Integral and computational representations of the extended Hurwitz–Lerch zeta function
Bibliographic record
Abstract
This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".