{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T12:21:55Z","timestamp":1740140515896,"version":"3.37.3"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005727","name":"Universidade de Coimbra","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005727","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mediterr. J. Math."],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of B\u00e4cklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.<\/jats:p>","DOI":"10.1007\/s00009-023-02307-3","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T16:52:30Z","timestamp":1676652750000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quadratic Decomposition of Bivariate Orthogonal Polynomials"],"prefix":"10.1007","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4685-1583","authenticated-orcid":false,"given":"Am\u00edlcar","family":"Branquinho","sequence":"first","affiliation":[]},{"given":"Ana","family":"Foulqui\u00e9-Moreno","sequence":"additional","affiliation":[]},{"given":"Teresa E.","family":"P\u00e9rez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,17]]},"reference":[{"key":"2307_CR1","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s11075-013-9747-2","volume":"66","author":"M Alfaro","year":"2014","unstructured":"Alfaro, M., Pe\u00f1a, A., P\u00e9rez, T.E., Rezola, M.L.: On linearly related orthogonal polynomials in several variables. Numer. Algorithms 66, 525\u2013553 (2014)","journal-title":"Numer. Algorithms"},{"issue":"3","key":"2307_CR2","first-page":"386","volume":"16","author":"L Carlitz","year":"1961","unstructured":"Carlitz, L.: The relationship of the Hermite to the Laguerre polynomials. Boll. Un. Mat. Ital. 16(3), 386\u2013390 (1961)","journal-title":"Boll. Un. Mat. Ital."},{"key":"2307_CR3","volume-title":"An Introduction to Orthogonal Polynomials, Mathematics and its Applications","author":"TS Chihara","year":"1978","unstructured":"Chihara, T.S.: An Introduction to Orthogonal Polynomials, Mathematics and its Applications, vol. 13. Gordon and Breach, New York (1978)"},{"issue":"12","key":"2307_CR4","doi-asserted-by":"publisher","first-page":"2243","DOI":"10.1016\/j.jat.2010.07.012","volume":"162","author":"MN de Jesus","year":"2010","unstructured":"de Jesus, M.N., Petronilho, J.: On orthogonal polynomials obtained via polynomial mappings. J. Approx. Theory 162(12), 2243\u20132277 (2010)","journal-title":"J. Approx. Theory"},{"key":"2307_CR5","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1524\/anly.1992.12.12.71","volume":"12","author":"K Douak","year":"1992","unstructured":"Douak, K., Maroni, P.: Les polyn\u00f4mes orthogonaux \u201cclassiques\u2019\u2019 de dimension deux. Analysis 12, 71\u2013107 (1992)","journal-title":"Analysis"},{"key":"2307_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107786134","volume-title":"Orthogonal Polynomials of Several Variables, 2nd edition, Encyclopedia of Mathematics and its Applications","author":"CF Dunkl","year":"2014","unstructured":"Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables, 2nd edition, Encyclopedia of Mathematics and its Applications, vol. 155. Cambridge Univ. Press, Cambridge (2014)"},{"issue":"2","key":"2307_CR7","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1090\/S0002-9947-1988-0951620-6","volume":"308","author":"J Geronimo","year":"1988","unstructured":"Geronimo, J., Van Assche, W.: Orthogonal polynomials on several intervals via a polynomial mapping. Trans. Am. Math. Soc. 308(2), 559\u2013581 (1988)","journal-title":"Trans. Am. Math. Soc."},{"doi-asserted-by":"crossref","unstructured":"Guemo Tefo, Y., Area, I., Foupouagnigni, M.: Bivariate symmetric discrete orthogonal polynomials. Advances in real and complex analysis with applications, 87-105, Trends Math., Birkh\u00e4user\/Springer, Singapore, (2017)","key":"2307_CR8","DOI":"10.1007\/978-981-10-4337-6_5"},{"key":"2307_CR9","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1016\/j.aim.2017.01.028","volume":"310","author":"P Iliev","year":"2017","unstructured":"Iliev, P., Xu, Y.: Connection coefficients for classical orthogonal polynomials of several variables. Adv. Math. 310, 290\u2013326 (2017)","journal-title":"Adv. Math."},{"doi-asserted-by":"crossref","unstructured":"Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their $$q$$-analogues. With a Foreword by Tom H. Koornwinder. Springer Monographs in Mathematics. Springer, Berlin (2010)","key":"2307_CR10","DOI":"10.1007\/978-3-642-05014-5"},{"issue":"2","key":"2307_CR11","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/j.exmath.2007.10.002","volume":"26","author":"AF Loureiro","year":"2008","unstructured":"Loureiro, A.F., Maroni, P.: Quadratic decomposition of Appell sequences. Expo. Math. 26(2), 177\u2013186 (2008)","journal-title":"Expo. Math."