{"paper":{"title":"Representations by probabilistic Frobenius-Euler and degenerate Frobenius-Euler polynomials","arxiv_id":"2508.17189","primary_cat":"math.NT","submitted_at":"2025-08-24T02:34:38+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.98,"id":"s0","own":false,"role":"background","text":"Let Y be a random variable whose moment generating function exists in a neighborhood of the origin.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.09,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"contribution","text":"The aim of this paper is to represent arbitrary polynomials in terms of probabilistic Frobenius-Euler polynomials associated with Y and probabilistic degenerate Frobenius-Euler polynomials associated with Y , and more generally of their higher-order counterparts.","hedge":{"p":0.87,"label":"asserted"},"own_p":0.51,"main_claim_p":0.31,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":2,"p":0.61,"id":"s2","own":true,"role":"contribution","text":"We derive explicit formulas with the help of umbral calculus and illustrate our results in the case of several random variables Y","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.3,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.36,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.61,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.63,"text":"We derive explicit formulas with the help of umbral calculus and illustrate our results in the case of several random variables Y","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":2962,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.04,"states_limitation":0.1,"human_participants":0.02,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.04,"compares_with_prior_results":0.06,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.82,"text":"We derive explicit formulas with the help of umbral calculus and illustrate our results in the case of several random variables Y","location":"abstract s2"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":3,"sentences_in_abstract":3},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}