Quantum (or q‐) operator equations and associated partial differential equations for bivariate Laguerre polynomials with applications to the q‐Hille‐Hardy type formulas
Bibliographic record
Abstract
Based on the extensive application of the ‐series and ‐polynomials including (for example) the ‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the ‐Laguerre polynomials. The mission of this paper is to find the general ‐operational equation together with the expansion issue of the bivariate ‐Laguerre polynomials from the perspective of ‐partial differential equations. We also give some applications including some ‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the ‐type generating functions for the bivariate ‐Laguerre polynomials by the technique based upon ‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate ‐Laguerre polynomials by applying ‐operational equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".