{"id":"https://openalex.org/W7220940573","doi":"https://doi.org/10.48550/arxiv.2610.07549","title":"Positivity of the basic hypergeometric series and the MacMahon identity","display_name":"Positivity of the basic hypergeometric series and the MacMahon identity","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220940573","doi":"https://doi.org/10.48550/arxiv.2610.07549"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.07549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07549","pdf_url":"https://arxiv.org/pdf/2610.07549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.07549","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157894921","display_name":"Alexandr Garbali","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Garbali, Alexandr","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5157894921"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9599999785423279,"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"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9599999785423279,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.039799999445676804,"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/T11191","display_name":"Mathematical functions and polynomials","score":9.999999747378752e-05,"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/basic-hypergeometric-series","display_name":"basic hypergeometric series","score":0.996999979019165},{"id":"https://openalex.org/keywords/q-binomial-coefficients","display_name":"q-binomial coefficients","score":0.8009999990463257},{"id":"https://openalex.org/keywords/multiset-permutations","display_name":"multiset permutations","score":0.9089999794960022},{"id":"https://openalex.org/keywords/inversion-statistic","display_name":"inversion statistic","score":0.9810000061988831},{"id":"https://openalex.org/keywords/generating-functions","display_name":"generating functions","score":0.7960000038146973},{"id":"https://openalex.org/keywords/binomial-series","display_name":"binomial series","score":0.7319999933242798}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8565000295639038},{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.815500020980835},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.6168000102043152},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.6022999882698059},{"id":"https://openalex.org/C197320386","wikidata":"https://www.wikidata.org/wiki/Q21028472","display_name":"Hypergeometric function","level":2,"score":0.5936999917030334},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.5875999927520752},{"id":"https://openalex.org/C176671685","wikidata":"https://www.wikidata.org/wiki/Q730600","display_name":"Hypergeometric distribution","level":2,"score":0.5831999778747559},{"id":"https://openalex.org/C92941272","wikidata":"https://www.wikidata.org/wiki/Q1062958","display_name":"Basic hypergeometric series","level":3,"score":0.5428000092506409},{"id":"https://openalex.org/C34718186","wikidata":"https://www.wikidata.org/wiki/Q209875","display_name":"Binomial coefficient","level":2,"score":0.5410000085830688},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.5033000111579895},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47350001335144043},{"id":"https://openalex.org/C1893757","wikidata":"https://www.wikidata.org/wiki/Q3653001","display_name":"Inversion (geology)","level":3,"score":0.44609999656677246},{"id":"https://openalex.org/C2781315470","wikidata":"https://www.wikidata.org/wiki/Q193623","display_name":"Binomial (polynomial)","level":2,"score":0.4447999894618988},{"id":"https://openalex.org/C60400383","wikidata":"https://www.wikidata.org/wiki/Q17028407","display_name":"Hypergeometric identity","level":5,"score":0.41839998960494995},{"id":"https://openalex.org/C201292218","wikidata":"https://www.wikidata.org/wiki/Q740970","display_name":"Recurrence relation","level":2,"score":0.3926999866962433},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.3896999955177307},{"id":"https://openalex.org/C158241908","wikidata":"https://www.wikidata.org/wiki/Q109744257","display_name":"Generalized hypergeometric function","level":3,"score":0.3869999945163727},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3702999949455261},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3702000081539154},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.35339999198913574},{"id":"https://openalex.org/C165175332","wikidata":"https://www.wikidata.org/wiki/Q5958021","display_name":"Hypergeometric function of a matrix argument","level":4,"score":0.35030001401901245},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3343000113964081},{"id":"https://openalex.org/C2776036281","wikidata":"https://www.wikidata.org/wiki/Q48769818","display_name":"Constraint (computer-aided design)","level":2,"score":0.3237999975681305},{"id":"https://openalex.org/C178197554","wikidata":"https://www.wikidata.org/wiki/Q1099110","display_name":"Sufficient statistic","level":2,"score":0.32010000944137573},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3093000054359436},{"id":"https://openalex.org/C73905626","wikidata":"https://www.wikidata.org/wiki/Q206925","display_name":"Power series","level":2,"score":0.303600013256073},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.2720000147819519},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2662999927997589},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.2556000053882599}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.07549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07549","pdf_url":"https://arxiv.org/pdf/2610.07549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"doi:10.48550/arxiv.2610.07549","doi":"https://doi.org/10.48550/arxiv.2610.07549","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.07549","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.07549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.07549","pdf_url":"https://arxiv.org/pdf/2610.07549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220940573.pdf","grobid_xml":"https://content.openalex.org/works/W7220940573.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4415343282"],"related_works":["https://openalex.org/W2129979922","https://openalex.org/W2957517418","https://openalex.org/W2052568162","https://openalex.org/W7101055304","https://openalex.org/W7160393050","https://openalex.org/W1996858000","https://openalex.org/W4313004969","https://openalex.org/W1996959888","https://openalex.org/W2794037826","https://openalex.org/W2149723041"],"abstract_inverted_index":{"The":[0,37,68,89],"basic":[1],"hypergeometric":[2],"series":[3,52],"${}_{n+1}\u03d5_{n}$":[4],"whose":[5],"upper":[6],"and":[7,62,82],"lower":[8],"parameters":[9],"are":[10],"non-negative":[11],"integer":[12],"powers":[13],"of":[14,43,60,65,75,85],"$q$":[15],"satisfying":[16],"a":[17,23,27,41,56,72,83,93],"certain":[18],"constraint":[19],"can":[20],"be,":[21],"after":[22],"normalisation,":[24],"computed":[25],"as":[26],"positive":[28],"polynomial.":[29,36],"We":[30],"derive":[31],"three":[32],"formulas":[33],"for":[34,78],"this":[35],"first":[38],"formula":[39,70,91],"is":[40,53,71,92],"generalisation":[42,84],"the":[44,79,86],"MacMahon":[45],"identity":[46],"in":[47],"which":[48],"an":[49],"infinite":[50],"binomial":[51],"identified":[54],"with":[55],"normalised":[57],"generating":[58,73],"function":[59,74],"des":[61],"maj":[63],"statistics":[64],"multiset":[66,76],"permutations.":[67],"second":[69],"permutations":[77],"inversion":[80],"statistic":[81,87],"mstc.":[88],"third":[90],"finite":[94],"sum":[95],"containing":[96],"$2n$":[97],"$q$-binomial":[98],"coefficients.":[99]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}