{"id":"https://openalex.org/W7204796115","doi":"https://doi.org/10.48550/arxiv.2608.27717","title":"Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals","display_name":"Dirichlet Beta Analogues, Sign Alternation, and Lerch Unification of Hyperbolic and Logarithmic Tangent Integrals","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204796115","doi":"https://doi.org/10.48550/arxiv.2608.27717"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.27717","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27717","pdf_url":"https://arxiv.org/pdf/2608.27717","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2608.27717","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123934152","display_name":"Luc Rams\u00e8s Talla Waffo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Waffo, Luc Rams\u00e8s Talla","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5123934152"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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":0.9932000041007996,"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":0.9932000041007996,"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":0.004999999888241291,"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":0.000699999975040555,"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/dirichlet-beta-function","display_name":"Dirichlet beta function","score":0.9869999885559082},{"id":"https://openalex.org/keywords/coefficient-extraction","display_name":"coefficient extraction","score":0.5929999947547913},{"id":"https://openalex.org/keywords/generalized-bernoulli-polynomials","display_name":"generalized Bernoulli polynomials","score":0.9290000200271606},{"id":"https://openalex.org/keywords/continuous-hahn-polynomials","display_name":"continuous Hahn polynomials","score":0.8980000019073486},{"id":"https://openalex.org/keywords/lerch-transcendent","display_name":"Lerch transcendent","score":0.7770000100135803}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.814300000667572},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5009999871253967},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5004000067710876},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.46639999747276306},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.4629000127315521},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44440001249313354},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39430001378059387},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.3734999895095825},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.3560999929904938},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.353300005197525},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3357999920845032},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.33219999074935913},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.3052999973297119},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2786000072956085},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C26826806","wikidata":"https://www.wikidata.org/wiki/Q438358","display_name":"Alternation (linguistics)","level":2,"score":0.272599995136261},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2632000148296356},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.27717","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27717","pdf_url":"https://arxiv.org/pdf/2608.27717","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.27717","doi":"https://doi.org/10.48550/arxiv.2608.27717","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.27717","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.27717","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.27717","pdf_url":"https://arxiv.org/pdf/2608.27717","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204796115.pdf","grobid_xml":"https://content.openalex.org/works/W7204796115.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W1244770280","https://openalex.org/W1662706126","https://openalex.org/W2582836336","https://openalex.org/W2950773481","https://openalex.org/W3022028954","https://openalex.org/W4254246953","https://openalex.org/W4416679650","https://openalex.org/W7085607328","https://openalex.org/W7168428236"],"related_works":["https://openalex.org/W7168428236","https://openalex.org/W4283581381","https://openalex.org/W4416679650","https://openalex.org/W7213428442","https://openalex.org/W2790599138","https://openalex.org/W7154827893","https://openalex.org/W2514684201","https://openalex.org/W4412454808","https://openalex.org/W4402517102","https://openalex.org/W7125825932"],"abstract_inverted_index":{"We":[0,72],"develop":[1],"a":[2,6,65,69,145,181],"Dirichlet-beta":[3],"counterpart":[4],"and":[5,15,68,98,108,136,159,175],"structural":[7,120],"extension":[8],"of":[9,144,185],"the":[10,26,41,83,87,96,116,133,140,156,162,169],"coefficient-extraction":[11],"framework":[12,179],"for":[13,78,123,155,161],"hyperbolic":[14],"logarithmic":[16,99,173],"tangent":[17,100],"integrals.":[18],"For":[19],"integers":[20],"$m\\geq":[21],"n\\geq1$":[22],"with":[23,104,128,149],"$m+n$":[24],"even,":[25],"zeta-type":[27],"formula":[28],"\\[":[29,44],"\\int_0^\\infty\\frac{\\tanh^{m+1}x}{x^{n+1}}\\,dx":[30],"=":[31,49],"(-1)^{(m-n)/2}":[32,50],"\\sum_{p=\\lceil":[33,51],"n/2\\rceil}^{(m+n)/2}":[34,52],"\\binom{2p}{n}(2^{2p+1}-1)":[35],"\\frac{\u03b6(2p+1)}{\u03c0^{2p}}":[36],"[u^{m+n-2p}](u\\cot":[37],"u)^{m+1}":[38],"\\]":[39,59],"admits":[40],"beta-type":[42],"analogue":[43],"\\int_0^\\infty":[45],"\\frac{\\tanh^m":[46],"x}{x^n\\cosh":[47],"x}\\,dx":[48],"2^{2p}\\binom{2p-1}{n-1}":[53],"\\frac{\u03b2(2p)}{\u03c0^{2p-1}}":[54],"[u^{m+n-2p}]":[55],"\\frac{u}{\\sin":[56],"u}(u\\cot":[57],"u)^m.":[58],"The":[60,151],"proof":[61],"uses":[62],"derivative":[63],"polynomials,":[64,94],"beta":[66,137,157],"kernel,":[67],"coefficient-collapse":[70],"argument.":[71],"then":[73],"establish":[74],"strict":[75],"sign":[76,88],"alternation":[77,118],"several":[79],"coefficient":[80],"families.":[81],"In":[82],"boundary":[84],"shifted-hyperbolic":[85],"case":[86,158,164],"is":[89,165],"controlled":[90],"by":[91],"generalized":[92],"Bernoulli":[93],"while":[95],"arctanh":[97],"families":[101],"are":[102,139],"identified":[103],"continuous":[105,109],"dual":[106],"Hahn":[107,110],"polynomials.":[111],"Their":[112],"zero":[113],"distributions":[114],"turn":[115],"observed":[117],"into":[119],"statements":[121],"valid":[122],"all":[124],"admissible":[125],"indices.":[126],"Finally,":[127],"$\u03bb(s)=(1-2^{-s})\u03b6(s)$,":[129],"we":[130],"show":[131],"that":[132],"parallel":[134],"zeta--lambda":[135,163],"identities":[138],"two":[141],"binary":[142],"specializations":[143],"single":[146],"Lerch-transcendent":[147],"scheme":[148],"$\\varepsilon\\in\\{0,1\\}$.":[150],"same":[152],"convention":[153],"$\\varepsilon=0$":[154],"$\\varepsilon=1$":[160],"used":[166],"throughout.":[167],"Besides":[168],"shifted":[170],"hyperbolic,":[171],"odd-$\\sinh$,":[172],"tangent,":[174],"$\\tanh$":[176],"families,":[177],"this":[178],"yields":[180],"further":[182],"unified":[183],"pair":[184],"reciprocal-arctanh":[186],"integral":[187],"formulae":[188],"on":[189],"$(0,1)$.":[190]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-01T00:00:00"}