{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T21:34:26Z","timestamp":1772832866978,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,7,3]],"date-time":"2021-07-03T00:00:00Z","timestamp":1625270400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,7,3]],"date-time":"2021-07-03T00:00:00Z","timestamp":1625270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100002322","name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002322","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e1 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2016\/02700-8"],"award-info":[{"award-number":["2016\/02700-8"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s40314-021-01516-4","type":"journal-article","created":{"date-parts":[[2021,7,3]],"date-time":"2021-07-03T13:02:21Z","timestamp":1625317341000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Orthogonal polynomials and M\u00f6bius transformations"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8343-7106","authenticated-orcid":false,"given":"R. S.","family":"Vieira","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8471-9407","authenticated-orcid":false,"given":"V.","family":"Botta","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,3]]},"reference":[{"key":"1516_CR1","volume-title":"Complex analysis: an introduction to the theory of analytic functions of one complex variable","author":"LV Ahlfors","year":"1979","unstructured":"Ahlfors LV (1979) Complex analysis: an introduction to the theory of analytic functions of one complex variable, 3rd edn. McGraw-Hill Education, New York","edition":"3"},{"key":"1516_CR2","volume-title":"Elements of computer algebra with applications","author":"AG Akritas","year":"1989","unstructured":"Akritas AG (1989) Elements of computer algebra with applications, vol 3. Wiley, New York"},{"issue":"7\u20138","key":"1516_CR3","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1016\/0375-9601(91)90711-G","volume":"156","author":"V Aldaya","year":"1991","unstructured":"Aldaya V, Bisquert J, Navarro-Salas J (1991) The quantum relativistic harmonic oscillator: generalized Hermite polynomials. Phys Lett A 156(7\u20138):381\u2013385. https:\/\/doi.org\/10.1016\/0375-9601(91)90711-G","journal-title":"Phys Lett A"},{"key":"1516_CR4","volume-title":"Sturm\u2013Liouville theory and its applications","author":"MA Al-Gwaiz","year":"2008","unstructured":"Al-Gwaiz MA (2008) Sturm\u2013Liouville theory and its applications, vol 7. Springer, Berlin"},{"issue":"2","key":"1516_CR5","first-page":"143","volume":"53","author":"D Alvarez-Castillo","year":"2007","unstructured":"Alvarez-Castillo D, Kirchbach M (2007) Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovski polynomials. Revista Mexicana de F\u00edsica E 53(2):143\u2013154","journal-title":"Revista Mexicana de F\u00edsica E"},{"issue":"2","key":"1516_CR6","first-page":"279","volume":"23","author":"G Bengochea","year":"2018","unstructured":"Bengochea G, Verde-Star L, Ortigueira M (2018) Operational method for the solution of ordinary differential equations using Hermite series. Math Commun 23(2):279\u2013293","journal-title":"Math Commun"},{"issue":"1","key":"1516_CR7","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1007\/BF01180560","volume":"29","author":"S Bochner","year":"1929","unstructured":"Bochner S (1929) \u00dcber Sturm\u2013Liouvillesche Polynomsysteme. Mathematische Zeitschrift 29(1):730\u2013736. https:\/\/doi.org\/10.1007\/BF01180560","journal-title":"Mathematische Zeitschrift"},{"key":"1516_CR8","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1090\/S0002-9904-1930-04888-0","volume":"36","author":"WC Brenke","year":"1930","unstructured":"Brenke WC (1930) On polynomial solutions of a class of linear differential equations of the second order. Bull Am Math Soc 36:77\u201384. https:\/\/doi.org\/10.1090\/S0002-9904-1930-04888-0","journal-title":"Bull Am Math