{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:36:06Z","timestamp":1747190166514,"version":"3.40.5"},"reference-count":7,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T00:00:00Z","timestamp":1595808000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2020,7,27]]},"abstract":"<jats:p>For <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>k<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>\u2124<\/mml:mi><\/mml:math>, the generalized Arakawa\u2013Kaneko zeta functions with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>a<\/mml:mi><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>b<\/mml:mi><\/mml:math>, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>c<\/mml:mi><\/mml:math> parameters are given by the Laplace-Mellin integral <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:msub><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>;<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>b<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mi mathvariant=\"normal\">\u0393<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><mml:mstyle displaystyle=\"true\"><mml:mrow><mml:msubsup><mml:mo stretchy=\"false\">\u222b<\/mml:mo><mml:mn>0<\/mml:mn><mml:mi>\u221e<\/mml:mi><\/mml:msubsup><mml:mrow><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mtext>Li<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mn>1<\/mml:mn><mml:mo>\u2212<\/mml:mo><mml:msup><mml:mrow><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>a<\/mml:mi><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfenced><mml:mo>\/<\/mml:mo><mml:msup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>\u2212<\/mml:mo><mml:msup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfenced><mml:msup><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>\u2212<\/mml:mo><mml:mi>x<\/mml:mi><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>\u2212<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mrow><\/mml:mstyle><mml:mtext>d<\/mml:mtext><mml:mi>t<\/mml:mi><\/mml:math>, where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>\u211c<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>x<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math> if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>k<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>, and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>\u211c<\/mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>x<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math> if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mi>k<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math>. In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>b<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math> parameters is obtained. Moreover, explicit, difference, and Raabe\u2019s formulas for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:msub><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>s<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>;<\/mml:mo><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>b<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> are derived.<\/jats:p>","DOI":"10.1155\/2020\/2041262","type":"journal-article","created":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T23:31:43Z","timestamp":1595892703000},"page":"1-6","source":"Crossref","is-referenced-by-count":0,"title":["On Generalized Arakawa\u2013Kaneko Zeta Functions with Parameters <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>b<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><\/mml:math>"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1003-7248","authenticated-orcid":true,"given":"Nestor G.","family":"Acala","sequence":"first","affiliation":[{"name":"Mathematics Department, College of Natural Sciences and Mathematics, Mindanao State University-Main Campus, Marawi, Lanao Del Sur, Philippines"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edward Rowe M.","family":"Aleluya","sequence":"additional","affiliation":[{"name":"Department of Physical Sciences and Mathematics, College of Science and Environment, Mindanao State University-Naawan, Naawan, Misamis Oriental, Philippines"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1017\/s0027763000006954"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.5802\/jtnb.197"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/s11139-009-9205-x"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.2206\/kyushujm.65.15"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.7153\/jca-06-10"},{"key":"6","first-page":"7","volume":"2","year":"2010","journal-title":"International Journal of Mathematical Combinatorics"},{"issue":"2","key":"7","first-page":"21","volume":"10","year":"2019","journal-title":"Journal of Inequalities and Special Functions"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/2041262.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/2041262.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2020\/2041262.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T23:31:48Z","timestamp":1595892708000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2020\/2041262\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,27]]},"references-count":7,"alternative-id":["2041262","2041262"],"URL":"https:\/\/doi.org\/10.1155\/2020\/2041262","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2020,7,27]]}}}