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Factorial Basis Method for q-Series Applications

Antonio Jiménez-Pastor, Ali Kemal Uncu

TL;DR

This work extends the factorial-basis framework to $q$-series by introducing $\\beta(n)$-factorial bases and product bases, enabling definite-sum representations and holonomic proofs of $q$-identities. The authors develop a compatible-operator theory in the $q$-setting, including $q$-binomial bases and $q$-power bases, and show how to combine bases via product bases to obtain nested sum representations and recurrences for summands. They apply the method to Rogers–Ramanujan-type identities, deriving new representations for bounded generating functions and recovering known results such as the corollary $d_j(a,-q;q)=a^j q^{3j^2+j}(q^3;q^6)_j$, all within a framework amenable to Zeilberger-style automatic proof. A SageMath prototype (pseries_basis) and accompanying resources illustrate practical, automated exploration and discovery of $q$-identities. Overall, the paper provides a computational approach that both proves classical $q$-series results and uncovers new ones, enhancing tools for $q$-combinatorics and partition theory.

Abstract

The Factorial Basis method, initially designed for quasi-triangular, shift-compatible factorial bases, provides solutions to linear recurrence equations in the form of definite-sums. This paper extends the Factorial Basis method to its q-analog, enabling its application in q-calculus. We demonstrate the adaptation of the method to q-sequences and its utility in the realm of q-combinatorics. The extended technique is employed to automatically prove established identities and unveil novel ones, particularly some associated with the Rogers-Ramanujan identities.

Factorial Basis Method for q-Series Applications

TL;DR

This work extends the factorial-basis framework to -series by introducing -factorial bases and product bases, enabling definite-sum representations and holonomic proofs of -identities. The authors develop a compatible-operator theory in the -setting, including -binomial bases and -power bases, and show how to combine bases via product bases to obtain nested sum representations and recurrences for summands. They apply the method to Rogers–Ramanujan-type identities, deriving new representations for bounded generating functions and recovering known results such as the corollary , all within a framework amenable to Zeilberger-style automatic proof. A SageMath prototype (pseries_basis) and accompanying resources illustrate practical, automated exploration and discovery of -identities. Overall, the paper provides a computational approach that both proves classical -series results and uncovers new ones, enhancing tools for -combinatorics and partition theory.

Abstract

The Factorial Basis method, initially designed for quasi-triangular, shift-compatible factorial bases, provides solutions to linear recurrence equations in the form of definite-sums. This paper extends the Factorial Basis method to its q-analog, enabling its application in q-calculus. We demonstrate the adaptation of the method to q-sequences and its utility in the realm of q-combinatorics. The extended technique is employed to automatically prove established identities and unveil novel ones, particularly some associated with the Rogers-Ramanujan identities.
Paper Structure (6 sections, 16 theorems, 65 equations)

This paper contains 6 sections, 16 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Compatible operators
  3. b(n)-factorial basis and q-binomial bases
  4. Product bases
  5. Examples
  6. Conclusions

Key Result

Theorem 1.1

Let $i=1$ or $i=2$. For every positive integer $n$, the number of partitions of size $n$ such that differences between consecutive parts are $\geq 2$ and the smallest part of the partition is $\geq i$ equals the number of partitions of size $n$ into parts congruent to $\pm i$ modulo 5.

Theorems & Definitions (24)

  • Theorem 1.1: Rogers-Ramanujan identities, combinatorial version
  • Theorem 1.2: Rogers-Ramanujan identities, $q$-series version
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • definition 1
  • Theorem 2.1: cf. JimenezPastor2023
  • Theorem 2.2: cf. JimenezPastor2023
  • definition 2
  • ...and 14 more