{"id":"https://openalex.org/W2937203737","doi":"https://doi.org/10.3390/sym11040538","title":"A Note on the Truncated-Exponential Based Apostol-Type Polynomials","display_name":"A Note on the Truncated-Exponential Based Apostol-Type Polynomials","publication_year":2019,"publication_date":"2019-04-15","ids":{"openalex":"https://openalex.org/W2937203737","doi":"https://doi.org/10.3390/sym11040538","mag":"2937203737"},"language":"en","primary_location":{"id":"doi:10.3390/sym11040538","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11040538","pdf_url":"https://www.mdpi.com/2073-8994/11/4/538/pdf?version=1555315570","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/4/538/pdf?version=1555315570","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079512627","display_name":"H. M. Srivastava","orcid":"https://orcid.org/0000-0002-9277-8092"},"institutions":[{"id":"https://openalex.org/I212119943","display_name":"University of Victoria","ror":"https://ror.org/04s5mat29","country_code":"CA","type":"education","lineage":["https://openalex.org/I212119943"]},{"id":"https://openalex.org/I184693016","display_name":"China Medical University","ror":"https://ror.org/00v408z34","country_code":"TW","type":"education","lineage":["https://openalex.org/I184693016"]},{"id":"https://openalex.org/I4210126829","display_name":"China Medical University Hospital","ror":"https://ror.org/0368s4g32","country_code":"TW","type":"healthcare","lineage":["https://openalex.org/I4210126829"]}],"countries":["CA","TW"],"is_corresponding":true,"raw_author_name":"H. M. Srivastava","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada","institution_ids":["https://openalex.org/I212119943"]},{"raw_affiliation_string":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan","institution_ids":["https://openalex.org/I4210126829","https://openalex.org/I184693016"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025679750","display_name":"Serkan Arac\u0131","orcid":"https://orcid.org/0000-0002-3950-6864"},"institutions":[{"id":"https://openalex.org/I4210154391","display_name":"Hasan Kalyoncu University","ror":"https://ror.org/054g2pw49","country_code":"TR","type":"education","lineage":["https://openalex.org/I4210154391"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Serkan Araci","raw_affiliation_strings":["Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey"],"affiliations":[{"raw_affiliation_string":"Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey","institution_ids":["https://openalex.org/I4210154391"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067814181","display_name":"Waseem Ahmad Khan","orcid":"https://orcid.org/0000-0002-4681-9885"},"institutions":[{"id":"https://openalex.org/I189949046","display_name":"Integral University","ror":"https://ror.org/039zd5s34","country_code":"IN","type":"education","lineage":["https://openalex.org/I189949046"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Waseem A. Khan","raw_affiliation_strings":["Department of Mathematics, Integral University, Lucknow 226026, Uttar Pradesh, India"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Integral University, Lucknow 226026, Uttar Pradesh, India","institution_ids":["https://openalex.org/I189949046"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067030524","display_name":"Mehmet A\u00e7\u0131kg\u00f6z","orcid":"https://orcid.org/0000-0003-1091-9697"},"institutions":[{"id":"https://openalex.org/I59738393","display_name":"Gaziantep University","ror":"https://ror.org/020vvc407","country_code":"TR","type":"education","lineage":["https://openalex.org/I59738393"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Mehmet Acikg\u00f6z","raw_affiliation_strings":["Department of Mathematics, Faculty of Science and Arts, Gaziantep University, TR-27310 Gaziantep, Turkey"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science and Arts, Gaziantep University, TR-27310 Gaziantep, Turkey","institution_ids":["https://openalex.org/I59738393"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5079512627"],"corresponding_institution_ids":["https://openalex.org/I184693016","https://openalex.org/I212119943","https://openalex.org/I4210126829"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":8.6372,"has_fulltext":true,"cited_by_count":32,"citation_normalized_percentile":{"value":0.98564492,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"11","issue":"4","first_page":"538","last_page":"538"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9951000213623047,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8278011083602905},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.6129822731018066},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.612640380859375},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.6029635071754456},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.5923780798912048},{"id":"https://openalex.org/keywords/wilson-polynomials","display_name":"Wilson polynomials","score":0.5742003321647644},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5299077033996582},{"id":"https://openalex.org/keywords/hahn-polynomials","display_name":"Hahn polynomials","score":0.4948720335960388},{"id":"https://openalex.org/keywords/macdonald-polynomials","display_name":"Macdonald polynomials","score":0.48727238178253174},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4847116470336914},{"id":"https://openalex.org/keywords/hermite-polynomials","display_name":"Hermite polynomials","score":0.45771920680999756},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.42868366837501526},{"id":"https://openalex.org/keywords/exponential-polynomial","display_name":"Exponential polynomial","score":0.41810885071754456},{"id":"https://openalex.org/keywords/exponential-function","display_name":"Exponential