Soc"},{"key":"1516_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511530050","volume-title":"Orthogonal rational functions","author":"A Bultheel","year":"1999","unstructured":"Bultheel A, Gonz\u00e1lez-Vera P, Hendriksen E, Njastad O (1999) Orthogonal rational functions, vol 5. Cambridge University Press, Cambridge"},{"issue":"4","key":"1516_CR10","doi-asserted-by":"publisher","first-page":"765","DOI":"10.2307\/2033295","volume":"8","author":"T Chihara","year":"1957","unstructured":"Chihara T (1957) On quasi-orthogonal polynomials. Proc Am Math Soc 8(4):765\u2013767. https:\/\/doi.org\/10.2307\/2033295","journal-title":"Proc Am Math Soc"},{"key":"1516_CR11","volume-title":"An introduction to orthogonal polynomials","author":"TS Chihara","year":"2011","unstructured":"Chihara TS (2011) An introduction to orthogonal polynomials. Dover Publications, Garden City"},{"issue":"1\u20132","key":"1516_CR12","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/S0377-0427(00)00633-6","volume":"133","author":"JS Dehesa","year":"2001","unstructured":"Dehesa JS, Mart\u00ednez-Finkelshtein A, S\u00e1nchez-Ruiz J (2001) Quantum information entropies and orthogonal polynomials. J Comput Appl Math 133(1\u20132):23\u201346. https:\/\/doi.org\/10.1016\/S0377-0427(00)00633-6","journal-title":"J Comput Appl Math"},{"key":"1516_CR13","volume-title":"Complex variables: harmonic and analytic functions","author":"FJ Flanigan","year":"1972","unstructured":"Flanigan FJ (1972) Complex variables: harmonic and analytic functions. Dover Publications, Garden City"},{"issue":"1","key":"1516_CR14","doi-asserted-by":"publisher","first-page":"352","DOI":"10.1016\/j.jmaa.2009.05.052","volume":"359","author":"D G\u00f3mez-Ullate","year":"2009","unstructured":"G\u00f3mez-Ullate D, Kamran N, Milson R (2009) An extended class of orthogonal polynomials defined by a Sturm\u2013Liouville problem. J Math Anal Appl 359(1):352\u2013367. https:\/\/doi.org\/10.1016\/j.jmaa.2009.05.052","journal-title":"J Math Anal Appl"},{"issue":"5","key":"1516_CR15","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1016\/j.jat.2009.11.002","volume":"162","author":"D G\u00f3mez-Ullate","year":"2010","unstructured":"G\u00f3mez-Ullate D, Kamran N, Milson R (2010) An extension of Bochner\u2019s problem: exceptional invariant subspaces. J Approx Theory 162(5):987\u20131006. https:\/\/doi.org\/10.1016\/j.jat.2009.11.002","journal-title":"J Approx Theory"},{"issue":"1\u20132","key":"1516_CR16","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1080\/10652469908819215","volume":"8","author":"MX He","year":"1999","unstructured":"He MX, Natalini P (1999) Relativistic Jacobi polynomials. Integral Transforms Spec Funct 8(1\u20132):43\u201356. https:\/\/doi.org\/10.1080\/10652469908819215","journal-title":"Integral Transforms Spec Funct"},{"issue":"12","key":"1516_CR17","doi-asserted-by":"publisher","first-page":"3199","DOI":"10.1088\/0305-4470\/29\/12\/023","volume":"29","author":"ME Ismail","year":"1996","unstructured":"Ismail ME (1996) Relativistic orthogonal polynomials are Jacobi polynomials. J Phys A: Math Gen 29(12):3199. https:\/\/doi.org\/10.1088\/0305-4470\/29\/12\/023","journal-title":"J Phys A: Math Gen"},{"key":"1516_CR18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325982","volume-title":"Classical and quantum orthogonal polynomials in one variable","author":"M Ismail","year":"2005","unstructured":"Ismail M (2005) Classical and quantum orthogonal polynomials in one variable. Cambridge University Press, Cambridge. https:\/\/doi.org\/10.1017\/CBO9781107325982"},{"issue":"2","key":"1516_CR19","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1112\/blms\/21.2.113","volume":"21","author":"WB Jones","year":"1989","unstructured":"Jones WB, Nj\u00e5stad O, Thron W (1989) Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull Lond Math Soc 21(2):113\u2013152. https:\/\/doi.org\/10.1112\/blms\/21.2.113","journal-title":"Bull Lond Math