arXiv subjects

George E. Andrews

Publications and source records attributed to George E. Andrews.

46 records · Page 3Linked to original sources

arXiv2011

The Jacobi-Stirling Numbers

The Jacobi-Stirling numbers were discovered as a result of a problem involving the spectral theory of powers of the classical second-order Jacobi differential expression. Specifically, these numbers are the coefficients of integral composite powers of the Jacobi expression in Lagrangian symmetric form. Quite remarkably, they share many properties with the classical Stirling numbers of the second kind which, as shown in LW, are the coefficients of integral powers of the Laguerre differential expression. In this paper, we establish several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations thereby extending and supplementing known contributions to the literature of Andrews-Littlejohn, Andrews-Gawronski-Littlejohn, Egge, Gelineau-Zeng, and Mongelli.

math.CO↗

arXiv2010

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.

math.NT↗

arXiv2005

On the Number of Distinct Multinomial Coefficients

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials.

math.CO↗

arXiv2004

Baxter Algebras and Hopf Algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

math.RA↗

arXiv2003

A new partition identity coming from complex dynamics

We present a new identity involving compositions (i.e. ordered partitions of natural numbers). The Formula has its origin in complex dynamical systems and appears when counting, in the polynomial family $\{f_c:z \mapsto z^d + c \}$, periodic critical orbits with equivalent itineraries. We give two different proofs of the identity; one following the original approach in dynamics and another with purely combinatorial methods.

math.CO↗

arXiv2003

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.

math.CO↗

arXiv2001

The WP - Bailey Tree and its Implications

Our object is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. We begin by observing how the WP-Bailey tree naturally entails a finite number of classical q-hypergeometric transformation formulas. We then show how to move beyond this closed set of results and in the process we explicate heretofore mysterious identities of D. Bressoud. Next, we use WP-Bailey pairs to provide a new proof of recent formula of A. Kirillov. Finally, we discuss the relation between our approach and that of W. Burge.

math.CO↗

arXiv2000

A four parameter generalization of Gollnitz's (BIG) partition theorem

We announce a new four parameter partition theorem from which the (big) theorem of Gollnitz follows by setting any one of the parameters equal to 0. This settles a problem of Andrews who asked whether there exists a result that goes beyond the partition theorem of Gollnitz. We state a four parameter q-series identity (key identity) which is the generating function form of this theorem. In a subsequent paper, the proof of the new four parameter key identity will be given.

math.CO↗

arXiv2000

ad-nilpotent $\frak b$-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number $C_n$. These (q,t)-Catalan numbers count on the one hand ad-nilpotent ideals with respect to dimension and class of nilpotence, and on the other hand admit interpretations in terms of natural statistics on Dyck paths.

math.RA↗

arXiv1998

An A$_2$ Bailey lemma and Rogers--Ramanujan-type identities

Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A_2 version of the classical Bailey lemma. We apply our result, which is distinct from the A_2 Bailey lemma of Milne and Lilly, to derive Rogers-Ramanujan-type identities for characters of the W_3 algebra.

math.QA↗