Frank Garvan - Profile on Academia.edu (original) (raw)

Papers by Frank Garvan

Advances in Applied Mathematics, Jan 1, 2008

Let π denote a partition into parts λ 1 ≥ λ 2 ≥ λ 3 . . .. In a 2006 paper we defined BG-rank(π) as

Journal of Combinatorial Theory, Series A, Jan 1, 2002

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new poly... more We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's "modular" partitions with modulus 2. This way we find a new combinatorial proof of Gauss's famous identity. We give a direct combinatorial proof that for n > 1 the partitions of n with crank k are equinumerous with partitions of n with crank −k.

Constructive Approximation, 1993

The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iteration a.+ 1 = (... more The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iteration a.+ 1 = (a. + bn)/2 and b.+ 1 = axfa~,b, with a0:= 1 and b 0 := x. The common limit is 2F1( 89 89 1; 1 -x2) -1 and the convergence is quadratic.

On the Andrews-Stanley refirnement of Ramanujan''s congruence modulo 5 and generalization

Hypergeometric analogues of the arithmetic-goemetirc mean iteration| NOVA. The University of Newcastle's Digital Repository

Cubic analogues of the Jacobian theta function θ (z, q)| NOVA. The University of Newcastle's Digital Repository

Some cubic modular identities of Ramanujan| NOVA. The University of Newcastle's Digital Repository

MR2177487 (2006h: 05023) 05A17

Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E

ABSTRACT We continue to investigate spt-type functions that arise from Bailey pairs. We prove sim... more ABSTRACT We continue to investigate spt-type functions that arise from Bailey pairs. We prove simple Ramanujan type congruences for these functions which can be explained by a spt-crank-type function. The spt-crank-type functions are constructed by adding an extra variable zzz into the generating functions. We find these generating functions to have interesting representations as either infinite products or as Hecke-Rogers-type double series. These series reduce nicely when zzz is a certain root of unity and allow us to deduce congruences. Additionally we find dissections when zzz is a certain root of unity to explain the congruences. Our double sum and product formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.

Contemporary Mathematics, 1994

There are four values of s for which the hypergeometric function 2 F 1 ( 1 2 − s, 1 2 + s; 1; ·) ... more There are four values of s for which the hypergeometric function 2 F 1 ( 1 2 − s, 1 2 + s; 1; ·) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions θ 3 (q), θ 4 (q) and is related to the arithmetic-geometric mean iteration of Gauss and Legendre. Analogues of the arithmetic-geometric mean are given for the remaining cases. The case s = 1 6 and its relationship to the work of Ramanujan is highlighted. The work presented includes various pieces of joint work with combinations of the following:

This is a survey about one of the most important achievements in optimization in Banach space the... more This is a survey about one of the most important achievements in optimization in Banach space theory, namely, James' weak compactness theorem, its relatives and its applications. We present here a good number of topics related to James' weak compactness theorem and try to keep the technicalities needed as simple as possible: Simons' inequality is our preferred tool. Besides the expected applications to measures of weak noncompactess, compactness with respect to boundaries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons' inequality on unbounded sets that appear as the epigraph of some fixed function f . Applications to variational problems for f and to risk measures associated with its Fenchel conjugate f * are studied.

HECKE-TYPE CONGRUENCES FOR ANDREWS' SPT-FUNCTION MODULO 16 AND 32

International Journal of Number Theory, 2014

ABSTRACT Inspired by recent congruences by N. Andersen [Int. J. Number Theory 9, No. 3, 713–728 (... more ABSTRACT Inspired by recent congruences by N. Andersen [Int. J. Number Theory 9, No. 3, 713–728 (2013; Zbl 1286.11162)] with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author [Congruences for Andrews’ spt -function modulo 32760 and extension of Atkin’s Hecke-type partition congruences. Borwein, Jonathan M. (ed.) et al., Number theory and related fields. In memory of Alf van der Poorten. Springer Proc. Math. Stat. 43, 165–185 (2013; Zbl 1286.11164), Trans. Am. Math. Soc. 364, No. 9, 4847–4873 (2012; Zbl 1286.11165)] for the function spt (n). We show that a normalized form of the generating function of spt (n) is an eigenform modulo 32 for the Hecke operators T(ℓ 2 ) for primes ℓ≥5 with ℓ≡1,11,17,19(mod24), and an eigenform modulo 16 for ℓ≡13,23(mod24).

Acta Arithmetica, 2014

showed that the generating functions of the sptoverpartition functions spt (n), spt 1 (n), spt 2 ... more showed that the generating functions of the sptoverpartition functions spt (n), spt 1 (n), spt 2 (n), and M2spt (n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang [7] defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences mod 5 and 7 for spt (n). Chen, Ji, and Zang [17] were able to define this spt-crank in terms of ordinary partitions. In this paper we define spt-cranks in terms of vector partitions that combinatorially explain the known simple congruences for all the spt-overpartition functions as well as new simple congruences. For all the overpartition functions except M2spt (n) we are able to define the spt-crank purely in terms of marked overpartitions. The proofs of the congruences depend on Bailey's Lemma and the difference formulas for the Dyson rank of an overpartition [24] and the M 2 -rank of a partition without repeated odd parts .

Transactions of the American Mathematical Society, 1994

There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: 4 3 = 4 4 + ... more There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: 4 3 = 4 4 + 4 2 . It is

Combinatorial interpretations of congruences for the spt-function

The Ramanujan Journal, 2012

ABSTRACT Let spt(n) denote the total number of appearances of the smallest parts in all the parti... more ABSTRACT Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. In 1988, the second author gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujan-type congruences for the spt-function mod 5, 7 and 13. We give new combinatorial interpretations of the spt-congruences mod 5 and 7. These are in terms of the same crank but for a restricted set of vector partitions. The proof depends on relating the spt-crank with the crank of vector partitions and the Dyson rank of ordinary partitions. We derive a number of identities for spt-crank modulo 5 and 7. We prove the surprising result that all the spt-crank coefficients are nonnegative.

Preface

The Ramanujan Journal, 2013

We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained... more We show how Rank-Crank type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank-Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson respectively. The first author's Lambert series identities are common generalizations. We also show how Atkin and Swinnerton-Dyer's proof using elliptic functions can be extended to prove these generalized Lambert series identities.

Mathematics of Computation, 1990

Journal of Symbolic Computation, 1995

Recently Berndt, Bhargava and Garvan were able to prove all of Ramanujan's results in the noteboo... more Recently Berndt, Bhargava and Garvan were able to prove all of Ramanujan's results in the notebooks on his theories of elliptic functions to alternative bases. In this paper we show how we used MAPLE to understand, prove and generalize some of Ramanujan's results.