We provide a proof of the Borwein Conjecture, which states that the coefficients of (q;q)3n/(q3;q3)n have a repeating sign pattern of +−−, using analytic methods. The proof is done by utilizing an ...expansion by Andrews to extract the “main part” of the coefficients, and then bound the various “error terms” that arise from this expansion.
As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. ...By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.
A general class of Schmidt theorems Andrews, George; Keith, William
Journal of number theory,
June 2023, 2023-06-00, Letnik:
247
Journal Article
Recenzirano
Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with ...colored partitions satisfying certain specifications may be a generally useful tool for establishing sum-product q-series identities, several examples of which are given.
We establish the asymptotic normality of the dimension of large‐size random Fishburn matrices by a complex‐analytic approach. The corresponding dual problem of size distribution under large dimension ...is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to {0,1}$$ \left\{0,1\right\} $$ or nonnegative integers ℕ0$$ {\mathbb{N}}_0 $$. The analytic saddle‐point approach we apply, based on a powerful transformation for q$$ q $$‐series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.
Generalizations of POD and PED partitions Ballantine, Cristina; Welch, Amanda
Discrete mathematics,
November 2024, 2024-11-00, Letnik:
347, Številka:
11
Journal Article
Recenzirano
Partitions with even (respectively odd) parts distinct and all other parts unrestricted are often referred to as PED (respectively POD) partitions. In this article, we generalize these notions and ...study sets of partitions in which parts with fixed residue(s) modulo r are distinct while all other parts are unrestricted. We also study partitions in which parts divisible by r (respectively congruent to r modulo 2r) must occur with multiplicity greater than one.
Encouraged by Bauer's series and Ramanujan's formulae for 1/π, we find three double series for π. One of them is∑k=1∞(8k+1)(12)k(14)k(34)kk!39k∑i=1k{1(2i−1)2−136i2}=3π54. We also establish ...q-analogues of the three double series in this paper.
We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers–Ramanujan type identities for alternating knots as conjectured by ...Garoufalidis, Lê and Zagier.
An expansion formula into bivariate formal power series is established that implies, as consequences, three remarkable identities of partial theta functions due to Warnaar (2003), Schilling–Warnaar ...(2002) and Andrews–Warnaar (2007).
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