encouraging. If P_6(x) and P_7(x) also matched, I could be reasonably confident the recurrence was correct.

But here’s the trick that saved me countless hours: I would often work with specific numerical values of x rather than the full symbolic expressions initially. For instance, I might evaluate everything at x=1 and x=2 first. The arithmetic is much simpler with concrete numbers, so I could detect errors more quickly. If the recurrence held numerically at several test points, I’d then verify it symbolically. This catches most mistakes before you’ve invested days in elaborate algebraic manipulation.

I also developed what I suppose you’d call a “modular checking” approach, though we didn’t use that terminology then. When setting up the linear system, I would work out the equations one row at a time, checking each row’s arithmetic before proceeding to the next. If I found an error later, I only needed to revisit one section rather than re-doing everything from scratch. I kept meticulous worksheets with each calculation step labelled – rather like showing your work in a student’s exercise book, but much more detailed.

Another practical matter: I would often solve the linear system twice using slightly different methods. For smaller systems, I might use direct elimination one time and then verify using a different elimination order. The two approaches should yield identical coefficients. When they didn’t – and this happened more often than I’d care to admit – I knew I’d made an arithmetic error somewhere and could work backward to locate it.

Regarding shortcuts that didn’t make it into publications, I discovered that certain patterns in the hypergeometric parameters would simplify the calculations considerably. For instance, when specific parameter relationships held, entire rows of the coefficient matrix would contain many zeros, making the linear system much easier to solve. I would check for these special cases first before launching into the general calculation. This wasn’t mathematically novel enough to merit publication, but it probably saved me weeks of computation time across various problems.

I also kept what I called my “suspicious results” notebook. If a recurrence relation came out with coefficients that seemed unusually complicated or irregular, I would flag it for extra verification. Experience taught me that the correct answer usually had a certain elegance to it – not always, mind you, but often enough that anomalous complexity served as a warning sign.

As for relevance to modern debugging, I suspect these habits remain useful even with computers doing the arithmetic. When your software produces an unexpected result, the instinct to test it against known cases, to work incrementally and check as you go, to look for patterns that suggest where errors might hide – these habits of mind transcend the particular tools we use. The computer may be faster than I ever was, but it’s just as capable of propagating a mistake if you don’t verify its work carefully.

The real skill wasn’t just executing the algorithm, you see. It was developing the judgment to know when something had gone wrong and the patience to track down exactly where. Those qualities, I believe, are as valuable now as they were in 1945.

Nathaniel Carter, 47, high school mathematics teacher, Portland, Oregon, USA
You mentioned that teaching undergraduates at Mercyhurst for three decades was deeply fulfilling, but I wonder – did you ever try to introduce your algorithmic approach to students who weren’t pursuing advanced degrees? Could a bright undergraduate, or even a determined high school student, learn to apply Sister Celine’s Method by hand? I’m thinking about how we might make powerful mathematical ideas accessible earlier in students’ education, rather than waiting until graduate school.

Thank you kindly, Mr Carter, for your thoughtful question – and for your own dedication as a teacher. It gives me great joy to hear from those working with young minds, as I considered classroom teaching not just a duty but a privilege. You ask whether my algorithmic approach was ever tried with undergraduates, or if it could be reached even by determined high school students with an appetite for challenge. Let me reflect a bit on that – thinking both in terms of practicality and, as I’ve always said, the spirit of learning.

At Mercyhurst, where I taught after completing my doctorate, we served students with a wide range of mathematical backgrounds. Many intended for careers in teaching themselves, some aiming for industry, and the rare few with true scholarly bent. In my own courses, especially for those in the upper years, I did try to bring in what you might call “real mathematics” – not merely mechanical rules, but honest-to-goodness proofs and the thrill of chasing an idea to its conclusion. My method for recurrence relations, although rooted in research, seemed to me an excellent showcase for what mathematics could achieve when guided by logic rather than guesswork alone.

When adapted to the right context, even advanced techniques can be taught in the undergraduate classroom. For example, instead of abstract polynomials with unfamiliar Greek letters, I would sometimes frame problems in terms of the binomial theorem – something they encountered in their first year. Then, by asking students to find relationships between sums or to look for patterns among coefficients, I could introduce the steps behind my method gradually. This required moving at a deliberate pace, never leaving anyone behind.

