5,918 research outputs found
MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms
We investigate a relationship between MacMahon's generalized sum-of-divisors
functions and Chebyshev polynomials of the first kind. This determines a
recurrence relation to compute these functions, as well as proving a conjecture
of MacMahon about their general form by relating them to quasi-modular forms.
These functions arise as solutions to a curve-counting problem on Abelian
surfaces.Comment: 6 Page
Double series representations for Schur's partition function and related identities
We prove new double summation hypergeometric -series representations for
several families of partitions, including those that appear in the famous
product identities of G\"ollnitz, Gordon, and Schur. We give several different
proofs for our results, using bijective partitions mappings and modular
diagrams, the theory of -difference equations and recurrences, and the
theories of summation and transformation for -series. We also consider a
general family of similar double series and highlight a number of other
interesting special cases.Comment: 19 page
q,k-generalized gamma and beta functions
We introduce the q,k-generalized Pochhammer symbol. We construct
and , the q,k-generalized gamma and beta fuctions, and
show that they satisfy properties that generalize those satisfied by the
classical gamma and beta functions. Moreover, we provide integral
representations for and Comment: 17 page
Partitions with fixed differences between largest and smallest parts
We study the number of partitions of with difference between
largest and smallest parts. Our main result is an explicit formula for the
generating function . Somewhat
surprisingly, is a rational function for ; equivalently,
is a quasipolynomial in for fixed . Our result generalizes to
partitions with an arbitrary number of specified distances.Comment: 5 page
A new four parameter q-series identity and its partition implications
We prove a new four parameter q-hypergeometric series identity from which the
three parameter key identity for the Goellnitz theorem due to Alladi, Andrews,
and Gordon, follows as a special case by setting one of the parameters equal to
0. The new identity is equivalent to a four parameter partition theorem which
extends the deep theorem of Goellnitz and thereby settles a problem raised by
Andrews thirty years ago. Some consequences including a quadruple product
extension of Jacobi's triple product identity, and prospects of future research
are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end
of Sec. 5, minor stylistic changes, typos eliminated. To appear in
Inventiones Mathematica
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