{"id":"https://openalex.org/W4389920824","doi":"https://doi.org/10.3390/math11244978","title":"On Some Formulas for the Lauricella Function","display_name":"On Some Formulas for the Lauricella Function","publication_year":2023,"publication_date":"2023-12-16","ids":{"openalex":"https://openalex.org/W4389920824","doi":"https://doi.org/10.3390/math11244978"},"language":"en","primary_location":{"id":"doi:10.3390/math11244978","doi":"https://doi.org/10.3390/math11244978","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11244978","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"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":"Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/math11244978","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059637367","display_name":"\u0410\u0439\u043d\u0443\u0440 \u0420\u044b\u0441\u043a\u0430\u043d","orcid":"https://orcid.org/0000-0002-8764-4751","observed_orcids":["https://orcid.org/0000-0002-8764-4751"]},"institutions":[{"id":"https://openalex.org/I4210128045","display_name":"Abai Kazakh National Pedagogical University","ror":"https://ror.org/02bvvhn54","country_code":"KZ","type":"education","lineage":["https://openalex.org/I4210128045"]}],"countries":["KZ"],"is_corresponding":true,"raw_author_name":"Ainur Ryskan","raw_affiliation_strings":["Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, 86 Tole Bi Street, Almaty 050012, Kazakhstan"],"raw_orcid":"https://orcid.org/0000-0002-8764-4751","affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, 86 Tole Bi Street, Almaty 050012, Kazakhstan","institution_ids":["https://openalex.org/I4210128045"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010614240","display_name":"Tuhtasin Gulamjanovich Ergashev","orcid":"https://orcid.org/0000-0003-3542-8309","observed_orcids":["https://orcid.org/0000-0003-3542-8309"]},"institutions":[{"id":"https://openalex.org/I32597200","display_name":"Ghent University","ror":"https://ror.org/00cv9y106","country_code":"BE","type":"education","lineage":["https://openalex.org/I32597200"]},{"id":"https://openalex.org/I4210109462","display_name":"Tashkent Institute of Irrigation and Agricultural Mechanization Engineers","ror":"https://ror.org/01s4mx151","country_code":"UZ","type":"education","lineage":["https://openalex.org/I4210109462"]}],"countries":["BE","UZ"],"is_corresponding":true,"raw_author_name":"Tuhtasin Ergashev","raw_affiliation_strings":["Department of Higher Mathematics, National Research University \u201cTIIAME\u201d, 39 Kari-Niyazi Street, Tashkent 100000, Uzbekistan","Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, 9000 Gent, Belgium"],"raw_orcid":"https://orcid.org/0000-0003-3542-8309","affiliations":[{"raw_affiliation_string":"Department of Higher Mathematics, National Research University \u201cTIIAME\u201d, 39 Kari-Niyazi Street, Tashkent 100000, Uzbekistan","institution_ids":["https://openalex.org/I4210109462"]},{"raw_affiliation_string":"Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, 9000 Gent, Belgium","institution_ids":["https://openalex.org/I32597200"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5010614240","https://openalex.org/A5059637367"],"corresponding_institution_ids":["https://openalex.org/I32597200","https://openalex.org/I4210109462","https://openalex.org/I4210128045"],"apc_list":{"value":2100,"currency":"CHF","value_usd":2336},"apc_paid":{"value":2100,"currency":"CHF","value_usd":2336},"fwci":1.286,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.73923943,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"11","issue":"24","first_page":"4978","last_page":"4978"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9937000274658203,"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"}},"topics":[{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.9937000274658203,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.003599999938160181,"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.0005000000237487257,"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/lauricella-functions","display_name":"Lauricella functions","score":0.8560000061988831},{"id":"https://openalex.org/keywords/limit-formulas","display_name":"limit formulas","score":0.9430000185966492},{"id":"https://openalex.org/keywords/expansion-formulas","display_name":"expansion formulas","score":0.953000009059906},{"id":"https://openalex.org/keywords/recurrence-relations","display_name":"recurrence relations","score":0.9919999837875366},{"id":"https://openalex.org/keywords/integral-representation","display_name":"integral representation","score":0.9990000128746033},{"id":"https://openalex.org/keywords/differentiation-formulas","display_name":"differentiation formulas","score":0.9950000047683716},{"id":"https://openalex.org/keywords/finite-sums","display_name":"finite sums","score":0.7239999771118164},{"id":"https://openalex.org/keywords/infinite-series","display_name":"infinite series","score":0.9100000262260437},{"id":"https://openalex.org/keywords/gaussian-hypergeometric-function","display_name":"Gauss hypergeometric function","score":0.8610000014305115}],"concepts":[{"id":"https://openalex.org/C122343491","wikidata":"https://www.wikidata.org/wiki/Q3754612","display_name":"Lauricella hypergeometric series","level":5,"score":0.8960131406784058},{"id":"https://openalex.org/C197320386","wikidata":"https://www.wikidata.org/wiki/Q21028472","display_name":"Hypergeometric