},{"issue":"7","key":"2307_CR12","doi-asserted-by":"publisher","first-page":"888","DOI":"10.1016\/j.jat.2010.07.009","volume":"163","author":"AF Loureiro","year":"2011","unstructured":"Loureiro, A.F., Maroni, P.: Quadratic decomposition of Laguerre polynomials via lowering operators. J. Approx. Theory 163(7), 888\u2013903 (2011)","journal-title":"J. Approx. Theory"},{"issue":"11","key":"2307_CR13","doi-asserted-by":"publisher","first-page":"1309","DOI":"10.1080\/10236190902810393","volume":"16","author":"A Macedo","year":"2010","unstructured":"Macedo, A., Maroni, P.: General quadratic decomposition. J. Differ. Equ. Appl. 16(11), 1309\u20131329 (2010)","journal-title":"J. Differ. Equ. Appl."},{"issue":"1","key":"2307_CR14","first-page":"81","volume":"56","author":"F Marcell\u00e1n","year":"1999","unstructured":"Marcell\u00e1n, F., Petronilho, J.: Orthogonal polynomials and quadratic transformations. Port. Math. 56(1), 81\u2013113 (1999)","journal-title":"Port. Math."},{"unstructured":"P. Maroni, Sur la d\u00e9composition quadratique d\u2019une suite de polyn\u00f4mes orthogonaux. I., Riv. Mat. Pura ed Apl. 6 (1990), 19-53","key":"2307_CR15"},{"issue":"3","key":"2307_CR16","first-page":"305","volume":"50","author":"P Maroni","year":"1993","unstructured":"Maroni, P.: Sur la d\u00e9composition quadratique d\u2019une suite de polyn\u00f4mes orthogonaux II. Port. Math. 50(3), 305\u2013329 (1993)","journal-title":"Port. Math."},{"issue":"9","key":"2307_CR17","doi-asserted-by":"publisher","first-page":"1519","DOI":"10.1080\/10236198.2011.579118","volume":"18","author":"P Maroni","year":"2012","unstructured":"Maroni, P., Tounsi, M.I.: Quadratic decomposition of symmetric semi-classical polynomial sequences of even class: an example from the case $$s = 2$$. J. Differ. Equ. Appl. 18(9), 1519\u20131530 (2012)","journal-title":"J. Differ. Equ. Appl."},{"doi-asserted-by":"crossref","unstructured":"Mesquita, T.A., da Rocha, Z.: Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases, Opusc. Math. 32(4), 675-687, (2012)","key":"2307_CR18","DOI":"10.7494\/OpMath.2012.32.4.675"},{"key":"2307_CR19","volume-title":"Orthogonal polynomials","author":"G Szeg\u0151","year":"1975","unstructured":"Szeg\u0151, G.: Orthogonal polynomials, 4th edn. Amer. Math. Soc, Providence, RI (1975)","edition":"4"},{"issue":"4","key":"2307_CR20","doi-asserted-by":"publisher","first-page":"1649","DOI":"10.1007\/s41980-021-00605-8","volume":"48","author":"YG Tefo","year":"2022","unstructured":"Tefo, Y.G., Akta\u015f, R., Area, I., G\u00fcldogan Lekesiz, E.: On a Symmetric Generalization of Bivariate Sturm-Liouville Problems. Bull. Iran. Math. Soc 48(4), 1649\u20131665 (2022)","journal-title":"Bull. Iran. Math. Soc"},{"issue":"2","key":"2307_CR21","doi-asserted-by":"publisher","first-page":"169","DOI":"10.4310\/MAA.1998.v5.n2.a5","volume":"5","author":"Y Xu","year":"1998","unstructured":"Xu, Y.: Orthogonal polynomials and cubature formulae on spheres and on simplices. Methods Appl. Anal. 5(2), 169\u2013184 (1998)","journal-title":"Methods Appl. Anal."},{"issue":"3","key":"2307_CR22","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1007\/s003650010036","volume":"17","author":"Y Xu","year":"2001","unstructured":"Xu, Y.: Orthogonal polynomials on the ball and the simplex for weight functions with reflection symmetries. Constr. Approx. 17(3), 383\u2013412 (2001)","journal-title":"Constr. Approx."}],"container-title":["Mediterranean Journal of Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00009-023-02307-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00009-023-02307-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00009-023-02307-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T20:05:33Z","timestamp":1683835533000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00009-023-02307-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,17]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["2307"],"URL":"https:\/\/doi.org\/10.1007\/s00009-023-02307-3","relation":{},"ISSN":["1660-5446","1660-5454"],"issn-type":[{"type":"print","value":"1660-5446"},{"type":"electronic","value":"1660-5454"}],"subject":[],"published":{"date-parts":[[2023,2,17]]},"assertion":[{"value":"22 February 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 September 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"118"}}