{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T13:40:05Z","timestamp":1709214005358},"reference-count":29,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2008,8]]},"abstract":"<jats:p>Let K be any unital commutative \u211a-algebra and z = (z<jats:sub>1<\/jats:sub>, z<jats:sub>2<\/jats:sub>, \u2026, z<jats:sub>n<\/jats:sub>) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]\u3008\u3008z\u3009\u3009 the formal power series algebra of z over K[[t]]. In [29], for each automorphism F<jats:sub>t<\/jats:sub>(z) = z - H<jats:sub>t<\/jats:sub>(z) of K[[t]]\u3008\u3008z\u3009\u3009 with H<jats:sub>t=0<\/jats:sub>(z) = 0 and o(H(z)) \u2265 1, a [Formula: see text] (noncommutative symmetric) system [28] \u03a9<jats:sub>F<jats:sub>t<\/jats:sub><\/jats:sub>has been constructed. Consequently, we get a Hopf algebra homomorphism [Formula: see text] from the Hopf algebra [Formula: see text] [9] of NCSFs (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the [Formula: see text] system \u03a9<jats:sub>F<jats:sub>t<\/jats:sub><\/jats:sub>by using some identities of NCSFs derived in [9] and the homomorphism [Formula: see text]. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system \u03a9<jats:sub>F<jats:sub>t<\/jats:sub><\/jats:sub>for the Taylor series expansions of u(F<jats:sub>t<\/jats:sub>) and [Formula: see text]; the D-Log and the formal flow of F<jats:sub>t<\/jats:sub>and inversion formulas for the inverse map of F<jats:sub>t<\/jats:sub>. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.<\/jats:p>","DOI":"10.1142\/s0218196708004615","type":"journal-article","created":{"date-parts":[[2008,8,25]],"date-time":"2008-08-25T09:25:17Z","timestamp":1219656317000},"page":"869-899","source":"Crossref","is-referenced-by-count":1,"title":["NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND THE INVERSION PROBLEM"],"prefix":"10.1142","volume":"18","author":[{"given":"WENHUA","family":"ZHAO","sequence":"first","affiliation":[{"name":"Department of Mathematics, Illinois State University, Normal, IL 61790-4520, USA"}]}],"member":"219","published-online":{"date-parts":[[2012,1,25]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1982-15032-7"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-05-07570-2"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196702001139"},{"key":"rf4","first-page":"159","volume":"1","author":"Duchamp G.","journal-title":"Discrete Math. 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