{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:50:34Z","timestamp":1760151034223,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,15]],"date-time":"2022-02-15T00:00:00Z","timestamp":1644883200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education","award":["NRF-2020R111A1A01052440"],"award-info":[{"award-number":["NRF-2020R111A1A01052440"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The literature has an astonishingly large number of integral formulae involving a range of special functions. In this paper, by using three Beta function formulae, we aim to establish three integral formulas whose integrands are products of the generalized hypergeometric series p+1Fp and the integrands of the three Beta function formulae. Among the many particular instances for our formulae, several are stated clearly. Moreover, an intriguing inequality that emerges throughout the proving procedure is shown. It is worth noting that the three integral formulae shown here may be expanded further by using a variety of more generalized special functions than p+1Fp. Symmetry occurs naturally in the Beta and p+1Fp functions, which are two of the most important functions discussed in this study.<\/jats:p>","DOI":"10.3390\/sym14020389","type":"journal-article","created":{"date-parts":[[2022,2,15]],"date-time":"2022-02-15T22:44:57Z","timestamp":1644965097000},"page":"389","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Certain Integral Formulae Associated with the Product of Generalized Hypergeometric Series and Several Elementary Functions Derived from Formulas for the Beta Function"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7240-7737","authenticated-orcid":false,"given":"Junesang","family":"Choi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Dongguk University, Gyeongju 38066, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2153-0524","authenticated-orcid":false,"given":"Shantha Kumari","family":"Kurumujji","sequence":"additional","affiliation":[{"name":"Department of Mathematics, AJ Institute of Engineering and Technology (Visvesvaraya Technological University (VTU)), Mangaluru 575006, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1217-963X","authenticated-orcid":false,"given":"Adem","family":"Kilicman","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Institute for Mathematical Research, University Putra Malaysia, UPM, Serdang 43400, Selangor, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3902-3050","authenticated-orcid":false,"given":"Arjun Kumar","family":"Rathie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Vedant College of Engineering & Technology (Rajasthan Technical University), Kota 324010, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,15]]},"reference":[{"key":"ref_1","unstructured":"Rainville, E.D. (1960). Special Functions, Macmillan Company."},{"key":"ref_2","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.","DOI":"10.1017\/CBO9781107325937"},{"key":"ref_3","unstructured":"Bailey, W.N. (1935). Generalized Hypergeometric Series, Cambridge University Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier.","DOI":"10.1016\/B978-0-12-385218-2.00002-5"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Choi, J., Qureshi, M.I., Bhat, A.H., and Majid, J. (2021). Reduction formulas for generalized hypergeometric series associated with new sequences and applications. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040150"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Choi, J. (2021). Certain applications of generalized Kummer\u2019s summation formulas for 2F1. Symmetry, 13.","DOI":"10.3390\/sym13081538"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1112\/plms\/s1-35.1.284","article-title":"Summation of a certain series","volume":"35","author":"Dixon","year":"1902","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_8","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953). Higher Transcendental Functions, McGraw-Hill Book Company."},{"key":"ref_9","first-page":"278","article-title":"A new proof of the classical Watson\u2019s summation theorem","volume":"11","author":"Rakha","year":"2011","journal-title":"Appl. Math. E-Notes"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1080\/10652469.2010.549487","article-title":"Generalization of classical summation theorems for the series 2F1 and 3F2 with applications","volume":"22","author":"Rakha","year":"2011","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_11","unstructured":"Prudnikov, A.P., Brychkov, Y.A., and Marichev, O.I. (1990). Integrals and Series. More Special Functions, Gordon and Breach Science Publishers."},{"key":"ref_12","unstructured":"Slater, L.J. (1966). Generalized Hypergeometric Functions, Cambridge University Press."},{"key":"ref_13","first-page":"13","article-title":"A note on generalized hypergeometric series","volume":"23","author":"Watson","year":"1925","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_14","first-page":"32","article-title":"Dixon\u2019s theorem on generalized hypergeometric functions","volume":"22","author":"Watson","year":"1924","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1112\/plms\/s2-23.1.104","article-title":"A group of generalized hypergeometric series: Relations between 120 allied series of the type F(a, b, c; e, f)","volume":"23","author":"Whipple","year":"1925","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1112\/plms\/s2-25.1.525","article-title":"Well-poised series and other generalized hypergeometric series","volume":"25","author":"Whipple","year":"1926","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_17","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1954). Tables of Integral Transforms, McGraw-Hill Book Company."},{"key":"ref_18","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Table of Integrals, Series, and Products, Academic Press. [6th ed.]."},{"key":"ref_19","first-page":"31","article-title":"On the sum of certain Appell\u2019s series","volume":"20","author":"Lavoie","year":"1969","journal-title":"Ganita"},{"key":"ref_20","unstructured":"Edwards, J. (1922). A Treatise on the Integral Calculus with Applications, Examples and Problems, Macmillan. [1st ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1007\/BF01450936","article-title":"Beta function formulae and integrals involving E-function","volume":"142","author":"MacRobert","year":"1961","journal-title":"Math. Annalen"},{"key":"ref_22","unstructured":"Wade, W.R. (2010). An Introduction to Analysis, Pearson Education Inc.. [4th ed.]."},{"key":"ref_23","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, John Wiley and Sons."},{"key":"ref_24","unstructured":"Srivastava, H.M., and Karlsson, P.W. (1985). Multiple Gaussian Hypergeometric Series, John Wiley and Sons."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., Saxena, R.K., and Haubold, H.J. (2010). The H-Function, Theory and Applications, Springer.","DOI":"10.1007\/978-1-4419-0916-9"},{"key":"ref_26","first-page":"366","article-title":"Formal solution of certain new pair of dual integral equations involving H-function","volume":"52","author":"Saxena","year":"1982","journal-title":"Proc. Nat. Acad. Sci. India Sect."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4109","DOI":"10.1088\/0305-4470\/20\/13\/019","article-title":"New properties of generalized hypergeometric series derivable from Feynman integrals. I. Transformation and reduction formulae","volume":"20","year":"1987","journal-title":"J. Phys. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4119","DOI":"10.1088\/0305-4470\/20\/13\/020","article-title":"New properties of generalized hypergeometric series derivable from Feynman integrals. II. A generalisation of the H function","volume":"20","year":"1987","journal-title":"J. Phys. A"},{"key":"ref_29","first-page":"401","article-title":"Open problem: Who knows about the \u2135-function?","volume":"1","author":"Baumann","year":"1998","journal-title":"Appl. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Gangha, V.G., Mayr, E.W., and Vorozhtsov, W.G. (2001). Fractional driftless Fokker-Planck equation with power law diffusion coefficients. Computer Algebra in Scientific Computing (CASC Konstanz 2001), Springer.","DOI":"10.1007\/978-3-642-56666-0"},{"key":"ref_31","first-page":"41","article-title":"Beta supper-functions on supper-Grassmannians","volume":"1","author":"Im","year":"2019","journal-title":"Lett. Math. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/389\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:20:22Z","timestamp":1760134822000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/389"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,15]]},"references-count":31,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020389"],"URL":"https:\/\/doi.org\/10.3390\/sym14020389","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,2,15]]}}}