function","score":0.4056558907032013},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20381739735603333}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8278011083602905},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.6129822731018066},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.612640380859375},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.6029635071754456},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.5923780798912048},{"id":"https://openalex.org/C43176163","wikidata":"https://www.wikidata.org/wiki/Q8023263","display_name":"Wilson polynomials","level":4,"score":0.5742003321647644},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5299077033996582},{"id":"https://openalex.org/C61349858","wikidata":"https://www.wikidata.org/wiki/Q1568814","display_name":"Hahn polynomials","level":5,"score":0.4948720335960388},{"id":"https://openalex.org/C171686973","wikidata":"https://www.wikidata.org/wiki/Q17098676","display_name":"Macdonald polynomials","level":4,"score":0.48727238178253174},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4847116470336914},{"id":"https://openalex.org/C201362023","wikidata":"https://www.wikidata.org/wiki/Q658574","display_name":"Hermite polynomials","level":2,"score":0.45771920680999756},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.42868366837501526},{"id":"https://openalex.org/C85384003","wikidata":"https://www.wikidata.org/wiki/Q5421525","display_name":"Exponential polynomial","level":3,"score":0.41810885071754456},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4056558907032013},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20381739735603333},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/sym11040538","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11040538","pdf_url":"https://www.mdpi.com/2073-8994/11/4/538/pdf?version=1555315570","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:f893a9d2bacf46fc8d362d416d073268","is_oa":true,"landing_page_url":"https://doaj.org/article/f893a9d2bacf46fc8d362d416d073268","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 11, Iss 4, p 538 (2019)","raw_type":"article"},{"id":"pmh:oai:dspace.library.uvic.ca:1828/10841","is_oa":false,"landing_page_url":"http://hdl.handle.net/1828/10841","pdf_url":null,"source":{"id":"https://openalex.org/S4306401509","display_name":"UVic\u2019s Research and Learning Repository (University of Victoria)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I212119943","host_organization_name":"University of Victoria","host_organization_lineage":["https://openalex.org/I212119943"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:openaccess.hku.edu.tr:20.500.11782/548","is_oa":true,"landing_page_url":"https://hdl.handle.net/20.500.11782/548","pdf_url":null,"source":{"id":"https://openalex.org/S4377196685","display_name":"DSpace-HK\u00dc (Hasan Kalyoncu University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210154391","host_organization_name":"Hasan Kalyoncu University","host_organization_lineage":["https://openalex.org/I4210154391"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.3390/sym11040538","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11040538","pdf_url":"https://www.mdpi.com/2073-8994/11/4/538/pdf?version=1555315570","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2937203737.pdf","grobid_xml":"https://content.openalex.org/works/W2937203737.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W273849276","https://openalex.org/W1497283712","https://openalex.org/W1499718604","https://openalex.org/W1501425978","https://openalex.org/W1519625436","https://openalex.org/W1966131895","https://openalex.org/W1978153654","https://openalex.org/W1985158692","https://openalex.org/W2007891558","https://openalex.org/W2008439130","https://openalex.org/W2018273723","https://openalex.org/W2018552887","https://openalex.org/W2032733792","https://openalex.org/W2034149193","https://openalex.org/W2035458920","https://openalex.org/W2044270276","https://openalex.org/W2049221554","https://openalex.org/W2053783010","https://openalex.org/W2072818124","https://openalex.org/W2086306912","https://openalex.org/W2086996530","https://openalex.org/W2102185360","https://openalex.org/W2128297633","https://openalex.org/W2129840940","https://openalex.org/W2140451685","https://openalex.org/W2151233001","https://openalex.org/W2152891299","https://openalex.org/W2159190998","https://openalex.org/W2164415850","https://openalex.org/W2282149267","https://openalex.org/W2312568493","https://openalex.org/W2352088301","https://openalex.org/W2485900856","https://openalex.org/W2554334727","https://openalex.org/W2579972501","https://openalex.org/W3112909250","https://openalex.org/W4231062687","https://openalex.org/W4239234298","https://openalex.org/W6630210061","https://openalex.org/W6678826555","https://openalex.org/W6698516097","https://openalex.org/W6732274189","https://openalex.org/W7036569420"],"related_works":["https://openalex.org/W2070537393","https://openalex.org/W2132314122","https://openalex.org/W2418221131","https://openalex.org/W2065361703","https://openalex.org/W4287241472","https://openalex.org/W1825644555","https://openalex.org/W4362728862","https://openalex.org/W3103228020","https://openalex.org/W1997096699","https://openalex.org/W2792593666"],"abstract_inverted_index":{"In":[0,16],"this":[1,24],"paper,":[2],"we":[3,18,52],"propose":[4],"to":[5],"investigate":[6],"the":[7,20,30,59],"truncated-exponential-based":[8],"Apostol-type":[9,32],"polynomials":[10,28],"and":[11,29,41],"derive":[12],"their":[13,46],"various":[14],"properties.":[15],"particular,":[17],"establish":[19],"operational":[21],"correspondence":[22],"between":[23],"new":[25],"family":[26],"of":[27,58],"familiar":[31],"polynomials.":[33,68],"We":[34],"also":[35],"obtain":[36],"some":[37],"implicit":[38],"summation":[39],"formulas":[40,62],"symmetric":[42],"identities":[43,64],"by":[44],"using":[45],"generating":[47],"functions.":[48],"The":[49],"results,":[50],"which":[51],"have":[53],"derived":[54],"here,":[55],"provide":[56],"generalizations":[57],"corresponding":[60],"known":[61],"including":[63],"involving":[65],"generalized":[66],"Hermite-Bernoulli":[67]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":4}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}