KimStantonSingleDenominator

Abstract

Integer polynomial denominators of contiguous basic hypergeometric coefficients.

Definition 1.1 (gamma).

$$\forall n\in \mathbb{N},\mathrm{gamma}\left(n\right)=(PowerSeries.coeff\left(n,(F\left((a\cdot q),b,(c\cdot q),q\right)\cdot (F\left(a,b,c,q\right){)}^{-1})\right))$$

Formalization. D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.gamma (✓ std3).

Citation. Jang Soo Kim and Dennis Stanton (2021). Three families of q-Lommel polynomials. DOI: 10.48550/arXiv.2105.10096. URL: https://arxiv.org/abs/2105.10096v3.

Commentary.

Printed pages 22–23 of arXiv:2105.10096v3: Let ∑ₙ≥₀ γ′ₙ(a,b,c) zⁿ = ₂φ₁(aq,b;cq;q,z)/₂φ₁(a,b;c;q,z).

Definition 1.2 (claim).

$$\mathrm{claim}=(\forall n\in \mathbb{N},\exists P\in \mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right),\frac{\mathrm{gamma}\left(n\right)}{(1-c)}=\frac{\mathrm{algebraMap}\left((\mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right)),(\mathrm{FractionRing}\left((\mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right))\right)),P\right)}{D\left(n\right)})$$

Formalization. D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.claim (✓ std3).

Citation. Jang Soo Kim and Dennis Stanton (2021). Three families of q-Lommel polynomials. DOI: 10.48550/arXiv.2105.10096. URL: https://arxiv.org/abs/2105.10096v3.

Commentary.

Printed page 22–23 of arXiv:2105.10096v3, Conjecture 8.4. Let ∑ₙ≥₀ γ′ₙ(a,b,c) zⁿ = ₂φ₁(aq,b;cq;q,z)/₂φ₁(a,b;c;q,z). Then γ′ₙ(a,b,c)/(1−c) = P′ₙ(a,b,c)/∏ₖ₌₀ⁿ(1−cqᵏ)^⌊(n+1)/(k+1)⌋, for some polynomial P′ₙ(a,b,c) in a,b,c,q with integer coefficients. The variables a, b, c, q are the four Fin 4 polynomial variables over the integers; the coefficient index is every natural number, including zero. The series is formal over the fraction field.

Theorem 1.3 (result).

$$\forall n\in \mathbb{N},\exists P\in \mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right),\frac{\mathrm{gamma}\left(n\right)}{(1-c)}=\frac{\mathrm{algebraMap}\left((\mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right)),(\mathrm{FractionRing}\left((\mathrm{MvPolynomial}\left((\mathrm{Fin}\left(4\right)),\mathbb{Z}\right))\right)),P\right)}{D\left(n\right)}$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.result (✓ std3). ∎

Resolves. Problems/ks2105-single-2phi1denom (proved) by D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.result.

Source. Repository-derived.

Acknowledgement. Jang Soo Kim and Dennis Stanton (2021). Three families of q-Lommel polynomials. DOI: 10.48550/arXiv.2105.10096. URL: https://arxiv.org/abs/2105.10096v3.

Commentary.

The recurrence puts the ratio coefficients in the localization generated by the lower-parameter factors. The pole onset bounds clear their floor exponent denominator; localization descent supplies an integer polynomial in all four variables.

References

• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.claim
• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.gamma
• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/KimStantonSingleDenominator.result
• Dependency: D5/S3/Combinatorics/MotzkinHankel/KimStantonDenominatorCore