Now, addressing the possibility for high schoolers, I do believe there are young people gifted enough – and stubborn enough, in the best sense – to follow the process by hand. It would be a weighty undertaking, certainly. Much would depend on intuition with sequences and facility with arithmetic, as the calculations can grow tedious. But as my own experience in the rural schools of Pennsylvania taught me, given encouragement and a bit of healthy ambition, even the largest mountains may be climbed, one careful step at a time. I often think of Mr. Gauss, who as a schoolboy astonished his teacher by summing the first hundred numbers in moments – sometimes all one needs is the right nudge to uncover deeper ability.

Of course, the limitations of paper and pencil and the weariness that comes from rechecking arithmetic leave some at a disadvantage compared to those working with modern machines. In that era, one learned economy of thought and respect for each calculation; mistakes were not so easily mended with a keystroke. But the method can still be understood in principle, which, I believe, is more important than memorising formulas by rote.

If I had to offer advice to colleagues today, it would be this: share with your students not only finished results, but the reasoning behind them, the failures and wrong turns, the simple idea that led to something more. Invite them to play with problems, to feel the honest confusion that precedes true understanding. Let them see that even work which later becomes the domain of mathematicians and machines was, at its heart, simply the product of thoughtful labour and a desire to understand. Mathematics, after all, is a living thing. And sometimes, the spark that lights a lifelong curiosity is found in the challenge itself – the slow, careful working-out of a problem by hand, pencil smudges and all.

Lebo Maseko, 29, data scientist and education advocate, Johannesburg, South Africa
Sister Celine, when Professors Wilf and Zeilberger finally tracked you down in 1993, what did that conversation feel like on a personal level? Not just the mathematics, but the human moment – were you angry that it had taken so long? Relieved? Vindicated? I ask because many of us working in male-dominated technical fields today struggle with the tension between wanting recognition for our work and not wanting to appear too concerned with credit. How did you reconcile those feelings?

Miss Maseko, you’ve asked me about the human moment – not the mathematics, but what it felt like when those two professors arrived at the motherhouse after all those years. That’s a question that requires some unpacking, as the feelings were rather more complicated than one might expect.

When Professor Wilf first telephoned and explained who he was and why he wished to speak with me, my initial reaction was simply bewilderment. Here was a mathematician from the University of Pennsylvania – a distinguished scholar – asking about work I’d done nearly half a century earlier as a young doctoral student. My first thought, quite honestly, was that he must have confused me with someone else. When he assured me he had the right person, I felt a peculiar flutter of nervousness, as though I were being called to the principal’s office after decades away from school.

Was I angry it had taken so long? No, not angry. Anger would suggest I’d been waiting all those years expecting recognition that never came. But I hadn’t been waiting. I’d been living a full life – teaching, serving my community, participating in the daily rhythms of religious life. The work I’d done in 1945 was complete; I’d solved the problems I set out to solve, published my findings, and moved on. One doesn’t spend forty years nursing grievances over things that might have been.

Relieved? Perhaps closer to that, though relief suggests a burden lifted. It was more a quiet satisfaction – the feeling you get when you discover that something you’d made years ago, something you’d rather forgotten about, turns out to have been useful after all. Like finding that an old quilt you’d stitched has been keeping someone warm all this time without your knowing.

Vindicated? Now that’s an interesting word, and I must be careful here. Vindication implies one has been wronged and seeks redress. I don’t believe I was wronged, not in any deliberate sense. The mathematical community didn’t suppress my work or deny its validity. They simply didn’t notice it – which is quite different. In a field as vast as mathematics, with thousands of papers published yearly, some ideas take time to find their moment. Mine happened to take thirty-three years.

But if I’m being entirely truthful with you, Miss Maseko – and your question deserves nothing less – there was a small, quiet part of me that felt something like… not vindication exactly, but perhaps recognition of effort honestly given. I had worked very hard on that thesis. I had seen something others hadn’t seen, developed a method that genuinely solved a class of problems more efficiently than previous approaches. To have that confirmed by scholars who understood its significance, who could see what I’d been reaching toward – yes, that brought a particular kind of satisfaction.

The tension you describe – between wanting recognition and not wanting to appear too concerned with credit – is one I knew well, though we framed it differently in my generation. For women of my era, especially religious women, there was an expectation of humility that sometimes tipped into self-effacement. We were taught to serve without seeking praise, to work without demanding acknowledgment. These are not bad values in themselves, but they can become problematic when they’re applied selectively – when men’s contributions are celebrated whilst women’s are expected to remain modestly hidden.

