{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:49:33Z","timestamp":1760150973741,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,21]],"date-time":"2022-01-21T00:00:00Z","timestamp":1642723200000},"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":["Symmetry"],"abstract":"<jats:p>A four-dimensional integral containing g(x,y,z,t)Cn(\u03bb)(x) is derived. Cn(\u03bb)(x) is the Gegenbauer polynomial, g(x,y,z,t) is a product of the generalized logarithm quotient functions and the integral is taken over the region 0\u2264x\u22641,0\u2264y\u22641,0\u2264z\u22641,0\u2264t\u22641. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz\u2013Lerch zeta functions is asymmetrical. All the results in this work are new.<\/jats:p>","DOI":"10.3390\/sym14020205","type":"journal-article","created":{"date-parts":[[2022,1,23]],"date-time":"2022-01-23T20:36:27Z","timestamp":1642970187000},"page":"205","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Quadruple Integral Containing the Gegenbauer Polynomial Cn(\u03bb)(x): Derivation and Evaluation"],"prefix":"10.3390","volume":"14","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 M3J 1P3, 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 M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1112\/jlms\/s2-33.1.133","article-title":"An integral of products of Ultraspherical functions and a q-extension","volume":"33","author":"Askey","year":"1986","journal-title":"J. Lond. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1090\/S0002-9939-1963-0156171-5","article-title":"A Class of Integral Equations Involving Ultraspherical Polynomials as Kernel","volume":"14","author":"Srivastava","year":"1963","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/jlms\/s2-6.1.1","article-title":"Integral Representations for Ultraspherical Polynomials","volume":"s2-6","author":"Bingham","year":"1972","journal-title":"J. Lond. Math. Soc."},{"doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and its Applications, Cambridge University Press.","key":"ref_4","DOI":"10.1017\/CBO9781107325937"},{"unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Digital Library of Mathematical Functions.","key":"ref_5"},{"key":"ref_6","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"},{"doi-asserted-by":"crossref","unstructured":"Reynolds, R., and Stauffer, A. (2021). Quadruple Integral Involving the Logarithm and Product of Bessel Functions Expressed in Terms of the Lerch Function. Axioms, 10.","key":"ref_7","DOI":"10.3390\/axioms10040324"},{"unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Tables of Integrals, Series and Products, Academic Press. [6th ed.].","key":"ref_8"},{"doi-asserted-by":"crossref","unstructured":"Harrison, M., and Waldron, P. (2011). Mathematics for Economics and Finance, Taylor & Francis.","key":"ref_9","DOI":"10.4324\/9780203829998"},{"unstructured":"Laurincikas, A., and Garunkstis, R. (2013). The Lerch Zeta-Function, Springer Science & Business Media.","key":"ref_10"},{"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.].","key":"ref_11","DOI":"10.1007\/978-0-387-48807-3"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/205\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:05:05Z","timestamp":1760133905000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/205"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,21]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020205"],"URL":"https:\/\/doi.org\/10.3390\/sym14020205","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,21]]}}}