arXiv:2105.10096v3
Three families of -Lommel polynomials
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Abstract
Three -versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a general theorem when moments coincide with orthogonal polynomial moments. The combinatorial results use weighted Motzkin paths, Schröder paths, and parallelogram polyominoes.
Dependence graphs
Proof lineage
On -real and -complex numbers
A statement-restricted proof-dependence graph for arXiv:2508.08440v2. It covers the paper's main analytic continuation and convergence theorems, exact $q$-Bessel formulas, monotonicity and almost-everywhere singularity, the unit-circle Jones/Burau bounds, the mod-$m$ classification results, and the hypergeometric construction of $q$-complex numbers. Expansion follows only claims whose truth can affect one of those marked results and stops at published books. Historical citations, comparisons, conjectures, and alternate routes are documented separately as non-edges.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
Current report
Detailed mathematical audit
01Statements2 reported findingsNo statement issues found
The three families of -Lommel polynomials, their explicit representations and recurrences, the continued-fraction formulas, the associated Askey-Wilson identifications, and the stated combinatorial interpretations are coherent at the stated generic-parameter scope. The specialized continued fraction in Equation (6.4), which is the MathAudit dependency, is correct.
Theorem 5.3 and Equation (6.4)No issue found
The basic-Bessel continued fraction has the stated normalization
PDF pages 14–19 · Theorem 5.3 and Equation (6.4) · arXiv:2105.10096v3
Theorem 5.3 gives the ratio of contiguous basic hypergeometric solutions as the Jacobi continued fraction determined by the displayed three-term recurrence. The specialization in Section 6 cancels the common prefactor and yields Equation (6.4) with numerator terms and denominator shifts . Substituting and gives the exact formula quoted as Lemma 3.1 in arXiv:2508.08440v2.
Exact arXiv version 3 ↗Sections 2–4 and 7–8No issue found
The recurrence and combinatorial families agree
PDF pages 3–13 and 20–24 · arXiv:2105.10096v3
For each family, the explicit polynomial sum satisfies the same recurrence and initial values as the corresponding denominator polynomials. The weighted Motzkin-, Schröder-, and polyomino-enumeration formulas reproduce those recurrences, including the endpoint weights, so the algebraic and combinatorial descriptions are compatible.
02Proofs2 reported findingsNo proof issues found
The recurrence comparisons, finite-continuant identities, basic-hypergeometric contiguous relation, and weighted-path bijections close the advertised arguments. The proof-critical Equation (6.4) is independently derived inside the paper rather than merely attributed to an inaccessible predecessor.
Proof of Theorem 5.3No issue found in the reviewed proof
Coefficient comparison proves the required contiguous relation
PDF pages 14–16 · arXiv:2105.10096v3
After normalizing the two contiguous series, coefficient comparison gives the displayed three-term relation. Pincherle's continued-fraction mechanism then identifies the solution ratio because the minimal solution and the initial quotient have the required formal expansions. Specializing the parameters produces the Section 6 fraction without an unproved limit interchange.
Exact arXiv version 3 ↗Combinatorial proofsNo issue found in the reviewed proof
The path decompositions preserve every recurrence weight
PDF pages 7–13 and 20–24 · arXiv:2105.10096v3
The first-return decompositions separate level, up-down, and long-horizontal steps with exactly the recurrence coefficients printed for the polynomial families. Boundary cases match the initial polynomials, and the subsequent path-to-polyomino maps preserve the area and step statistics used in the generating functions.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.