{"id":"https://openalex.org/W4281637664","doi":"https://doi.org/10.3390/sym14061119","title":"A Note on q-analogue of Degenerate Catalan Numbers Associated with p-adic Integral on Zp","display_name":"A Note on q-analogue of Degenerate Catalan Numbers Associated with p-adic Integral on Zp","publication_year":2022,"publication_date":"2022-05-29","ids":{"openalex":"https://openalex.org/W4281637664","doi":"https://doi.org/10.3390/sym14061119"},"language":"en","primary_location":{"id":"doi:10.3390/sym14061119","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061119","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1119/pdf?version=1653815777","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/2073-8994/14/6/1119/pdf?version=1653815777","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5067814181","display_name":"Waseem Ahmad Khan","orcid":"https://orcid.org/0000-0002-4681-9885"},"institutions":[{"id":"https://openalex.org/I138564716","display_name":"Prince Mohammad bin Fahd University","ror":"https://ror.org/03d64na34","country_code":"SA","type":"education","lineage":["https://openalex.org/I138564716"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Waseem A. Khan","raw_affiliation_strings":["Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-4681-9885","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia","institution_ids":["https://openalex.org/I138564716"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5067814181"],"corresponding_institution_ids":["https://openalex.org/I138564716"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":11.1872,"has_fulltext":true,"cited_by_count":20,"citation_normalized_percentile":{"value":0.99214027,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"14","issue":"6","first_page":"1119","last_page":"1119"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":1.0,"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":1.0,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9943000078201294,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.9111901521682739},{"id":"https://openalex.org/keywords/catalan-number","display_name":"Catalan number","score":0.9087712168693542},{"id":"https://openalex.org/keywords/stirling-number","display_name":"Stirling number","score":0.751274824142456},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7048020362854004},{"id":"https://openalex.org/keywords/stirling-numbers-of-the-first-kind","display_name":"Stirling numbers of the first kind","score":0.5451852083206177},{"id":"https://openalex.org/keywords/catalan","display_name":"Catalan","score":0.5446214079856873},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5222904682159424},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.49886012077331543},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43607833981513977},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3656524419784546},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.363755464553833},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.18503329157829285},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10567116737365723}],"concepts":[{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.9111901521682739},{"id":"https://openalex.org/C43929256","wikidata":"https://www.wikidata.org/wiki/Q270513","display_name":"Catalan number","level":2,"score":0.9087712168693542},{"id":"https://openalex.org/C168902406","wikidata":"https://www.wikidata.org/wiki/Q80918","display_name":"Stirling number","level":2,"score":0.751274824142456},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7048020362854004},{"id":"https://openalex.org/C93765064","wikidata":"https://www.wikidata.org/wiki/Q2657556","display_name":"Stirling numbers of the first kind","level":3,"score":0.5451852083206177},{"id":"https://openalex.org/C164105321","wikidata":"https://www.wikidata.org/wiki/Q7026","display_name":"Catalan","level":2,"score":0.5446214079856873},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5222904682159424},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49886012077331543},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43607833981513977},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3656524419784546},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.363755464553833},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.18503329157829285},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10567116737365723},{"id":"https://openalex.org/C15708023","wikidata":"https://www.wikidata.org/wiki/Q80083","display_name":"Humanities","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"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":3,"locations":[{"id":"doi:10.3390/sym14061119","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061119","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1119/pdf?version=1653815777","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:10519507e97242b98e5824372d0002e8","is_oa":true,"landing_page_url":"https://doaj.org/article/10519507e97242b98e5824372d0002e8","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,"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":"Symmetry, Vol 14, Iss 6, p 1119 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/6/1119/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14061119","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14061119","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14061119","pdf_url":"https://www.mdpi.com/2073-8994/14/6/1119/pdf?version=1653815777","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":[{"id":"https://openalex.org/F4320328105","display_name":"Prince Mohammad Bin Fahd University","ror":"https://ror.org/03d64na34"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4281637664.pdf","grobid_xml":"https://content.openalex.org/works/W4281637664.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1974467810","https://openalex.org/W2064497635","https://openalex.org/W2166223445","https://openalex.org/W2412261190","https://openalex.org/W2522881025","https://openalex.org/W2882994152","https://openalex.org/W2963702694","https://openalex.org/W3104187149","https://openalex.org/W3168269546","https://openalex.org/W4280616181","https://openalex.org/W6715210130","https://openalex.org/W6796551118"],"related_works":["https://openalex.org/W3088269311","https://openalex.org/W3203759461","https://openalex.org/W4281748020","https://openalex.org/W4281637664","https://openalex.org/W4313403335","https://openalex.org/W4298205293","https://openalex.org/W4394876590","https://openalex.org/W2294731123","https://openalex.org/W3131323730","https://openalex.org/W2737298300"],"abstract_inverted_index":{"In":[0],"this":[1,49],"paper,":[2],"we":[3,39],"introduce":[4],"q-analogues":[5],"of":[6,15,32,48,51],"degenerate":[7,29],"Catalan":[8],"numbers":[9,31],"and":[10,22,35,46,53],"polynomials":[11,52],"with":[12,27],"the":[13,28,33],"help":[14],"a":[16,42],"fermionic":[17],"p-adic":[18],"q-integrals":[19],"on":[20],"Zp":[21],"establish":[23],"some":[24],"new":[25,44],"connections":[26],"Stirling":[30],"first":[34],"second":[36],"kinds.":[37],"Furthermore,":[38],"also":[40],"find":[41],"few":[43],"identities":[45],"results":[47],"type":[50],"numbers.":[54]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":4}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2022-06-13T00:00:00"}