I think what I learned, sitting with Professor Wilf in that retirement home parlour in 1993, was that it’s possible to hold both truths simultaneously. You can be genuinely content with a life of service and teaching, genuinely at peace with choices made for reasons of faith and community – and still feel a warm glow of satisfaction when someone recognises that your intellectual work had merit. These feelings need not cancel each other out.

My advice to you and others facing similar tensions would be this: do your work with integrity, for its own sake first. But don’t apologise for wanting that work to be seen and credited appropriately. There’s no virtue in false modesty, and there’s no vice in standing firm when attribution matters. The line between humility and erasure is thinner than many would have us believe.

Carlos Paredes, 52, historian of science, Buenos Aires, Argentina
What if ENIAC and other early computers had been available for mathematical research during your doctoral years, say 1943-1946? If you’d been able to implement and demonstrate your algorithm on a working computer whilst still at Michigan, do you think the trajectory of your career – and of computational mathematics more broadly – would have unfolded differently? Would we perhaps be calling it “Fasenmyer theory” today rather than WZ theory?

Mr Paredes, what a fascinating question to contemplate – one of those “what if” scenarios that historians love to explore. You ask what might have happened if ENIAC or similar computing machines had been available for mathematical research during my doctoral years at Michigan. It’s a tantalising thought, and I’ve wondered about it myself from time to time.

Let me begin with the practical realities. ENIAC was unveiled in February 1946, just as I was completing my dissertation. But even had it been operational a year or two earlier, access would have been extraordinarily difficult. The machine occupied an entire room at the University of Pennsylvania, required a team of specialists to operate, and was primarily devoted to military calculations – ballistics tables and the like. It wasn’t as though a graduate student from Michigan could simply telephone Philadelphia and ask to run some polynomial calculations. The very notion seems almost quaint now, but these machines were national resources, carefully guarded and allocated.

That said, let’s imagine a world where I had somehow gained access to computing resources capable of implementing my algorithm. What would have changed?

First and most obviously, I could have produced far more examples. My published papers demonstrated the method on a handful of polynomial families – Legendre, Jacobi, Bateman’s polynomials. With a computer, I might have worked through dozens or even hundreds of cases, building a comprehensive catalogue of recurrence relations. This would have made the power and generality of the approach much more evident. Instead of saying “here’s a method that works,” I could have said “here’s a method that works, and here are fifty examples proving it.”

Second, the speed and reliability of computation might have encouraged me to pursue more ambitious generalisations. Some extensions of the method that I considered but set aside as too computationally intensive – involving multiple summations or more complex parameter relationships – could have been explored. The algorithm’s true scope might have been established much earlier.

As for whether my career trajectory would have changed – that’s harder to say. Even with impressive computational results, I was still a Catholic nun whose vocation called her to teaching and community service. The Sisters of Mercy had invested in my education expecting I would return to Mercyhurst, and I had every intention of honouring that commitment. No amount of computational success would have altered those fundamental priorities.

However – and here’s where it becomes intriguing – if my thesis had included extensive computational demonstrations of the method’s power, it might have attracted more immediate attention from the mathematical community. Perhaps other researchers would have built upon it sooner, rather than the work languishing for three decades. In that case, yes, we might speak of “Fasenmyer’s algorithm” or even “Fasenmyer theory” rather than WZ theory. The foundational credit might have been harder to overlook if the work had been more visibly impactful from the start.

But I must be honest about another possibility: even with computational results, the algorithmic nature of the work might still have been undervalued in 1946. The mathematical culture of that era prized elegant theorems and deep analytical insights. Algorithms were viewed as useful tools, certainly, but not as mathematics in the highest sense. It took the computer revolution of the 1970s and 1980s to change that perspective, to make algorithmic thinking central rather than peripheral to mathematical practice.

So perhaps the real question is not whether I personally would have received more recognition, but whether the entire field of algorithmic mathematics might have developed a decade or two earlier. If my computational demonstrations had inspired others to think algorithmically about mathematical problems in the late 1940s – if that seed had been planted when computers were first becoming available – then perhaps the computer algebra revolution wouldn’t have waited until the 1970s.

That, Mr Paredes, would have been a genuine shift in mathematical history. Whether my name would be more prominently attached to it, I cannot say. But the work itself might have found its moment sooner, which is perhaps what matters most in the end.

