The generalized Borwein conjecture. I. The Burge transform

We give simple elementary proofs of Bressoud's and Schur's polynomial versions of the Rogers-Ramanujan identities

Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be applied to prove many instances of Bressoud's generalized Borwein conjecture.

We revisit Bressoud's generalized Borwein conjecture. Making use of new positivity-preserving transformations for q-binomial coefficients we establish the truth of infinitely many cases of the Bressoud conjecture. In addition, we prove new bounded version of Lebesgue's identity and of Euler's Pentagonal Number Theorem. Finally, we discuss new companions to Andrews-Gordon mod 21 and Bressoud mod 20 identities.

We contribute to the zoo of dubious identities established by J.M. and P.B. Borwein in their 1992 paper, "Strange Series and High Precision Fraud" with five new entries, each of a different variety than the last. Some of these identities are again a high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way.

We derive by analytic means a number of bilateral identities of the Rogers--Ramanujan type. Our results include bilateral extensions of the Rogers--Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multiseries including multilateral extensions of the Andrews--Gordon identities, of the Andrews--Bressoud generalization of the G\"ollnitz--Gordon identities, of Bressoud's even modulus ide...

We provide a proof of the Borwein Conjecture using analytic methods.

In this paper, we conjecture an extension to Bressoud's 1996 generalization of Borwein's famous 1990 conjecture. We then state three infinite hierarchies of non-negative $q$-series identities which are interesting examples of our proposed conjecture and Bressoud's generalized conjecture. Finally, using certain positivity-preserving transformations for $q$-binomial coefficients, we prove the non-negativity of the three infinite families.

What follows is a lightly edited version of the author's unpublished master's essay, submitted in partial fulfillment of the requirements of the degree of Master of Arts at the Pennsylvania State University, dated June 1994, written under the supervision of Professor George E. Andrews. It was retyped by the author on November 23, 2022. Obvious typographical errors in the original were corrected without comment; hopefully not too many new errors were introduced during the rety...

Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new q-binomial transformations are also applied to obtain multisum Rogers--Ramanujan ide...

In this paper we want to prove some formulas listed by S. Ramanujan in his paper "Modular equations and approximations to $\pi$" \cite{24} with an elementary method.