{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:35:57Z","timestamp":1760236557307,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,30]],"date-time":"2021-11-30T00:00:00Z","timestamp":1638230400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council","doi-asserted-by":"publisher","award":["504070"],"award-info":[{"award-number":["504070"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan\u2019s G and Ap\u00e9ry\u2019s \u03b6(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.<\/jats:p>","DOI":"10.3390\/axioms10040324","type":"journal-article","created":{"date-parts":[[2021,11,30]],"date-time":"2021-11-30T04:48:37Z","timestamp":1638247717000},"page":"324","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4230-9925","authenticated-orcid":false,"given":"Robert","family":"Reynolds","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7252-5004","authenticated-orcid":false,"given":"Allan","family":"Stauffer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,30]]},"reference":[{"key":"ref_1","first-page":"105","article-title":"Archive for History of Exact Sciences","volume":"Volume 49","author":"Jacques","year":"1995","journal-title":"On the Early History of Bessel Functions"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Shishkina, E., and Sitnik, S. 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Soc."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Falconer, K. (1985). The Geometry of Fractal Sets (Cambridge Tracts in Mathematics), Cambridge University Press.","DOI":"10.1017\/CBO9780511623738"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/BF02844037","article-title":"Multiple integrals involving product of modified Bessel functions of the second kind","volume":"14","author":"Ragab","year":"1965","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"235","DOI":"10.12988\/imf.2020.91272","article-title":"A Method for Evaluating Definite Integrals in Terms of Special Functions with Examples","volume":"15","author":"Reynolds","year":"2020","journal-title":"Int. Math. Forum"},{"key":"ref_9","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Tables of Integrals, Series and Products, Academic Press. [6th ed.]."},{"key":"ref_10","unstructured":"Gr\u00f6bner, W., and Hofreiter, N. (1973). Integraltafel, Zweiter Teil, Bestimmte Integrale, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Oldham, K.B., Myland, J.C., and Spanier, J. (2009). An Atlas of Functions: With Equator, the Atlas Function Calculator, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-48807-3"},{"key":"ref_12","unstructured":"Lewin, L. (1981). Polylogarithms and Associated Functions, North-Holland Publishing Co."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/324\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:37:41Z","timestamp":1760168261000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/10\/4\/324"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,30]]},"references-count":12,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["axioms10040324"],"URL":"https:\/\/doi.org\/10.3390\/axioms10040324","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,11,30]]}}}