Media

Areas of Expertise

Combinatorics

Number Theory

Education

University of Kentucky

Ph.D.

Mathematics

2002

The Pennsylvania State University

M.A.

Mathematics

1994

Rutgers University

B.A.

Mathematics

1989

Affiliations

• American Mathematical Society
• Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
• American Statistical Association

Links

• Scholar Profile
• Personal Website

Articles

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

Journal of Combinatorial Theory, Series A

Kanade, Shashank; Lepowsky, James; Russell, Matthew C.; Sills, Andrew V.

2017 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.

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

Journal of Difference Equations and Applications

Andrew V. Sills & Doron Zeilberger

2013 In his book 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 towards proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question.

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Ramanujan-Slater type identities related to the moduli 18 and 24

Journal of Mathematical Analysis and Applications

Andrew V. Sills & James McLaughlin

2008 We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.

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Identities of the Rogers-Ramanujan-Slater type

Int. J Number Theory 3

Andrew V. Sills

2007 It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers–Ramanujan identities (L. J. Slater, Further identities of the Rogers–Ramanujan type, Proc. London Math Soc. (2)54 (1952) 147–167) can be easily derived using just three multiparameter Bailey pairs and their associated q-difference equations. As a bonus, new Rogers–Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities.

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Disturbing the Dyson conjecture, in a generally GOOD way

Journal of Combinational Theory, Series A

Andrew V. Sills

2006 In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding q-analogs are supplied. Finally, perturbed versions of the q-Dixon summation formula are presented.

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Finite Rogers-Ramanujan type identities

Electron J. Combin

Andrew V. Sills

2003 Polynomial generalizations of all 130 of the identities in Slater's list of identities of the Rogers-Ramanujan type are presented. Furthermore, duality relationships among many of the identities are derived. Some of the these polynomial identities were previously known but many are new.

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