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Andrew V. Sills

Publications and source records attributed to Andrew V. Sills.

51 records · Page 3Linked to original sources
arXiv2018

On Series Expansions of Capparelli's Infinite Product

Using Lie theory, Stefano Capparelli conjectured an interesting Rogers-Ramanujan type partition identity in his 1988 Rutgers Ph.D. thesis. The first proof was given by George Andrews, using combinatorial methods. Later, Capparelli was able to provide a Lie theoretic proof. Most combinatorial Rogers-Ramanujan type identities (e.g. the Göllnitz-Gordon identities, Gordon's combinatorial generalization of the Rogers-Ramanujan identities, etc.) have an analytic counterpart. The main purpose of this paper is to provide two new series representations for the infinite product associated with Capparelli's conjecture. Some additional related identities, including new infinite families are also presented.

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arXiv2018

Identities of the Rogers--Ramanujan--Bailey Type

A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W. N. Bailey in his paper, "Identities of the Rogers-Ramanujan type," [Proc. London Math. Soc. (2), 50 (1949), 421--435]. This leads to the derivation of a number of elegant new Rogers-Ramanujan type identities.

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arXiv2018

$q$-Difference Equations and Identities of the Rogers--Ramanujan--Bailey Type

In a recent paper, I defined the "standard multiparameter Bailey pair" (SMPBP) and demonstrated that all of the classical Bailey pairs considered by W.N. Bailey in his famous paper (\textit{Proc. London Math. Soc. (2)}, \textbf{50} (1948), 1--10) arose as special cases of the SMPBP. Additionally, I was able to find a number of new Rogers-Ramanujan type identities. From a given Bailey pair, normally only one or two Rogers-Ramanujan type identities follow immediately. In this present work, I present the set of $q$-difference equations associated with the SMPBP, and use these $q$-difference equations to deduce the complete families of Rogers-Ramanujan type identities.

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arXiv2018

Disturbing the Dyson Conjecture (in a GOOD Way)

We present a case study in {\it experimental} yet {\it rigorous} mathematics by describing an algorithm, fully implemented in both Mathematica and Maple, that {\it automatically conjectures}, and then {\it automatically proves}, closed-form expressions extending Dyson's celebrated constant term conjecture.

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arXiv2018

The exponential distribution analog of the Grubbs--Weaver method

Grubbs and Weaver (JASA 42 (1947) 224--241) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically for samples of size up to 100, and conjectured that their "rule of eights" is valid for all sample sizes. Here we examine the analogous problem where the underlying distribution is exponential and find that a "rule of fours" yields optimality and prove the result rigorously.

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arXiv2018

On identities of the Rogers--Ramanujan type

A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated $q$-difference equations points to a connection with a mild extension of Gordon's combinatorial generalization of the Rogers-Ramanujan identities (\emph{Amer. J. Math.}, 83 (1961), 393--399). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater's list (\emph{Proc. London Math. Soc. (2)} 54 (1952), 147--167), as well as the new identities presented here. A list of 26 new double sum--product Rogers-Ramanujan type identities are included as an appendix.

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arXiv2018

A classical $q$-hypergeometric approach to the standard $A_2^{(2)}$ standard modules

This is a written expansion of the talk delivered by the author at the International Conference on Number Theory in Honor of Krishna Alladi for his 60th Birthday, held at the University of Florida, March 17--21, 2016. Here we derive Bailey pairs that give rise to Rogers--Ramanujan type identities which are the principally specialized character of the $A_2^{(2)}$ standard module $(\ell - 2i+2)Λ_0 + (i-1)Λ_1$ for any level $\ell$, and $i=1, 2$.}

math.CA↗
arXiv2018

A formula for the partition function that "counts"

We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then study $D(n,k)$ as a quasipolynomial. We consider the natural polynomial approximation $\tilde{D}(n,k)$ to the quasipolynomial representation of $D(n,k)$. Numerically, the sum $\sum_{1\leq k \leq \sqrt{n}} \tilde{D}(n,k)$ appears to be extremely close to the initial term of the Hardy--Ramanujan--Rademacher convergent series for $p(n)$.

math.CO↗
arXiv2014

Ghost series and a motivated proof of the Andrews-Bressoud identities

We present what we call a "motivated proof" of the Andrews-Bressoud partition identities for even moduli. A "motivated proof" of the Rogers-Ramanujan identities was given by G. E. Andrews and R. J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a "motivated proof" of the somewhat analogous Göllnitz-Gordon-Andrews identities has been found. In the present work, we introduce "shelves" of formal series incorporating what we call "ghost series," which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that these new series will provide insight into the ongoing program of vertex-algebraic categorification of the various "motivated proofs."

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arXiv2013

Compositions, Partitions, and Fibonacci Numbers

A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions of n + 1 into parts greater than one. Some commentary about the history of partitions and compositions is provided.

math.CO↗
arXiv2013

A Rademacher Type Formula For Partitions and Overpartitions

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of $n$ and the Zuckerman formula for the Fourier coefficients of $\vartheta_4(0\vert τ)^{-1}$ is presented.

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arXiv2011

Rademacher's infinite partial fraction conjecture is (almost certainly) false

In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most $N$ parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made toward proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we provide a fast algorithm for calculating the Rademacher coefficients, a large amount of data, direct formulas for certain collections of Rademacher coefficients, and overwhelming evidence against the truth of the conjecture. While the limits of the sequences of Rademacher coefficients do not exist (the sequences oscillate and attain arbitrarily large positive and negative values), the sequences do get very close to Rademacher's conjectured limits for certain (predictable) indices in the sequences.

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