Results 11  20
of
41
Some more identities of RogersRamanujan type
 Ramanujan J
"... Abstract. In this we paper we prove several new identities of the RogersRamanujanSlater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including seriesseries identities, Bailey pairs, a theorem of Watson on basic hypergeometric se ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. In this we paper we prove several new identities of the RogersRamanujanSlater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including seriesseries identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods. 1. introduction The most famous of the “qseries=product ” identities are the RogersRamanujan identities: ∞ ∑ qn2 ∞ ∏ 1 (q; q)n (1 − q5j+1)(1 − q5j+4),
Some new Transformations for Bailey pairs and WPBailey Pairs
 Central European Journal of Mathematics
"... Abstract. We derive several new transformations relating WPBailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert se ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. We derive several new transformations relating WPBailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series. 1.
SOME IMPLICATIONS OF THE WPBAILEY TREE
"... Abstract. We consider a special case of a WPBailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WPBailey pairs, and use them to derive some new transformations for basic hypergeometric series. Finally, we b ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. We consider a special case of a WPBailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WPBailey pairs, and use them to derive some new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WPBailey pairs (αn(a, k), βn(a, k)), in which αn(a, k) is independent of k, for generalizations of identities of the RogersRamanujan type. 1.
Asymptotics for rank and crank moments
 Bull. London Math. Soc
"... Abstract. Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to two of the authors that re ned a conjecture of Garvan. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to two of the authors that re ned a conjecture of Garvan. Garvan's original conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The re ned version provides precise asymptotic estimates for both the moments and their di erences. Our proof uses the HardyRamanujan circle method, multiple sums of Bernoulli polynomials, and the theory of quasimock theta functions.
Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation
"... Abstract. We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized twodimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization b ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Abstract. We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized twodimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik’s kpercolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Froböse percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without kgaps, which are also related to certain infinite probability processes relevant to these percolation models. One of our key technical probability results is also of independent interest. We prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no “kgap ” patterns, which interpolates the general Markov chain solution that arises in the case that all of the probabilities are equal.
qORTHOGONAL POLYNOMIALS, ROGERSRAMANUJAN IDENTITIES, AND MOCK THETA FUNCTIONS
"... In honor of Professor A.A Karatsuba’s 75th birthday Abstract. In this paper, we examine the role that qorthogonal polynomials can play in the application of Bailey pairs. The use of specializations of qorthogonal polynomials reveals new instances of mock theta functions. 1. ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In honor of Professor A.A Karatsuba’s 75th birthday Abstract. In this paper, we examine the role that qorthogonal polynomials can play in the application of Bailey pairs. The use of specializations of qorthogonal polynomials reveals new instances of mock theta functions. 1.
kRUN OVERPARTITIONS AND MOCK THETA FUNCTIONS
"... Abstract. In this paper we introduce krun overpartitions as natural analogs to partitions without ksequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of results for krun overpartitions, beginning with ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. In this paper we introduce krun overpartitions as natural analogs to partitions without ksequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of results for krun overpartitions, beginning with a double summation qhypergeometric series representation for the generating functions. In the special case of 1run overpartitions we further relate the generating function to one of Ramanujan’s mock theta functions. Finally, we describe the relationship between krun overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of krun overpartitions of size n.
Superconformal algebras and mock theta functions
 J. Phys. A: Math. Theor
"... Abstract. It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been develo ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been developed to analyze modular properties of mock theta functions. We consider the case of N = 4 superconformal algebra at general levels and obtain the decomposition of characters of BPS representations into a sum of simple Jacobi forms and an infinite series of nonBPS representations. We apply our method to study elliptic genera of hyperKähler manifolds in higher dimensions. In particular we determine the elliptic genera in the case of complex 4 dimensions of the Hilbert scheme of points on K3 surfaces K [2] and complex tori A [[3]]. 1.
THE BAILEY CHAIN AND MOCK THETA FUNCTIONS
"... Abstract. We use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of qhypergeometric multisums which are mock theta functions (in the modern sense). We also prove identities involving some of these multisums and certain classical mock theta funct ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract. We use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to explicitly construct families of qhypergeometric multisums which are mock theta functions (in the modern sense). We also prove identities involving some of these multisums and certain classical mock theta functions.