Bailey pairs and quantum q-series identities. I. The classical identities

Area of Science:

• Number Theory
• Combinatorics
• Special Functions

Background:

• q-series identities are fundamental in number theory and combinatorics.
• Mock theta functions and quadratic forms have deep connections to q-series.
• Previous work by Cohen and Bryson-Ono-Pitman-Rhoades established key identities.

Purpose of the Study:

• To prove specific q-series identities at roots of unity.
• To utilize Bailey pairs and quadratic forms in the proofs.
• To generalize these identities into infinite families.

Main Methods:

• Application of Bailey pairs and the standard Bailey lemma.
• Incorporation of quadratic forms from mock theta function research.
• Utilizing changes-of-base identities by Bressoud-Ismail-Stanton.
• Employing the Bailey chain to generate infinite families.

Main Results:

• Successful proof of q-series identities at roots of unity.
• Demonstration of the utility of quadratic forms in this context.
• Establishment of infinite families of q-series identities.

Conclusions:

• Bailey pairs provide a powerful tool for proving q-series identities.
• The methods used offer a unified approach to related identities.
• This work deepens the understanding of mock theta functions and q-series.