Somchai Rattanakosin, 41, quantum physicist using special functions, Bangkok, Thailand
Sister Celine, you mentioned that hypergeometric functions appear naturally in quantum mechanics and other areas of physics. I work with these functions daily when solving Schrödinger equations for complex systems. Could you explain what drew you specifically to Bateman polynomials in your 1949 paper? Were you aware of any physical applications at the time, or was the choice purely mathematical? I’m curious whether the connection between your pure mathematical work and practical physics applications was something you considered, or whether that came as a surprise later.

Dr Rattanakosin, how wonderful to hear from someone applying these functions to actual physical problems – quantum mechanics, no less. Your question about Bateman polynomials is particularly dear to me, as that 1949 paper was my attempt to make the algorithmic method accessible to a broader mathematical audience through a concrete, worked example.

Let me be forthright: my choice of Bateman polynomials was primarily mathematical rather than motivated by physical applications. In the mid-1940s, when I was developing my method, I was working within the realm of pure mathematics – special functions, orthogonal polynomials, the beautiful machinery of hypergeometric series. My focus was on the structure of these functions themselves, the relationships among them, and the problem of finding recurrence relations efficiently.

Bateman had introduced his polynomial family in the 1930s whilst studying certain integral representations and generating functions. They possessed interesting properties – they were a generalisation of Legendre polynomials, involving two free parameters rather than one. This made them an excellent test case for my algorithm. The additional parameters meant the calculations were more involved than for simpler families, which helped demonstrate that the method could handle genuinely complicated cases. Yet they weren’t so obscure that readers would be unfamiliar with them entirely.

The Monthly was the publication I chose deliberately. The American Mathematical Monthly reached a wide audience – not just research specialists, but teachers at colleges and universities across the country. I wanted to show that finding pure recurrence relations need not be an arcane art requiring special tricks or exceptional insight. Anyone who could follow step-by-step instructions and carry out the arithmetic could apply the method successfully. Bateman’s polynomials provided the right level of complexity to make that point convincing.

Now, as to whether I was aware of physical applications at the time – here I must confess my knowledge was rather limited. I knew that Legendre polynomials arose in solving Laplace’s equation in spherical coordinates, and that various orthogonal polynomial families appeared in mathematical physics more generally. Professor Rainville had certainly discussed some of these connections in his lectures. But the specific quantum mechanical applications, the detailed ways these functions emerge when solving Schrödinger equations for complex systems – that wasn’t my area of expertise.

The connection between pure mathematical work and practical physics applications was something I appreciated in principle, though it rarely guided my day-to-day research. I took a certain pleasure in knowing that the functions I studied weren’t merely abstract curiosities, that they had honest work to do in the world. But I was fundamentally a pure mathematician by temperament. The beauty of the relationships among special functions, the elegance of a well-constructed proof – these were what drove my interest.

That said, I don’t think I ever doubted that useful mathematics would eventually find its applications, even if I couldn’t predict exactly where. History had shown this pattern repeatedly. Gauss’s work on differential geometry, developed purely for its mathematical interest, later became essential for Einstein’s general relativity. Fourier’s investigations of heat equations gave us tools now used throughout engineering and physics. Mathematics has a peculiar way of proving itself useful decades or even centuries after its initial development.

So when you tell me that you work with hypergeometric functions daily whilst solving quantum mechanical problems, that my algorithmic approach helps you find the relationships you need – yes, that brings considerable satisfaction. Not because I foresaw these particular applications, but because it confirms something I always believed: that careful mathematical work, honestly done, contributes to human knowledge in ways we cannot always anticipate.

The surprise, if any, came not from discovering that the functions had physical importance – that was already known in general terms – but from learning how central algorithmic methods would become to working scientists. In 1949, I imagined mathematicians using my technique to establish theoretical results about polynomial families. I didn’t fully grasp that physicists, engineers, and applied scientists would one day rely on automated versions of these algorithms as everyday research tools.

That transformation – from a technique for proving theorems to an essential component of computational physics – required the computer revolution I couldn’t quite envision. Your work represents that future I helped to build but never fully saw.

Reflection

Sister Mary Celine Fasenmyer passed away on 27th December 1996, at the age of ninety, just three years after Herbert Wilf’s camera captured her reflections in that Erie retirement home. In the span between her rediscovery and her death, she witnessed something rare: the chance to see her youthful work finally find its rightful place in mathematical history, to understand that the algorithm she’d painstakingly developed by hand in 1945 had become foundational infrastructure for an entire generation of computational mathematicians.

