{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:31:51Z","timestamp":1760059911284,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Indefinite integrals of products of exponential functions, power functions and generalized hypergeometric functions of some types are considered. Necessary and sufficient conditions are established for the algebraic independence of large sets of such functions (for various parameters) and their derivatives, as well as their values. All the algebraic relations between these functions are written out explicitly.<\/jats:p>","DOI":"10.3390\/axioms14080572","type":"journal-article","created":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T08:13:31Z","timestamp":1753431211000},"page":"572","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Algebraic Independence of the Values of Functions That Are Certain Integrals Involving the 1F1(1; \u03bb + 1; z) Hypergeometric Function"],"prefix":"10.3390","volume":"14","author":[{"given":"Vasily","family":"Gorelov","sequence":"first","affiliation":[{"name":"Moscow Power Engineering Institute, National Research University, 111250 Moscow, Russia"}]},{"given":"Gennady","family":"Voronov","sequence":"additional","affiliation":[{"name":"Department of Higher Mathematics, MIREA, Russian Technological University, 119454 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Gorelov, V.A. (2023). On the Algebraic Independence of the Values of Functions Associated with Hypergeometric Functions. Axioms, 12.","DOI":"10.3390\/axioms12010036"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1134\/S0001434625030046","article-title":"On algebraic properties of functions related to hypergeometric functions","volume":"117","author":"Gorelov","year":"2025","journal-title":"Math. Notes"},{"key":"ref_3","first-page":"1","article-title":"\u00dcber einige Anwendungen Diophantischer Approximationen","volume":"1","author":"Siegel","year":"1929","journal-title":"Abh. Preuss. Acad. Wiss. Phys.-Math. Kl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Siegel, C.L. (1949). Transcendental Numbers, Princeton University Press.","DOI":"10.1515\/9781400882359"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Shidlovsky, A.B. (1989). Transcendental Numbers, Walter de Gruyter.","DOI":"10.1515\/9783110889055"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"903","DOI":"10.4007\/annals.2021.194.3.7","article-title":"A non-hypergeometric E-function","volume":"194","author":"Fresan","year":"2021","journal-title":"Ann. of Math."},{"key":"ref_7","unstructured":"Fischler, S., and Rivoal, T. (2020, June 04). On Siegel\u2019s Problem for E-Functions. Available online: https:\/\/arxiv.org\/pdf\/1910.06817.pdf."},{"key":"ref_8","first-page":"705","article-title":"S\u00e9ries Gevrey de type arithm\u00e9tique","volume":"151","author":"Andre","year":"2000","journal-title":"I. Th\u00e9oremes de puret\u00e9 et de dualit\u00e9 Ann. of Math."},{"key":"ref_9","first-page":"283","article-title":"On transcendence and algebraic independence of the values of some functions","volume":"11","author":"Shidlovsky","year":"1959","journal-title":"Trans. Moscow Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1134\/S0001434616050059","article-title":"On the algebraic properties of solutions of inhomogeneous hypergeometric equations","volume":"99","author":"Gorelov","year":"2016","journal-title":"Math. Notes"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF02421605","article-title":"Sur les relations alg\u00e9briques entre les int\u00e9grales ind\u00e9finies","volume":"78","author":"Ostrowski","year":"1946","journal-title":"Acta Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.2307\/2373294","article-title":"Algebraic groups and algebraic dependence","volume":"90","author":"Kolchin","year":"1968","journal-title":"Amer. J. Math."},{"key":"ref_13","unstructured":"Wolfram, S. (2002). The Mathematica Book, Addison-Wesley. [5th ed.]."},{"key":"ref_14","first-page":"40","article-title":"Mathematical modeling of the influence of relaxation processes on the temperature field in an elastic half-space","volume":"5","author":"Dzhemesyuk","year":"2017","journal-title":"Rus. Technol. J."},{"key":"ref_15","unstructured":"Riddle, A. (1994). Applied Electronic Engineering with Mathematica, Wesley."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/572\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:15:46Z","timestamp":1760033746000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/572"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,25]]},"references-count":15,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080572"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080572","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,25]]}}}