{"id":"https://openalex.org/W1479753667","doi":"https://doi.org/10.37236/3693","title":"Congruences of Finite Summations of the Coefficients in certain Generating Functions","display_name":"Congruences of Finite Summations of the Coefficients in certain Generating Functions","publication_year":2014,"publication_date":"2014-06-09","ids":{"openalex":"https://openalex.org/W1479753667","doi":"https://doi.org/10.37236/3693","mag":"1479753667"},"language":"en","primary_location":{"id":"doi:10.37236/3693","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3693","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i2p45/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i2p45/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5113483021","display_name":"Po-Yi Huang","orcid":null},"institutions":[{"id":"https://openalex.org/I91807558","display_name":"National Cheng Kung University","ror":"https://ror.org/01b8kcc49","country_code":"TW","type":"education","lineage":["https://openalex.org/I91807558"]}],"countries":["TW"],"is_corresponding":true,"raw_author_name":"Po-Yi Huang","raw_affiliation_strings":["Department of Mathematics\r\nNational Cheng Kung University, Taiwan","Department of Mathematics#R##N#National Cheng Kung University, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics\r\nNational Cheng Kung University, Taiwan","institution_ids":["https://openalex.org/I91807558"]},{"raw_affiliation_string":"Department of Mathematics#R##N#National Cheng Kung University, Taiwan","institution_ids":["https://openalex.org/I91807558"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017034112","display_name":"Shu-Chung Liu","orcid":null},"institutions":[{"id":"https://openalex.org/I145876445","display_name":"University of Education, Winneba","ror":"https://ror.org/00y1ekh28","country_code":"GH","type":"education","lineage":["https://openalex.org/I145876445"]},{"id":"https://openalex.org/I4210154699","display_name":"National Hsinchu University of Education","ror":"https://ror.org/05prep819","country_code":"TW","type":"education","lineage":["https://openalex.org/I4210154699"]}],"countries":["GH","TW"],"is_corresponding":false,"raw_author_name":"Shu-Chung Liu","raw_affiliation_strings":["Department of Applied Mathematics\r\nNational Hsinchu University of Education, Taiwan","Department of Applied Mathematics#R##N#National Hsinchu University of Education, Taiwan"],"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics\r\nNational Hsinchu University of Education, Taiwan","institution_ids":["https://openalex.org/I4210154699"]},{"raw_affiliation_string":"Department of Applied Mathematics#R##N#National Hsinchu University of Education, Taiwan","institution_ids":["https://openalex.org/I145876445"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113502540","display_name":"Yeong-Nan Yeh","orcid":null},"institutions":[{"id":"https://openalex.org/I4210099334","display_name":"Institute of Mathematics, Academia Sinica","ror":"https://ror.org/0120q3031","country_code":"TW","type":"facility","lineage":["https://openalex.org/I4210099334","https://openalex.org/I84653119"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Yeong-Nan Yeh","raw_affiliation_strings":["Institute of Mathematics\r\nAcademia Sinica, Taiwan","Institute of Mathematics#R##N#Academia Sinica, Taiwan"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics\r\nAcademia Sinica, Taiwan","institution_ids":["https://openalex.org/I4210099334"]},{"raw_affiliation_string":"Institute of Mathematics#R##N#Academia Sinica, Taiwan","institution_ids":["https://openalex.org/I4210099334"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5113483021"],"corresponding_institution_ids":["https://openalex.org/I91807558"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.03181189,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"21","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11428","display_name":"Advanced Mathematical Identities","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9998000264167786,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/congruence-relation","display_name":"Congruence relation","score":0.9427182078361511},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8264118432998657},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7582254409790039},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.6564769744873047},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5282004475593567},{"id":"https://openalex.org/keywords/beta","display_name":"BETA (programming language)","score":0.4580878019332886},{"id":"https://openalex.org/keywords/generating-function","display_name":"Generating function","score":0.4195868968963623},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.41125935316085815},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3347983956336975},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1154269278049469},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.057913392782211304}],"concepts":[{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.9427182078361511},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8264118432998657},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7582254409790039},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.6564769744873047},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5282004475593567},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.4580878019332886},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.4195868968963623},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.41125935316085815},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3347983956336975},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1154269278049469},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.057913392782211304},{"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/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/3693","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3693","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i2p45/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/3693","is_oa":true,"landing_page_url":"https://doi.org/10.37236/3693","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i2p45/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8570293571","display_name":null,"funder_award_id":"101-2115-M-001-013-MY3","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"}],"funders":[{"id":"https://openalex.org/F4320321040","display_name":"National Science Council","ror":"https://ror.org/02kv4zf79"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1479753667.pdf","grobid_xml":"https://content.openalex.org/works/W1479753667.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W1970793819","https://openalex.org/W2048492341","https://openalex.org/W2048534739","https://openalex.org/W2048654814","https://openalex.org/W2048900920","https://openalex.org/W2052078837","https://openalex.org/W2059233545","https://openalex.org/W2066257237","https://openalex.org/W2067374988","https://openalex.org/W2080740693","https://openalex.org/W2103798576","https://openalex.org/W2121206741","https://openalex.org/W2123209697","https://openalex.org/W2126209209","https://openalex.org/W2128332871","https://openalex.org/W2130837115","https://openalex.org/W2131834133","https://openalex.org/W2139249899","https://openalex.org/W2963782622","https://openalex.org/W3105839574","https://openalex.org/W4249809825","https://openalex.org/W4285719527","https://openalex.org/W4402918723","https://openalex.org/W4403206322"],"related_works":["https://openalex.org/W2778946875","https://openalex.org/W2023974151","https://openalex.org/W2163207601","https://openalex.org/W4301322097","https://openalex.org/W3049507786","https://openalex.org/W2114281869","https://openalex.org/W2898484429","https://openalex.org/W4388583127","https://openalex.org/W2971526513","https://openalex.org/W2059792039"],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3],"develop":[4],"a":[5],"general":[6],"method":[7],"to":[8],"enumerate":[9,52],"the":[10,24,25,53],"congruences":[11,54],"of":[12,55],"finite":[13,58],"summations":[14,59],"$\\sum_{k=0}^{p-1}":[15],"\\frac{a_k}{m^k}":[16],"\\!\\pmod{p}$":[17,22],"and":[18,41,69],"$\\sum_{k=0}^{p-1-h}":[19],"\\frac{a_k":[20],"a_{k+h}}{B^k}":[21],"for":[23],"infinite":[26],"sequence":[27],"$\\{a_n\\}_{n\\ge":[28],"0}$":[29],"with":[30,47],"generating":[31,61],"functions":[32,62],"$(1+x":[33],"f(x))^\\frac{N}{2}$,":[34],"where":[35],"$f(x)$":[36],"is":[37,43],"an":[38,44],"integer":[39,46],"polynomial":[40],"$N$":[42],"odd":[45],"$|N|<":[48],"p$.":[49],"We":[50],"also":[51],"some":[56],"similar":[57],"involving":[60],"$\\frac{1-\\alpha":[63,70],"x":[64],"-\\sqrt{1-2(\\alpha+\\beta)x":[65],"+":[66],"Bx^2}}{\\beta":[67],"x}$":[68],"x-\\sqrt{1-2\\alpha":[71],"x+(\\alpha^2-4\\beta)x^2}}{2\\beta":[72],"x^2}$.":[73]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}