An extension of Van Vleck’s functional equation for the sine

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• 作者：B. Belaid ; E. Elhoucien
• 关键词：Key words and phrasessemigroup ; d’Alembert’s functional equation ; sine function ; Van Vleck ; involution ; multiplicative function ; homomorphism
• 刊名：Acta Mathematica Hungarica
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：150
• 期：1
• 页码：258-267
• 全文大小：501 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2632
• 卷排序：150

文摘

In [7] H. Stetkær obtained the solutions of Van Vleck’s functional equation$$ f (x\tau(y)z_0) - f(xyz_0) = 2f(x)f(y), \quad x, y \in S $$for the sine where S is a semigroup, \({\tau}\) is an involution of S and z<sub>0sub> is a fixed element in the center of S. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck’s functional equation for the sine$$ \mu(y)f (x\tau(y)z_0) - f(xyz_0) = 2f(x)f(y),\quad x, y \in S$$where \({\mu : S\to \mathbb{C}}\) is a multiplicative function such that \({\mu (x\tau(x))=1}\) for all \({{x\in S}}\). Furthermore, we obtain the solutions of a variant of Van Vleck’s functional equation$$ \mu(y)f (\sigma(y)xz_0) - f(xyz_0) = 2f(x)f(y), \quad x, y \in M $$for the sine on a monoid M, where \({\sigma}\) is an involutive automorphism of M.Key words and phrasessemigroupd’Alembert’s functional equationsine functionVan Vleckinvolutionmultiplicative functionhomomorphismMathematics Subject Classification39B3239B52References1.B. Bouikhalene and E. Elqorachi, A class of functional equations on monoids, manuscript (2016).2.E. Elqorachi and A. Redouani, Solutions and stability of a variant of Wilson’s functional equation, arXiv:1505.06512v1 [math.CA] (2015), Demonstratio Mathematicae (to appear).3.B. R. Ebanks and H. Stetkær, d’Alembert’s other functional equation on monoidswith involution, Aequationes Math., 89 (2015), 187–206.MathSciNetCrossRefMATHGoogle Scholar4.Pl. Kannappan, A functional equation for the cosine, Canad. Math. Bull., 2 (1968), 495–498.CrossRefMATHGoogle Scholar5.A. M. Perkins and P. K. Sahoo, On two functional equations with involution on groups related to sine and cosine functions, Aequationes Math. (2014). DOI:10.1007/s00010-014-0309-z.6.P. K. Sahoo, A functional equation with restricted argument related to cosine function, TOJIMS (2014).7.H. Stetkær, Van Vleck’s functional equation for the sine, Aequationes Math., (2014). DOI:10.1007/s00010-015-0349-z.8.H. Stetkær, d’Alembert’s functional equation on groups, in: Recent Developments in Functional Equations and Inequalities Banach Center Publ., vol. 99. Polish Acad. Sci. Inst. Math. (Warsaw, 2013), pp. 173–191.9.H. Stetkær, Functional Equations on Groups World Scientific Publishing Co (Singapore, 2013).10.H. Stetkær, A variant of d’Alembert’s functional equation, Aequationes Math. (2014), DOI 10.1007/s00010-014-0253-y.11.E. B. Vleck Van, A functional equation for the sine, Ann. Math. Second Ser., 11 (1910), 161–165.12.E. B. Vleck Van, A functional equation for the sine, Additional note, Ann. Math. Second Ser. 13 (1911–1912), 154.Copyright information© Akadémiai Kiadó, Budapest, Hungary 2016Authors and AffiliationsB. Belaid1E. Elhoucien2Email author1.Polydisciplinary FacultyUniversity Sultan Moulay SlimaneBeni-MellalMorocco2.Department of Mathematics, Faculty of SciencesUniversity Ibn ZohrAgadirMorocco About this article CrossMark Print ISSN 0236-5294 Online ISSN 1588-2632 Publisher Name Springer Netherlands About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s10474-016-0630-1_An extension of Van Vleck’s functi", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s10474-016-0630-1_An extension of Van Vleck’s functi", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips