{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T16:47:35Z","timestamp":1762015655545,"version":"3.41.2"},"reference-count":5,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:p> In the first part of the present paper, we have proved that, in order to find an explicit expression for the action, on the [Formula: see text]-orthogonal polynomials, of the 1-parameter unitary groups [Formula: see text] and [Formula: see text], the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable is required. In this paper we solve this problem. The solution is obtained in two steps. First we determine the explicit form of normal order in the more general framework of [Formula: see text]-mode-type interacting Fock spaces. Then this result is applied to solve the inverse normal order problem in the semi-circle case. <\/jats:p>","DOI":"10.1142\/s1230161222500184","type":"journal-article","created":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T03:26:24Z","timestamp":1675308384000},"source":"Crossref","is-referenced-by-count":1,"title":["The Quantum Mechanics Canonically Associated to Free Probability II: The Normal and Inverse Normal Order Problem"],"prefix":"10.1142","volume":"29","author":[{"given":"Luigi","family":"Accardi","sequence":"first","affiliation":[{"name":"Centro Vito Volterra, Universit\u00e0 di Roma \u201cTor Vergata\u201d, Via Columbia 2, 00133 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tarek","family":"Hamdi","sequence":"additional","affiliation":[{"name":"Department of Management Information Systems, College of Business Management, Qassim University, Ar Rass, Saudi Arabia"},{"name":"Laboratoire d\u2019Analyse Math\u00e9matiques et Applications LR11ES11, Universit\u00e9 de Tunis El-Manar, Tunisie"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yun Gang","family":"Lu","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bari, via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"S1230161222500184BIB001","doi-asserted-by":"crossref","first-page":"2250017","DOI":"10.1142\/S1230161222500172","volume":"29","author":"Accardi L.","year":"2022","journal-title":"Open Sys. Information Dyn."},{"key":"S1230161222500184BIB002","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/BF02099625","author":"Accardi L.","year":"1996","journal-title":"Commun. Math. Phys. 180"},{"key":"S1230161222500184BIB003","doi-asserted-by":"crossref","first-page":"2250003","DOI":"10.1142\/S0219025722500035","volume":"25","author":"Accardi L.","year":"2022","journal-title":"Infin. Dimen. Anal. Quant. Prob. Rel. Top."},{"key":"S1230161222500184BIB004","first-page":"1997","author":"Accardi L.","year":"1997","journal-title":"IIAS Rep."},{"key":"S1230161222500184BIB005","volume":"5","author":"Rainville E. D.","year":"1960","journal-title":"Special functions"}],"container-title":["Open Systems &amp; Information Dynamics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1230161222500184","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T03:26:37Z","timestamp":1675308397000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S1230161222500184"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12]]},"references-count":5,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["10.1142\/S1230161222500184"],"URL":"https:\/\/doi.org\/10.1142\/s1230161222500184","relation":{},"ISSN":["1230-1612","1793-7191"],"issn-type":[{"type":"print","value":"1230-1612"},{"type":"electronic","value":"1793-7191"}],"subject":[],"published":{"date-parts":[[2022,12]]},"article-number":"2250018"}}