function","level":2,"score":0.799517810344696},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7542035579681396},{"id":"https://openalex.org/C82240120","wikidata":"https://www.wikidata.org/wiki/Q190573","display_name":"Gamma function","level":2,"score":0.6655232906341553},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.618327260017395},{"id":"https://openalex.org/C60400383","wikidata":"https://www.wikidata.org/wiki/Q17028407","display_name":"Hypergeometric identity","level":5,"score":0.5989090800285339},{"id":"https://openalex.org/C181881257","wikidata":"https://www.wikidata.org/wiki/Q4780998","display_name":"Appell series","level":5,"score":0.5689830780029297},{"id":"https://openalex.org/C158241908","wikidata":"https://www.wikidata.org/wiki/Q109744257","display_name":"Generalized hypergeometric function","level":3,"score":0.564948558807373},{"id":"https://openalex.org/C182078607","wikidata":"https://www.wikidata.org/wiki/Q4907583","display_name":"Bilateral hypergeometric series","level":5,"score":0.5483700037002563},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.521155595779419},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5109943151473999},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5082761645317078},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.45426884293556213},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.41645199060440063},{"id":"https://openalex.org/C165175332","wikidata":"https://www.wikidata.org/wiki/Q5958021","display_name":"Hypergeometric function of a matrix argument","level":4,"score":0.36813104152679443},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33384567499160767},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2626100182533264},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15063169598579407},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/math11244978","doi":"https://doi.org/10.3390/math11244978","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11244978","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"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":"Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:gam:jmathe:v:11:y:2023:i:24:p:4978-:d:1301665","doi":null,"is_oa":false,"landing_page_url":"https://www.mdpi.com/2227-7390/11/24/4978/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},{"id":"pmh:oai:doaj.org/article:ffac456dae204576afc0bcb2e877bccf","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/ffac456dae204576afc0bcb2e877bccf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics, Vol 11, Iss 24, p 4978 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/math11244978","doi":"https://doi.org/10.3390/math11244978","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math11244978","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"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":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322726","display_name":"Ministry of Education and Science of the Republic of Kazakhstan","ror":"https://ror.org/03pj6ge82"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W2029223455","https://openalex.org/W2029640189","https://openalex.org/W2047009366","https://openalex.org/W2050787879","https://openalex.org/W2140915219","https://openalex.org/W2952142657","https://openalex.org/W2953722538","https://openalex.org/W2979950972","https://openalex.org/W3167148495","https://openalex.org/W4220762046","https://openalex.org/W4243264049","https://openalex.org/W4245330902","https://openalex.org/W6764634067"],"related_works":["https://openalex.org/W2039910938","https://openalex.org/W2768167360","https://openalex.org/W2074469687","https://openalex.org/W1988084663","https://openalex.org/W3167148495","https://openalex.org/W4394575854","https://openalex.org/W2039214402","https://openalex.org/W2228332305","https://openalex.org/W2029223455","https://openalex.org/W4385875564"],"abstract_inverted_index":{"Lauricella,":[0],"G.":[1],"in":[2],"1893":[3],"defined":[4],"four":[5],"multidimensional":[6],"hypergeometric":[7,122],"functions":[8,15,28],"FA,":[9],"FB,":[10],"FC":[11],"and":[12,36,71,75,90,96,105,116,119],"FD.":[13],"These":[14],"depended":[16],"on":[17],"three":[18],"variables":[19,35],"but":[20],"were":[21],"later":[22],"generalized":[23],"to":[24,67,86],"many":[25],"variables.":[26],"Lauricella\u2019s":[27,51],"are":[29,57,62,65,81,84,127],"infinite":[30,91],"sums":[31],"of":[32,34,40,78,98,108,114,124],"products":[33],"corresponding":[37],"parameters,":[38],"each":[39],"them":[41],"has":[42],"its":[43],"own":[44],"parameters.":[45],"In":[46,93],"the":[47,54,88,94,99,109,120],"present":[48],"work":[49],"for":[50],"function":[52,123],"FA(n),":[53],"limit":[55],"formulas":[56,61,80],"established,":[58],"some":[59],"expansion":[60],"obtained":[63,82,100],"that":[64,83],"used":[66,85],"write":[68],"recurrence":[69],"relations,":[70],"new":[72],"integral":[73,106],"representations":[74,107],"a":[76],"number":[77],"differentiation":[79],"obtain":[87],"finite":[89],"sums.":[92],"presentation":[95],"proof":[97],"formulas,":[101],"already":[102],"known":[103],"expansions":[104],"considered":[110],"FA(n)":[111],"function,":[112],"definitions":[113],"gamma":[115],"beta":[117],"functions,":[118],"Gaussian":[121],"one":[125],"variable":[126],"used.":[128]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2023-12-19T00:00:00"}