IdentityFinder and some new identities of Rogers-Ramanujan type

The Rogers-Ramanujan identities and various analogous identities (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with integer partitions. These identities (written in generating function form) are typically of the form "product side" equals "sum side," with the product side enumerating partitions obeying certain congruence conditions and the sum side obeying certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic computation to generate various such sum sides and then use Euler's algorithm to see which of them actually do produce elegant conjectured product sides. We not only rediscover many of the known identities but also discover some apparently new ones, as conjectures supported by strong mathematical evidence.Comment: 10 page

On a pair of identities from Ramanujan's lost notebook

Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.Comment: 17 page

A variant of IdentityFinder{\texttt{IdentityFinder}} and some new identities of Rogers-Ramanujan-MacMahon type

We report on findings of a variant of ${\texttt{IdentityFinder}}$ - a Maple program that was used by two of the authors to conjecture several new identities of Rogers-Ramanujan kind. In the present search, we modify the parametrization of the search space by taking into consideration several aspects of Lepowsky and Wilson's $Z$-algebraic mechanism and its variant by Meurman and Primc. We search for identities based on forbidding the appearance of "flat" partitions as sub-partitions. Several new identities of Rogers-Ramanujan-MacMahon type are found and proved.Comment: subsumes arXiv:1703.0471

One-Parameter Generalizations of Rogers-Ramanujan Type Identities

Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many new one-parameter generalizations of the Rogers-Ramanujan type identities, such as a generalization of the analytic versions of the first and second G\"{o}llnitz-Gordon partition identities, and generalizations of the first, second, and third Rogers-Selberg identities

The singular support of the Ising model

We prove a new Fermionic quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson-Slater $q$-series identity of Rogers-Ramanujan type and to Nahm sums for the matrix $\left( \begin{smallmatrix} 8 & 3 \\ 3 & 2 \end{smallmatrix} \right)$. We find, as consequences, an explicit monomial basis for the Ising model, and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identities of the Rogers-Ramanujan-Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge $1/2$. We give a combinatorial interpretation to the identity associated with the vacuum module.Comment: 19 page

A solution of Sun's $520 challenge concerning 520/pi

We prove a Ramanujan-type formula for $520/\pi$ conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for $1/\pi$.Comment: 17 page

Searching for modular companions

In this note, we report on the results of a computer search performed to find possible modular companions to certain $q$-series identities and conjectures. For the search, we use conditions arising from the asymptotics of Nahm sums. We focus on two sets of identities: Capparelli's identities, and certain partition conjectures made by the author jointly with Matthew C. Russell.Comment: 9 page

Mathematics in the Age of the Turing Machine

The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.Comment: 45 pages. This article will appear in "Turing's Legacy," ASL Lecture Notes in Logic, editor Rodney G. Downe

Rogers-Ramanujan-Slater Type Identities

In this survey article, we present an expanded version of Lucy Slater’s famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater’s papers, and older identities (such as those in Ramanujan’s lost notebook) which were not included in Slater’s papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities

qFunctions -- A Mathematica package for qq-series and partition theory applications

We describe the qFunctions Mathematica package for $q$-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for $q$-shift equations and recurrences for given $q$-series and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on $q$-differential, $q$-shift equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. Here, we also extend the classical method of the weighted words approach. Moreover, qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions.Comment: 17 page