{"id":"https://openalex.org/W4313515592","doi":"https://doi.org/10.3390/sym15010136","title":"A Note on Modified Degenerate Changhee\u2013Genocchi Polynomials of the Second Kind","display_name":"A Note on Modified Degenerate Changhee\u2013Genocchi Polynomials of the Second Kind","publication_year":2023,"publication_date":"2023-01-03","ids":{"openalex":"https://openalex.org/W4313515592","doi":"https://doi.org/10.3390/sym15010136"},"language":"en","primary_location":{"id":"doi:10.3390/sym15010136","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15010136","pdf_url":"https://www.mdpi.com/2073-8994/15/1/136/pdf?version=1673406986","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/2073-8994/15/1/136/pdf?version=1673406986","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 Ahmad 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"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046233988","display_name":"Maryam Salem Alatawi","orcid":"https://orcid.org/0000-0002-5567-7444"},"institutions":[{"id":"https://openalex.org/I72264486","display_name":"University of Tabuk","ror":"https://ror.org/04yej8x59","country_code":"SA","type":"education","lineage":["https://openalex.org/I72264486"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Maryam Salem Alatawi","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-5567-7444","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia","institution_ids":["https://openalex.org/I72264486"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"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":2.1428,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.83192759,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"15","issue":"1","first_page":"136","last_page":"136"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9994999766349792,"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":0.9994999766349792,"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/T13234","display_name":"advanced mathematical theories","score":0.9944000244140625,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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":0.9902999997138977,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/stirling-number","display_name":"Stirling number","score":0.7377691268920898},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.734975278377533},{"id":"https://openalex.org/keywords/recurrence-relation","display_name":"Recurrence relation","score":0.628777027130127},{"id":"https://openalex.org/keywords/degenerate-energy-levels","display_name":"Degenerate energy levels","score":0.6094919443130493},{"id":"https://openalex.org/keywords/difference-polynomials","display_name":"Difference polynomials","score":0.5745353698730469},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5471824407577515},{"id":"https://openalex.org/keywords/discrete-orthogonal-polynomials","display_name":"Discrete orthogonal polynomials","score":0.5153605341911316},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4917772114276886},{"id":"https://openalex.org/keywords/orthogonal-polynomials","display_name":"Orthogonal polynomials","score":0.4822417199611664},{"id":"https://openalex.org/keywords/classical-orthogonal-polynomials","display_name":"Classical orthogonal polynomials","score":0.4753548800945282},{"id":"https://openalex.org/keywords/wilson-polynomials","display_name":"Wilson polynomials","score":0.4648961126804352},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4596708416938782},{"id":"https://openalex.org/keywords/stirling-numbers-of-the-second-kind","display_name":"Stirling numbers of the second kind","score":0.44030100107192993},{"id":"https://openalex.org/keywords/gegenbauer-polynomials","display_name":"Gegenbauer polynomials","score":0.4402584433555603},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.41518735885620117},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.28596532344818115},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.27747127413749695},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05178999900817871}],"concepts":[{"id":"https://openalex.org/C168902406","wikidata":"https://www.wikidata.org/wiki/Q80918","display_name":"Stirling number","level":2,"score":0.7377691268920898},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.734975278377533},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.628777027130127},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.6094919443130493},{"id":"https://openalex.org/C86607863","wikidata":"https://www.wikidata.org/wiki/Q5275279","display_name":"Difference polynomials","level":3,"score":0.5745353698730469},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5471824407577515},{"id":"https://openalex.org/C52537462","wikidata":"https://www.wikidata.org/wiki/Q5282055","display_name":"Discrete orthogonal polynomials","level":3,"score":0.5153605341911316},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4917772114276886},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.4822417199611664},{"id":"https://openalex.org/C21736991","wikidata":"https://www.wikidata.org/wiki/Q17006917","display_name":"Classical orthogonal polynomials","level":3,"score":0.4753548800945282},{"id":"https://openalex.org/C43176163","wikidata":"https://www.wikidata.org/wiki/Q8023263","display_name":"Wilson polynomials","level":4,"score":0.4648961126804352},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4596708416938782},{"id":"https://openalex.org/C2284495","wikidata":"https://www.wikidata.org/wiki/Q2601117","display_name":"Stirling numbers of the second kind","level":2,"score":0.44030100107192993},{"id":"https://openalex.org/C78540521","wikidata":"https://www.wikidata.org/wiki/Q1498246","display_name":"Gegenbauer polynomials","level":4,"score":0.4402584433555603},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.41518735885620117},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.28596532344818115},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27747127413749695},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05178999900817871},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15010136","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15010136","pdf_url":"https://www.mdpi.com/2073-8994/15/1/136/pdf?version=1673406986","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:bf9ea9bea2dd4ce7abb43fdb4d04ce1f","is_oa":false,"landing_page_url":"https://doaj.org/article/bf9ea9bea2dd4ce7abb43fdb4d04ce1f","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 1, p 136 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/1/136/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15010136","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/sym15010136","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15010136","pdf_url":"https://www.mdpi.com/2073-8994/15/1/136/pdf?version=1673406986","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":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4313515592.pdf"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W2489155872","https://openalex.org/W2548582274","https://openalex.org/W2770784779","https://openalex.org/W2785994739","https://openalex.org/W2793089065","https://openalex.org/W2963214667","https://openalex.org/W2963965549","https://openalex.org/W3062268568","https://openalex.org/W4248062783","https://openalex.org/W4281637664","https://openalex.org/W4283701005","https://openalex.org/W4285253107","https://openalex.org/W4286462324","https://openalex.org/W4287009801","https://openalex.org/W4293215527","https://openalex.org/W4306404500"],"related_works":["https://openalex.org/W2441714773","https://openalex.org/W4293439290","https://openalex.org/W4213415692","https://openalex.org/W2942161250","https://openalex.org/W3045250558","https://openalex.org/W2149824583","https://openalex.org/W2884309141","https://openalex.org/W2093376952","https://openalex.org/W2963588781","https://openalex.org/W2010340235"],"abstract_inverted_index":{"In":[0],"this":[1,88],"study,":[2],"we":[3,52,68],"introduce":[4,69],"modified":[5,70],"degenerate":[6,72],"Changhee\u2013Genocchi":[7,73],"polynomials":[8,43,74],"of":[9,61,75,87,90],"the":[10,41,45,58,62,76],"second":[11,65,77],"kind,":[12],"and":[13,21,28,38,64,85],"analyze":[14],"some":[15,54,82],"properties":[16,86],"by":[17],"providing":[18],"several":[19],"relations":[20,27,39],"applications.":[22],"We":[23,79],"first":[24,63],"attain":[25],"diverse":[26],"formulas":[29],"covering":[30],"addition":[31],"formulas,":[32,37],"recurrence":[33],"rules,":[34],"implicit":[35],"summation":[36],"with":[40],"earlier":[42],"in":[44],"literature.":[46],"By":[47],"using":[48],"their":[49],"generating":[50],"function,":[51],"derive":[53,81],"new":[55,83],"relations,":[56],"including":[57],"Stirling":[59],"numbers":[60],"kinds.":[66],"Moreover,":[67],"higher-order":[71],"kind.":[78],"also":[80],"identities":[84],"type":[89],"polynomials.":[91]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2023-01-06T00:00:00"}