Soc"},{"key":"1516_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jat.2013.10.003","volume":"178","author":"K Jordaan","year":"2014","unstructured":"Jordaan K, To\u00f3kos F (2014) Orthogonality and asymptotics of pseudo-Jacobi polynomials for non-classical parameters. J Approx Theory 178:1\u201312. https:\/\/doi.org\/10.1016\/j.jat.2013.10.003","journal-title":"J Approx Theory"},{"key":"1516_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05014-5","volume-title":"Hypergeometric orthogonal polynomials and their $$q$$-analogues","author":"R Koekoek","year":"2010","unstructured":"Koekoek R, Lesky PA, Swarttouw RF (2010) Hypergeometric orthogonal polynomials and their $$q$$-analogues. Springer Science & Business Media, Berlin"},{"issue":"1","key":"1516_CR22","doi-asserted-by":"publisher","first-page":"100","DOI":"10.2307\/1990516","volume":"65","author":"HL Krall","year":"1949","unstructured":"Krall HL, Frink O (1949) A new class of orthogonal polynomials: the Bessel polynomials. Trans Am Math Soc 65(1):100\u2013115. https:\/\/doi.org\/10.2307\/1990516","journal-title":"Trans Am Math Soc"},{"key":"1516_CR23","first-page":"1","volume":"19","author":"ABJ Kuijlaars","year":"2005","unstructured":"Kuijlaars ABJ, Mart\u00ednez-Finkelshtein A, Orive R (2005) Orthogonality of Jacobi polynomials with general parameters. Electron Trans Numer Anal 19:1\u201317","journal-title":"Electron Trans Numer Anal"},{"key":"1516_CR24","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.laa.2014.05.013","volume":"470","author":"DS Mackey","year":"2015","unstructured":"Mackey DS, Mackey N, Mehl C, Mehrmann V (2015) M\u00f6bius transformations of matrix polynomials. Linear Algebra Appl 470:120\u2013184","journal-title":"Linear Algebra Appl"},{"key":"1516_CR25","first-page":"299","volume":"16","author":"P Natalini","year":"1996","unstructured":"Natalini P (1996) The relativistic Laguerre polynomials. Rendiconti di Matematica Ser VII 16:299\u2013313","journal-title":"Rendiconti di Matematica Ser VII"},{"key":"1516_CR26","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1016\/j.amc.2016.03.036","volume":"287","author":"I P\u0103v\u0103loiu","year":"2016","unstructured":"P\u0103v\u0103loiu I, C\u0103tina\u015f E (2016) On a robust Aitken\u2013Newton method based on the Hermite polynomial. Appl Math Comput 287:224\u2013231. https:\/\/doi.org\/10.1016\/j.amc.2016.03.036","journal-title":"Appl Math Comput"},{"issue":"3","key":"1516_CR27","doi-asserted-by":"publisher","first-page":"253","DOI":"10.2478\/s11534-007-0018-5","volume":"5","author":"A Raposo","year":"2007","unstructured":"Raposo A, Weber H, Alvarez-Castillo D, Kirchbach M (2007) Romanovski polynomials in selected physics problems. Open Phys 5(3):253\u2013284. https:\/\/doi.org\/10.2478\/s11534-007-0018-5","journal-title":"Open Phys"},{"issue":"1023","key":"1516_CR28","first-page":"1023","volume":"188","author":"V Romanovski","year":"1929","unstructured":"Romanovski V (1929) Sur quelques classes nouvelles de polynomes orthogonaux. Comptes Rendus de l\u2019Acad\u00e9mie des Sciences de Paris 188(1023):1023\u20131025","journal-title":"Comptes Rendus de l\u2019Acad\u00e9mie des Sciences de Paris"},{"issue":"1","key":"1516_CR29","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1112\/plms\/s1-16.1.245","volume":"s1\u201316","author":"EJ Routh","year":"1885","unstructured":"Routh EJ (1885) On some properties of certain solutions of a differential equation of the second order. Proc Lond Math Soc s1\u201316(1):245\u2013262. https:\/\/doi.org\/10.1112\/plms\/s1-16.1.245","journal-title":"Proc Lond Math Soc"},{"key":"1516_CR30","volume-title":"Orthogonal polynomials on the unit circle part I: classical theory","author":"B Simon","year":"2004","unstructured":"Simon B (2004a) Orthogonal polynomials on the unit circle part I: classical theory. American Mathematical Society, Providence"},{"key":"1516_CR31","volume-title":"Orthogonal polynomials on the unit circle