Some Families of the Incomplete H-Functions and the Incomplete $$\overline H $$-Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications
Bibliographic record
Abstract
Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin–Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ(s, x) and Γ(s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ p,q m,n (z) and Γ p,q m,n (z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ q (γ) [z] and p Ψ q (Γ) [z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann–Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete $$\overline H $$ H ¯ -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".