{"id":"https://openalex.org/W7156788314","doi":"https://doi.org/10.48550/arxiv.2604.23206","title":"A Proof of Bala's General-$m$ Representation of the Harmonic Numbers","display_name":"A Proof of Bala's General-$m$ Representation of the Harmonic Numbers","publication_year":2026,"publication_date":"2026-04-25","ids":{"openalex":"https://openalex.org/W7156788314","doi":"https://doi.org/10.48550/arxiv.2604.23206"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.23206","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.23206","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.23206","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5134752296","display_name":"Tong Niu","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Niu, Tong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5134752296"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.32829999923706055,"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":0.32829999923706055,"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/T11166","display_name":"Analytic Number Theory Research","score":0.1177000030875206,"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.07940000295639038,"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/integer","display_name":"Integer (computer science)","score":0.7904999852180481},{"id":"https://openalex.org/keywords/identity","display_name":"Identity (music)","score":0.609000027179718},{"id":"https://openalex.org/keywords/substitution","display_name":"Substitution (logic)","score":0.588699996471405},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.5697000026702881},{"id":"https://openalex.org/keywords/harmonic","display_name":"Harmonic","score":0.40459999442100525},{"id":"https://openalex.org/keywords/harmonic-number","display_name":"Harmonic number","score":0.39469999074935913},{"id":"https://openalex.org/keywords/multiple","display_name":"Multiple","score":0.3901999890804291}],"concepts":[{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.7904999852180481},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.784600019454956},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.609000027179718},{"id":"https://openalex.org/C2778220771","wikidata":"https://www.wikidata.org/wiki/Q1522579","display_name":"Substitution (logic)","level":2,"score":0.588699996471405},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5697000026702881},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5275999903678894},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.40459999442100525},{"id":"https://openalex.org/C28581250","wikidata":"https://www.wikidata.org/wiki/Q13407133","display_name":"Harmonic number","level":3,"score":0.39469999074935913},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.3901999890804291},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3874000012874603},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.36550000309944153},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.2892000079154968},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.27570000290870667},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27300000190734863},{"id":"https://openalex.org/C56863137","wikidata":"https://www.wikidata.org/wiki/Q743139","display_name":"Half-integer","level":2,"score":0.266400009393692},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.2535000145435333}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.23206","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.23206","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.23206","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.23206","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"For":[0],"every":[1,6],"nonzero":[2,52],"integer":[3,7,53],"$m$":[4,54],"and":[5,44,68,101],"$n":[8],"\\ge":[9],"1$,":[10],"the":[11,25,55,62,102,109],"$n$\\textsuperscript{th}":[12],"harmonic":[13],"number":[14],"$H_n":[15],"=":[16,42,46,86,115,119],"1":[17],"+":[18,20,22,34],"\\tfrac12":[19],"\\dots":[21],"\\tfrac1n$":[23],"satisfies":[24],"identity":[26,56],"\\[":[27],"H_n":[28],"\\;=\\;":[29],"\\frac{1}{m}\\,\\sum_{k=1}^{n}":[30],"\\frac{(-1)^{k+1}}{k}\\,":[31],"\\binom{m":[32],"k}{k}\\binom{n":[33],"(m-1)k}{n":[35],"-":[36],"k}.":[37],"\\]":[38],"The":[39,79,121],"cases":[40],"$m":[41,45,128],"1$":[43],"2$":[47],"are":[48],"classical;":[49],"for":[50],"general":[51],"was":[57],"conjectured":[58],"by":[59],"P.~Bala":[60],"in":[61,66,77],"OEIS":[63],"entry":[64],"A001008":[65],"2022":[67],"remained":[69],"open.":[70],"We":[71],"prove":[72],"it":[73],"here,":[74],"working":[75],"throughout":[76],"$\\QQ[[x]]$.":[78],"proof":[80],"reduces,":[81],"via":[82],"a":[83,92],"substitution":[84,108],"$u":[85],"x/(1-x)^m$,":[87],"to":[88,125],"two":[89],"formal-power-series":[90],"identities:":[91],"Lagrange--B\u00fcrmann":[93],"evaluation":[94],"of":[95,113],"$\\sum_{k\\ge1}":[96],"\\binom{mk}{k}":[97],"u^k":[98],"/":[99],"k$,":[100],"fixed-point":[103],"fact":[104],"that":[105,107],"under":[106],"unique":[110],"solution":[111],"$v(u)$":[112],"$v":[114,118],"u(1-v)^{m}$":[116],"is":[117],"x$.":[120],"argument":[122],"extends":[123],"verbatim":[124],"arbitrary":[126],"complex":[127],"\\ne":[129],"0$.":[130]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-29T00:00:00"}