part II: spectral theory","author":"B Simon","year":"2004","unstructured":"Simon B (2004b) Orthogonal polynomials on the unit circle part II: spectral theory. American Mathematical Society, Providence"},{"issue":"1","key":"1516_CR32","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s002200050772","volume":"210","author":"V Spiridonov","year":"2000","unstructured":"Spiridonov V, Zhedanov A (2000) Spectral transformation chains and some new biorthogonal rational functions. Commun Math Phys 210(1):49\u201383. https:\/\/doi.org\/10.1007\/s002200050772","journal-title":"Commun Math Phys"},{"key":"1516_CR33","volume-title":"Orthogonal polynomials","author":"G Szeg\u0151","year":"1939","unstructured":"Szeg\u0151 G (1939) Orthogonal polynomials, vol 23. American Mathematical Society, Providence"},{"issue":"1","key":"1516_CR34","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/S0377-0427(98)00171-X","volume":"99","author":"V Totik","year":"1998","unstructured":"Totik V (1998) Orthogonal polynomials with respect to varying weights. J Comput Appl Math 99(1):373\u2013385. https:\/\/doi.org\/10.1016\/S0377-0427(98)00171-X","journal-title":"J Comput Appl Math"},{"key":"1516_CR35","doi-asserted-by":"publisher","unstructured":"Vieira RS (2019) Polynomials with symmetric zeros. In: Polynomials\u2014theory and application. IntechOpen. https:\/\/doi.org\/10.5772\/intechopen.82728","DOI":"10.5772\/intechopen.82728"},{"key":"1516_CR36","doi-asserted-by":"publisher","unstructured":"Vieira RS (2021) How to count the number of zeros that a polynomial has on the unit circle? J Comp Appl Math 384. https:\/\/doi.org\/10.1016\/j.cam.2020.113169","DOI":"10.1016\/j.cam.2020.113169"},{"issue":"2","key":"1516_CR37","doi-asserted-by":"publisher","first-page":"415","DOI":"10.2478\/s11533-007-0004-6","volume":"5","author":"HJ Weber","year":"2007","unstructured":"Weber HJ (2007a) Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula. Central Eur J Math 5(2):415\u2013427. https:\/\/doi.org\/10.2478\/s11533-007-0004-6","journal-title":"Central Eur J Math"},{"issue":"3","key":"1516_CR38","doi-asserted-by":"publisher","first-page":"581","DOI":"10.2478\/s11533-007-0014-4","volume":"5","author":"H Weber","year":"2007","unstructured":"Weber H (2007b) Connections between Romanovski and other polynomials. Open Math 5(3):581\u2013595. https:\/\/doi.org\/10.2478\/s11533-007-0014-4","journal-title":"Open Math"},{"issue":"6","key":"1516_CR39","doi-asserted-by":"publisher","first-page":"1128","DOI":"10.1016\/j.jfranklin.2011.05.003","volume":"348","author":"S Yal\u00e7inba\u015f","year":"2011","unstructured":"Yal\u00e7inba\u015f S, Aynig\u00fcl M, Sezer M (2011) A collocation method using Hermite polynomials for approximate solution of pantograph equations. J Frankl Inst 348(6):1128\u20131139. https:\/\/doi.org\/10.1016\/j.jfranklin.2011.05.003","journal-title":"J Frankl Inst"},{"issue":"2","key":"1516_CR40","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1006\/jath.1999.3339","volume":"101","author":"A Zhedanov","year":"1999","unstructured":"Zhedanov A (1999) Biorthogonal rational functions and the generalized eigenvalue problem. J Approx Theory 101(2):303\u2013329. https:\/\/doi.org\/10.1006\/jath.1999.3339","journal-title":"J Approx Theory"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01516-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01516-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01516-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T19:28:31Z","timestamp":1632511711000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01516-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,3]]},"references-count":40,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1516"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01516-4","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,3]]},"assertion":[{"value":"11 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"181"}}