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Finite Nilpotent q-Binomial Identity

Abstract

A finite q-difference recurrence yields the nilpotent q-binomial identity.

Theorem 1.1 (Nilpotent q-binomial transformation).

Lean statement: D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiHeine.nilpotent_q_binomial

Proof. Machine-checked in Lean as D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiHeine.nilpotent_q_binomial (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

Commentary.

Let R be a commutative ring, let a, q and z belong to R, and let N be natural. If q raised to N and z raised to N are zero, then the product over i below N of (1 minus a z q raised to i) times the inverse of (1 minus z q raised to i) equals the sum over r below N of z raised to r times the corresponding finite product with factors (1 minus a q raised to i) and inverses of (1 minus q raised to i plus 1).

References