Throughout our conversation – both the main interview and the thoughtful questions from our community – certain themes emerged with striking clarity. Fasenmyer’s story is one of profound temporal displacement: brilliant work arriving decades before the world possessed the tools to appreciate it fully. Her algorithm was perfectly conceived for computer automation, yet she developed it in an era when “computer” still meant a person, usually a woman, performing calculations by hand. This mismatch between innovation and readiness created a vulnerability that allowed her contribution to slip into obscurity for thirty-three years.

Yet what stands out most powerfully is her refusal to frame this as tragedy. Where others might see erasure or injustice, Fasenmyer saw the natural unfolding of a life lived according to different priorities. Her perspective challenges our contemporary assumptions about what constitutes a successful scientific career. She chose teaching over research, community over individual recognition, service over self-promotion – and maintained, convincingly, that these were genuine choices reflecting her deepest values, not compromises forced upon her by circumstance.

This represents a subtle departure from some historical accounts that portray her primarily as a victim of her era’s limitations on women in mathematics. Whilst those limitations were undeniably real – the lack of research positions for women, the expectation that female academics would prioritise teaching, the institutional structures that kept women’s work invisible – Fasenmyer herself seemed to embrace a more nuanced view. She acknowledged the obstacles whilst also asserting her agency in choosing a path that brought her genuine fulfilment. The tension between these truths deserves our careful attention.

The historical record contains notable gaps and uncertainties. We know relatively little about her day-to-day life at Mercyhurst, her relationships with students, or the specific ways she integrated her mathematical identity with her religious vocation. Her choice to publish only two papers after her dissertation remains somewhat mysterious – was this purely a matter of institutional expectations and limited time, or were there other factors at play? Did she attempt to continue research and encounter barriers, or did she genuinely feel the work was complete? These questions resist easy answers.

What we do know with certainty is the extraordinary afterlife of her work. When Doron Zeilberger rediscovered her 1945 thesis in 1978, he immediately recognised its significance and began the work of extending and formalising her approach. Collaborating with Herbert Wilf at the University of Pennsylvania, Zeilberger developed what became known as WZ theory – a powerful framework for proving and discovering hypergeometric identities that won them the prestigious Leroy P. Steele Prize in 1998. Today, implementations of Sister Celine’s Method and its descendants appear in Mathematica, Maple, and other computer algebra systems used by thousands of researchers worldwide. Every time a physicist verifies a combinatorial identity arising from quantum mechanics, every time a mathematician automates a tedious summation, they are benefiting from the algorithmic insight Fasenmyer articulated eight decades ago.

The questions posed by our community – from Alina Popescu’s interest in error-checking techniques to Somchai Rattanakosin’s inquiries about physical applications – revealed dimensions of Fasenmyer’s work that rarely appear in formal mathematical histories. Her practical wisdom about hand calculation, her pedagogical instincts about making advanced methods accessible, her philosophical reflections on recognition and attribution – these human elements enrich our understanding of how mathematical knowledge actually develops and circulates.

Lebo Maseko’s question about the emotional reality of delayed recognition prompted perhaps Fasenmyer’s most revealing response. Her acknowledgment that one can simultaneously feel at peace with one’s choices and still experience quiet satisfaction when recognition finally arrives speaks to the complex negotiations women in male-dominated fields have always faced. The expectation of self-effacing humility, the cultural pressure to minimise one’s own achievements, the fine line between legitimate modesty and complicit erasure – these remain pressing concerns for women in STEM today.

The parallels between Fasenmyer’s era and our own are both encouraging and sobering. Modern computer algebra has matured into a vibrant field where algorithmic thinking is celebrated rather than dismissed as “mere technique.” Women mathematicians now hold positions at major research universities, though still not in proportions approaching parity. Attribution practices have improved, yet foundational contributions – particularly those by women and marginalised groups – still risk being absorbed into later frameworks without proper acknowledgment. The WZ naming convention itself exemplifies this pattern: whilst Wilf and Zeilberger were scrupulous about crediting Fasenmyer, the theory’s name ensures that most users associate it with the men who formalised it rather than the woman who originated it.

