{"id":"https://openalex.org/W4387625548","doi":"https://doi.org/10.3934/mfc.2023041","title":"On extended $ k $ -generalized Mittag-Leffler function and its properties","display_name":"On extended $ k $ -generalized Mittag-Leffler function and its properties","publication_year":2023,"publication_date":"2023-10-13","ids":{"openalex":"https://openalex.org/W4387625548","doi":"https://doi.org/10.3934/mfc.2023041"},"language":"en","primary_location":{"id":"doi:10.3934/mfc.2023041","doi":"https://doi.org/10.3934/mfc.2023041","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2023041","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=6527baa06ff4320687800352","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/data/article/export-pdf?id=6527baa06ff4320687800352","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107875950","display_name":"Shilpi Jain","orcid":"https://orcid.org/0000-0002-0906-2801","observed_orcids":["https://orcid.org/0000-0002-0906-2801"]},"institutions":[{"id":"https://openalex.org/I4405274437","display_name":"Poornima Institute of Engineering and Technology","ror":"https://ror.org/01q4evf45","country_code":"IN","type":"education","lineage":["https://openalex.org/I4405274437"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Shilpi Jain","raw_affiliation_strings":["Department of Mathematics, Poornima College of Engineering, Jaipur 302012, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Poornima College of Engineering, Jaipur 302012, India","institution_ids":["https://openalex.org/I4405274437"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034891193","display_name":"B. B. Jaimini","orcid":"https://orcid.org/0000-0001-9689-7331","observed_orcids":["https://orcid.org/0000-0001-9689-7331"]},"institutions":[{"id":"https://openalex.org/I4405277463","display_name":"Government College Kota","ror":"https://ror.org/02sjap392","country_code":"IN","type":"education","lineage":["https://openalex.org/I4405277463"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"B.B. Jaimini","raw_affiliation_strings":["Department of Mathematics, Government College, Kota 324001, Rajasthan, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Government College, Kota 324001, Rajasthan, India","institution_ids":["https://openalex.org/I4405277463"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5093061826","display_name":"Meenu Buri","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Meenu Buri","raw_affiliation_strings":["Department of Mathematics, Government College, Jhunjhunu 333001, Rajasthan, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Government College, Jhunjhunu 333001, Rajasthan, India","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053042789","display_name":"Praveen Agarwal","orcid":"https://orcid.org/0000-0001-7556-8942","observed_orcids":["https://orcid.org/0000-0001-7556-8942"]},"institutions":[{"id":"https://openalex.org/I126527374","display_name":"Peoples' Friendship University of Russia","ror":"https://ror.org/02dn9h927","country_code":"RU","type":"education","lineage":["https://openalex.org/I126527374"]},{"id":"https://openalex.org/I182000528","display_name":"Ajman University","ror":"https://ror.org/01j1rma10","country_code":"AE","type":"education","lineage":["https://openalex.org/I182000528"]},{"id":"https://openalex.org/I4401726791","display_name":"Anand International College of Engineering","ror":"https://ror.org/05sy8fj60","country_code":"IN","type":"education","lineage":["https://openalex.org/I4401726791"]}],"countries":["AE","IN","RU"],"is_corresponding":false,"raw_author_name":"Praveen Agarwal","raw_affiliation_strings":["Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India","Nonlinear Dynamics Research Center(NDRC)Ajman University, Ajman,UAE","Peoples'Friendship University of Russia(RUDN University)6 Miklukho-Maklaya Street, Moscow, Russian Federation"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India","institution_ids":["https://openalex.org/I4401726791"]},{"raw_affiliation_string":"Nonlinear Dynamics Research Center(NDRC)Ajman University, Ajman,UAE","institution_ids":["https://openalex.org/I182000528"]},{"raw_affiliation_string":"Peoples'Friendship University of Russia(RUDN University)6 Miklukho-Maklaya Street, Moscow, Russian Federation","institution_ids":["https://openalex.org/I126527374"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5107875950"],"corresponding_institution_ids":["https://openalex.org/I4405274437"],"apc_list":null,"apc_paid":null,"fwci":1.0586,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.7022725,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":97},"biblio":{"volume":"8","issue":"4","first_page":"472","last_page":"479"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10288","display_name":"Fractional Differential Equations Solutions","score":0.33739998936653137,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.33739998936653137,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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":0.31690001487731934,"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":0.22310000658035278,"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/mittag-leffler-function","display_name":"Mittag-Leffler function","score":0.9079999923706055},{"id":"https://openalex.org/keywords/recurrence-relations","display_name":"recurrence relations","score":0.7480000257492065},{"id":"https://openalex.org/keywords/integral-representation","display_name":"integral