{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:38:09Z","timestamp":1760243889520,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T00:00:00Z","timestamp":1268956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Generating functions play important roles in theory of orthogonal polynomials. In particular, it is important to consider generating functions that have symmetry. This paper is a survey on generating functions that define unitary operators. First, classical generating functions that define unitary operators are discussed. Next, group theoretical approach to generating functions that have unitarity are discussed.<\/jats:p>","DOI":"10.3390\/sym2010345","type":"journal-article","created":{"date-parts":[[2010,3,21]],"date-time":"2010-03-21T13:10:46Z","timestamp":1269177046000},"page":"346-365","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetry in Generating Functions"],"prefix":"10.3390","volume":"2","author":[{"given":"Shigeru","family":"Watanabe","sequence":"first","affiliation":[{"name":"The University of Aizu, Ikki-machi Tsuruga, Aizu-Wakamatsu City, Fukushima 965-8580, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2010,3,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1002\/cpa.3160140303","article-title":"On a Hilbert space of analytic functions and an associated integral transform Part I","volume":"14","author":"Bargmann","year":"1961","journal-title":"Comm. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"421","DOI":"10.3836\/tjm\/1270132271","article-title":"Hilbert spaces of analytic functions and the Gegenbauer polynomials","volume":"13","author":"Watanabe","year":"1990","journal-title":"Tokyo J. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"351","DOI":"10.3836\/tjm\/1255958675","article-title":"Spherical harmonics on U(n)\/U(n \u2212 1) and associated Hilbert spaces of analytic functions","volume":"23","author":"Watanabe","year":"2000","journal-title":"Tokyo J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1023\/A:1011076027510","article-title":"q-analogue of the Watanabe unitary transform associated to the q-continuous Gegenbauer polynomials","volume":"53","author":"Essadiq","year":"2000","journal-title":"Lett. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/j.jfa.2006.02.008","article-title":"Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators\u2014the case of the compact type","volume":"238","author":"Watanabe","year":"2006","journal-title":"J. Funct. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/BF01337923","article-title":"Verallgemeinerung und L\u00f6sung der Diracschen statistischen Gleichung","volume":"49","author":"Fock","year":"1928","journal-title":"Z. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Temme, N.M. (1996). Special Functions, An Introduction to the Classical Functions of Mathematical physics, John Wiley & Sons, Inc.","DOI":"10.1002\/9781118032572"},{"key":"ref_8","unstructured":"Prudnikov, A.P., Brychkov, Y.A., and Marichev, O.I. (1986). Integrals and Series, Gordon and Breach."},{"key":"ref_9","first-page":"46","article-title":"Generating functions of the Jacobi polynomials and related Hilbert spaces of analytic functions","volume":"74A","author":"Watanabe","year":"1998","journal-title":"Proc. Japan Acad."},{"key":"ref_10","first-page":"1125","article-title":"Generating functions and integral representations for the spherical functions on some classical Gelfand pairs","volume":"33","author":"Watanabe","year":"1993","journal-title":"J. Math. Kyoto Univ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1090\/S0002-9947-1977-0447483-0","article-title":"Composition series and intertwining operators for the spherical principal series I","volume":"229","author":"Johnson","year":"1977","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Takeuchi, M. (1994). Modern Spherical Functions, Amer. Math. Soc.","DOI":"10.1090\/mmono\/135"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"317","DOI":"10.2307\/1968205","article-title":"On linear functional operations and the moment problem for a finite interval in one or several dimensions","volume":"34","author":"Hildebrandt","year":"1933","journal-title":"Ann. of Math."},{"key":"ref_14","unstructured":"Helgason, S. (1984). Groups and Geometric Analysis, Academic Press."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/2\/1\/346\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:01:49Z","timestamp":1760220109000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/2\/1\/346"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,19]]},"references-count":14,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,3]]}},"alternative-id":["sym2010345"],"URL":"https:\/\/doi.org\/10.3390\/sym2010345","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2010,3,19]]}}}