{"id":"https://openalex.org/W7213428442","doi":"https://doi.org/10.48550/arxiv.2609.16092","title":"From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials","display_name":"From Umbral Hyperbolic Integrals to a Cotangent Coefficient Formula for the Mittag Leffler Polynomials","publication_year":2026,"publication_date":"2026-09-14","ids":{"openalex":"https://openalex.org/W7213428442","doi":"https://doi.org/10.48550/arxiv.2609.16092"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.16092","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16092","pdf_url":"https://arxiv.org/pdf/2609.16092","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/2609.16092","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":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":0.73580002784729,"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":0.73580002784729,"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":0.24799999594688416,"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"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0052999998442828655,"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/umbral-calculus","display_name":"umbral calculus","score":0.8460000157356262},{"id":"https://openalex.org/keywords/dirichlet-beta-function","display_name":"Dirichlet beta function","score":0.7170000076293945},{"id":"https://openalex.org/keywords/meixner-polynomials","display_name":"Meixner polynomials","score":0.953000009059906},{"id":"https://openalex.org/keywords/generating-functions","display_name":"generating functions","score":0.6660000085830688},{"id":"https://openalex.org/keywords/formal-power-series","display_name":"formal power series","score":0.5350000262260437}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8704000115394592},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5975000262260437},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.5837000012397766},{"id":"https://openalex.org/C138187205","wikidata":"https://www.wikidata.org/wiki/Q131251","display_name":"Tangent","level":2,"score":0.5177000164985657},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5113999843597412},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4408000111579895},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.43860000371932983},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.36039999127388},{"id":"https://openalex.org/C10628310","wikidata":"https://www.wikidata.org/wiki/Q619458","display_name":"Orthogonal polynomials","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.33959999680519104},{"id":"https://openalex.org/C82240120","wikidata":"https://www.wikidata.org/wiki/Q190573","display_name":"Gamma function","level":2,"score":0.30169999599456787},{"id":"https://openalex.org/C118431652","wikidata":"https://www.wikidata.org/wiki/Q17098827","display_name":"Mehler\u2013Heine formula","level":5,"score":0.29440000653266907},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.2782999873161316},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.25679999589920044},{"id":"https://openalex.org/C68827188","wikidata":"https://www.wikidata.org/wiki/Q674533","display_name":"Inverse trigonometric functions","level":2,"score":0.2556000053882599}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.16092","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16092","pdf_url":"https://arxiv.org/pdf/2609.16092","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.2609.16092","doi":"https://doi.org/10.48550/arxiv.2609.16092","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.16092","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:2609.16092","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.16092","pdf_url":"https://arxiv.org/pdf/2609.16092","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213428442.pdf","grobid_xml":"https://content.openalex.org/works/W7213428442.grobid-xml"},"referenced_works_count":18,"referenced_works":["https://openalex.org/W186403072","https://openalex.org/W1244770280","https://openalex.org/W1985849796","https://openalex.org/W2001412859","https://openalex.org/W2093332696","https://openalex.org/W2154507350","https://openalex.org/W2313242044","https://openalex.org/W2567806540","https://openalex.org/W2798526604","https://openalex.org/W2912065219","https://openalex.org/W4212799367","https://openalex.org/W4386768358","https://openalex.org/W4394014594","https://openalex.org/W4416679650","https://openalex.org/W7085607328","https://openalex.org/W7130713871","https://openalex.org/W7168428236","https://openalex.org/W7204796115"],"related_works":["https://openalex.org/W7168428236","https://openalex.org/W7204796115","https://openalex.org/W4412477989","https://openalex.org/W4386306563","https://openalex.org/W6907483939","https://openalex.org/W2010685539","https://openalex.org/W4312230741","https://openalex.org/W6903654861","https://openalex.org/W3157413085","https://openalex.org/W4378802337"],"abstract_inverted_index":{"Let":[0],"\\(g_n(x)\\)":[1],"be":[2],"the":[3,13,39,58,67,72,81,118,130,139],"Mittag--Leffler":[4,83],"polynomials":[5],"defined":[6],"by":[7,33,61,99],"$$":[8,10,16,22,24,28,43,47,92,96,100,102,105,110],"\\sum_{n\\ge0}g_n(x)t^n=\\frac12\\left(\\frac{1+t}{1-t}\\right)^x.":[9],"We":[11,85],"derive":[12],"coefficient":[14],"formula":[15,74],"g_n(x)=\\frac1n\\sum_{j=0}^{\\lfloor":[17],"(n-1)/2\\rfloor}":[18],"(-1)^j\\frac{2^{n-2j-1}}{(n-2j-1)!}":[19],"[u^{2j}](u\\cot":[20],"u)^n\\,x^{n-2j},":[21],"equivalently":[23],"[u^{n-r}](u\\cot":[25],"u)^n":[26],"=(-1)^{(n-r)/2}\\frac{n(r-1)!}{2^{r-1}}[x^r]g_n(x).":[27],"The":[29,121],"identity":[30],"is":[31,54,124],"suggested":[32],"comparing":[34],"an":[35],"umbral":[36,82,115],"representation":[37,116],"of":[38,66,80,129],"hyperbolic":[40],"tangent":[41],"integral":[42,73],"K(a,b)=\\int_0^\\infty":[44],"\\frac{\\tanh^a":[45],"x}{x^b}\\,dx":[46],"with":[48],"a":[49,77,88,113,125,135],"compact":[50],"cotangent-coefficient":[51],"formula,":[52],"but":[53],"proved":[55,68],"independently":[56],"from":[57],"generating":[59],"function":[60],"formal":[62],"Lagrange--B\u00fcrmann":[63],"inversion.":[64],"Substitution":[65],"bridge":[69],"back":[70],"into":[71],"then":[75],"yields":[76],"rigorous":[78],"derivation":[79],"representation.":[84],"also":[86],"develop":[87],"Dirichlet--beta":[89],"analogue":[90],"for":[91,117],"I_{m,n}^\u03b2=\\int_0^\\infty\\frac{\\tanh^m":[93],"x}{x^n\\cosh":[94],"x}\\,dx.":[95],"Defining":[97],"\\(q_m(x)\\)":[98],"\\sum_{m\\ge0}q_m(x)t^m=(1-t^2)^{-1/2}\\exp(2x\\operatorname{artanh}t),":[101],"we":[103],"prove":[104],"[u^{m-r}]\\frac{u}{\\sin":[106],"u}(u\\cot":[107],"u)^m":[108],"=(-1)^{(m-r)/2}\\frac{r!}{2^r}[x^r]q_m(x).":[109],"This":[111],"gives":[112],"parallel":[114],"beta":[119],"integrals.":[120],"family":[122],"\\(q_m\\)":[123],"shifted":[126],"\\(c=-1,\u03b2=1\\)":[127],"specialization":[128],"classical":[131],"Meixner":[132],"family,":[133],"yielding":[134],"beta/Meixner":[136],"counterpart":[137],"to":[138],"zeta/Mittag--Leffler":[140],"correspondence.":[141]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-17T00:00:00"}