For young women pursuing scientific careers today, Fasenmyer’s story offers both inspiration and caution. Her intellectual achievement was undeniable – she saw something no one else had seen, developed a genuinely novel approach, and executed it with rigorous precision. No amount of institutional marginalisation or delayed recognition can diminish that accomplishment. Yet her visibility depended entirely on the advocacy of later researchers who chose to seek her out, credit her properly, and champion her legacy. Without Zeilberger’s curiosity and Wilf’s dedication to honouring her contribution, Fasenmyer might have died completely unknown, her algorithm attributed to others or lost entirely.

This underscores the critical importance of active mentorship, deliberate citation practices, and institutional structures that preserve and celebrate women’s contributions in real time rather than relying on future rediscovery. We cannot afford to lose another thirty years of brilliant work to obscurity. Every young mathematician whose thesis languishes unread, every algorithmic insight dismissed as insufficiently theoretical, every teaching-focused faculty member whose research contributions go unrecognised – these represent not just individual losses but collective impoverishment of the field.

Fasenmyer’s legacy also reminds us that impact and recognition operate on different timescales. The work that seemed most important in 1945 – the specific recurrence relations she derived for particular polynomial families – has been superseded by more general methods. But the underlying insight – that finding recurrence relations could be transformed from an art into an algorithm – proved far more durable and influential than anyone, including Fasenmyer herself, could have anticipated. This suggests a humbling lesson about the unpredictability of mathematical progress and the wisdom of supporting diverse approaches and ideas, even when their ultimate significance isn’t immediately apparent.

As we close this conversation, I find myself thinking about all the Sister Celines who never got their 1993 interview, whose work disappeared without champions to retrieve it, whose names we’ll never know because the trail went cold decades ago. Fasenmyer was fortunate – if we can call a thirty-three-year delay fortunate – to have lived long enough to see her vindication. How many others weren’t so lucky?

The challenge for those of us working in mathematics and STEM today is to ensure we don’t create another generation of invisible contributors. This means looking beyond prestigious institutions to find important work. It means valuing teaching and service alongside research productivity. It means questioning whose ideas get algorithmic names and whose become absorbed into infrastructure. It means actively seeking out the voices we might otherwise miss.

Sister Mary Celine Fasenmyer wrote the algorithm before computers existed to run it. Now, in an age where those computers execute her method millions of times daily, we owe her not just historical acknowledgment but a commitment to ensuring that the next woman who sees something no one else has seen won’t have to wait half a lifetime for the world to notice.

Who have we missed?

This series is all about recovering the voices history left behind – and I’d love your help finding the next one. If there’s a woman in STEM you think deserves to be interviewed in this way – whether a forgotten inventor, unsung technician, or overlooked researcher – please share her story.

Email me at voxmeditantis@gmail.com or leave a comment below with your suggestion – even just a name is a great start. Let’s keep uncovering the women who shaped science and innovation, one conversation at a time.

Editorial Note: This interview is a dramatised reconstruction created for educational and illustrative purposes. Sister Mary Celine Fasenmyer passed away in December 1996, making a contemporary conversation impossible. The dialogue, responses, and perspectives presented here have been carefully constructed based on historical sources including her published mathematical papers (1947, 1949), biographical accounts, the 1993 video interview conducted by Herbert Wilf, and documented facts about her life, work, and the mathematical context of her era.

Whilst we have striven for historical accuracy in representing her mathematical contributions, career trajectory, and the institutional landscape she inhabited, the specific words, reflections, and personal anecdotes attributed to Sister Celine in this piece are necessarily interpretative. We have drawn upon period-appropriate language, the known culture of mid-twentieth-century American Catholic religious communities, and the documented attitudes of women mathematicians of her generation to create what we hope is a plausible and respectful representation of how she might have responded to such questions.

The supplementary questions from our fictional community members represent the kinds of thoughtful inquiries that scholars, educators, and STEM professionals today might genuinely wish to pose. Sister Celine’s responses to these questions are likewise imagined, though grounded in what we know of her work, values, and the choices she made throughout her life. This approach allows us to explore dimensions of her experience – the emotional reality of delayed recognition, the practical techniques of hand calculation, the philosophical tensions between service and recognition – that formal historical records often leave unexamined.

Our aim is not to deceive, but to illuminate: to make Sister Mary Celine Fasenmyer’s remarkable story more accessible and resonant for contemporary readers whilst honouring the historical record that makes such reconstruction possible.

Bob Lynn | © 2025 Vox Meditantis. All rights reserved.