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Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False, Andrew Sills, Doron Zeilberger
Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False, Andrew Sills, Doron Zeilberger
Department of Mathematical Sciences Faculty Publications
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. In this paper, we present a recurrence (alias difference equation) which provides a fast algorithm for calculating the Rademacher …
Rademacher-Type Formulas For Restricted Partition And Overpartition Functions, Andrew Sills
Rademacher-Type Formulas For Restricted Partition And Overpartition Functions, Andrew Sills
Department of Mathematical Sciences Faculty Publications
A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.
A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills
A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills
Department of Mathematical Sciences Faculty Publications
We provide a bijective map from the partitions enumerated by the series side of the Rogers–Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers–Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.