representation","score":0.9580000042915344},{"id":"https://openalex.org/keywords/integral-transforms","display_name":"integral transforms","score":0.9679999947547913}],"concepts":[{"id":"https://openalex.org/C98724732","wikidata":"https://www.wikidata.org/wiki/Q1935235","display_name":"Mittag-Leffler function","level":3,"score":0.7152410745620728},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6633269786834717},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.6183227300643921},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5696595311164856},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5675162672996521},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5427144169807434},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5199559926986694},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.5102007389068604},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.44079506397247314},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.41562527418136597},{"id":"https://openalex.org/C118871108","wikidata":"https://www.wikidata.org/wiki/Q1750507","display_name":"Beta function (physics)","level":5,"score":0.4112367033958435},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3443857431411743},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3299877643585205},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16651135683059692},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09869459271430969},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.08944526314735413},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C108568745","wikidata":"https://www.wikidata.org/wiki/Q234181","display_name":"Quantum gravity","level":3,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C23409769","wikidata":"https://www.wikidata.org/wiki/Q7310750","display_name":"Relationship between string theory and quantum field theory","level":4,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/mfc.2023041","doi":"https://doi.org/10.3934/mfc.2023041","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2023041","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=6527baa06ff4320687800352","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/mfc.2023041","doi":"https://doi.org/10.3934/mfc.2023041","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2023041","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=6527baa06ff4320687800352","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","poland-20","russia-white-list-4"],"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1969582618","display_name":null,"funder_award_id":"02011/12/2020 NHBM (R.P)/RDII/7867","funder_id":"https://openalex.org/F4320323196","funder_display_name":"National Board for Higher Mathematics"},{"id":"https://openalex.org/G3600847317","display_name":null,"funder_award_id":"02011/12/2020","funder_id":"https://openalex.org/F4320323196","funder_display_name":"National Board for Higher Mathematics"}],"funders":[{"id":"https://openalex.org/F4320323196","display_name":"National Board for Higher Mathematics","ror":"https://ror.org/02m388s04"},{"id":"https://openalex.org/F4320329913","display_name":"RUDN University","ror":"https://ror.org/02dn9h927"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4387625548.pdf","grobid_xml":"https://content.openalex.org/works/W4387625548.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W2016585644","https://openalex.org/W2045350842","https://openalex.org/W2091714372","https://openalex.org/W2108338926","https://openalex.org/W2165128195","https://openalex.org/W2165469120","https://openalex.org/W2168326749","https://openalex.org/W2749156727","https://openalex.org/W2860407298","https://openalex.org/W2913812462","https://openalex.org/W2949897179","https://openalex.org/W3205001210","https://openalex.org/W3216059901"],"related_works":["https://openalex.org/W2950236431","https://openalex.org/W7123747150","https://openalex.org/W2142785017","https://openalex.org/W4405399672","https://openalex.org/W4381664502","https://openalex.org/W4412360827","https://openalex.org/W4412877195","https://openalex.org/W2595823523","https://openalex.org/W4213084462","https://openalex.org/W7125530589"],"abstract_inverted_index":{"In":[0],"this":[1,37],"current":[2],"paper,":[3],"we":[4],"are":[5,32],"using":[6],"the":[7,12,43,66,72],"concept":[8],"of":[9,11,42,55,80,97],"extension":[10],"beta":[13],"function":[14,23],"to":[15],"define":[16],"an":[17],"extended":[18,67,73],"$":[19,25,68,74],"k":[20,69,75],"$-generalized":[21],"Mittag-Leffler":[22],"(GMLf)":[24],"E_{k,":[26],"l,":[27],"m}^{\\rho,":[28],"\\sigma;c}(x;p)":[29],"$.":[30],"There":[31],"four":[33],"sections":[34],"included":[35],"in":[36],"paper":[38],"containing":[39],"some":[40,56],"properties":[41],"above-described":[44],"function,":[45],"like":[46],"derivatives,":[47],"integral":[48,51],"representation,":[49],"and":[50],"transform.":[52],"The":[53],"establishment":[54],"recurrence":[57],"relations":[58],"has":[59],"also":[60,64,91],"been":[61],"done.":[62],"We":[63],"derive":[65],"$-GMLf":[70],"from":[71],"$-Riemann-Liouville":[76],"(R-L)":[77],"fractional":[78],"derivative":[79],"generalized":[81],"MLf.":[82],"Numerous":[83],"former":[84],"results":[85],"studied":[86],"by":[87],"many":[88],"researchers":[89],"can":[90],"be":[92],"derived":[93],"as":[94],"special":[95],"cases":[96],"our":[98],"results.":[99]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2023-10-14T00:00:00"}