Pares coherentes generalizados de polinomios ortogonales en dos variables
En este trabajo nos centraremos en la obtención de pares x_k-coherentes de polinomios ortogonales en varias variables a partir de un sistema de polinomios ortogonales escogido inicialmente, concepto introducido en [28] de 2019 por Francisco Marcellán, Misael Marriaga, Teresa Pérez y Miguel Piñar, me...
- Autores:
-
Cortés Garzón, Juan Esteban
- Tipo de recurso:
- Fecha de publicación:
- 2023
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/84168
- Palabra clave:
- 510 - Matemáticas
Funciones ortogonales
Functions, orthogonal
Series, orthogonal
Series ortogonales
Polinomios Ortogonales
Varias Variables
Pares Coherentes
Programación
- Rights
- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
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| dc.title.spa.fl_str_mv |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| dc.title.translated.eng.fl_str_mv |
Generalized Coherent Pairs of Orthogonal Polynomials in Two Variables |
| title |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| spellingShingle |
Pares coherentes generalizados de polinomios ortogonales en dos variables 510 - Matemáticas Funciones ortogonales Functions, orthogonal Series, orthogonal Series ortogonales Polinomios Ortogonales Varias Variables Pares Coherentes Programación |
| title_short |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| title_full |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| title_fullStr |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| title_full_unstemmed |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| title_sort |
Pares coherentes generalizados de polinomios ortogonales en dos variables |
| dc.creator.fl_str_mv |
Cortés Garzón, Juan Esteban |
| dc.contributor.advisor.none.fl_str_mv |
Pinzón Cortés, Natalia Camila |
| dc.contributor.author.none.fl_str_mv |
Cortés Garzón, Juan Esteban |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas |
| topic |
510 - Matemáticas Funciones ortogonales Functions, orthogonal Series, orthogonal Series ortogonales Polinomios Ortogonales Varias Variables Pares Coherentes Programación |
| dc.subject.lemb.spa.fl_str_mv |
Funciones ortogonales |
| dc.subject.lemb.eng.fl_str_mv |
Functions, orthogonal Series, orthogonal |
| dc.subject.lemb.spe.fl_str_mv |
Series ortogonales |
| dc.subject.proposal.spa.fl_str_mv |
Polinomios Ortogonales Varias Variables Pares Coherentes Programación |
| description |
En este trabajo nos centraremos en la obtención de pares x_k-coherentes de polinomios ortogonales en varias variables a partir de un sistema de polinomios ortogonales escogido inicialmente, concepto introducido en [28] de 2019 por Francisco Marcellán, Misael Marriaga, Teresa Pérez y Miguel Piñar, mediante el uso de programación en el software Wolfram Mathematica. (Texto tomado de la fuente) |
| publishDate |
2023 |
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2023-07-07T20:15:55Z |
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2023-07-07T20:15:55Z |
| dc.date.issued.none.fl_str_mv |
2023-06 |
| dc.type.spa.fl_str_mv |
Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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info:eu-repo/semantics/acceptedVersion |
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Text |
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http://purl.org/redcol/resource_type/TM |
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acceptedVersion |
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https://repositorio.unal.edu.co/handle/unal/84168 |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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https://repositorio.unal.edu.co/handle/unal/84168 https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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spa |
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spa |
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Álvarez-Nodarse, Renato: Monogr. Semin. Mat. García Galdeano. Vol. 26: Polinomios hipergeométricos clásicos y q-polinomios. Zaragoza: Prensas Universitarias de Zaragoza, 2003. – ISBN 84–7733–637–7. Appell, P. ; Kampé de Fériet, J.: Fonctions hypergéométriques et hypersphériques. Polynômes d’Hermite. 1926. Area, Iván: Hypergeometric multivariate orthogonal polynomials. En: Orthogonal polynomials. Proceedings of the 2nd AIMS-Volkswagen Stiftung workshop on introduction to orthogonal polynomials and applications, Douala, Cameroon, October 5–12, 2018. Cham: Birkhäuser, 2020. – ISBN 978–3–030–36743–5; 978–3–030–36746–6; 978–3–030– 36744–2, p. 165–193. Area, Iván. ; Godoy, E. ; Ronveaux, A. ; Zarzo, A.: Bivariate second-order linear partial differential equations and orthogonal polynomial solutions. En: J. Math. Anal.Appl. 387 (2012), Nr. 2, p. 1188–1208. – ISSN 0022–247X. Barrio, Roberto ; Peña, Juan M. ; Sauer, Tomas: Three term recurrence for the evaluation of multivariate orthogonal polynomials. En: Journal of Approximation Theory 162 (2010), Nr. 2, p. 407–420. – ISSN 0021–9045. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, 1978. Delgado, Antonia M.: Ortogonalidad no estándar: Problemas Directos e Inversos, Doctoral Dissertation, Universidad Carlos III de Madrid. In Spanish, Tesis de Grado, 2006. Delgado, Antonia M. ; Geronimo, Jeffrey S. ; Iliev, Plamen ; Marcellán, Francisco: Two variable orthogonal polynomials and structured matrices. En: SIAM J. Matrix Anal. Appl. 28 (2006), Nr. 1, p. 118–147. – ISSN 0895–4798. Delgado, Antonia M. ; Marcellán, Francisco: Companion linear functionals and Sobolev inner products: a case study. En: Methods Appl. Anal. 11 (2004), Nr. 2, p. 237–266. Delgado, Antonia M. ; Marcellán, Francisco: On an extension of symmetric coherent pairs of orthogonal polynomials. En: J. Comput. Appl. Math. 178 (2005), Nr. 1-2, p. 155–168. Dunkl, Charles F. ; Xu, Yuan: Orthogonal polynomials of several variables. Cambridge: Cambridge University Press, 2001 (Encycl. Math. Appl.). ISSN 0953–4806. Engelis, G. K.: On some two-dimensional analogues of the classical orthogonal polynomials. En: Latv. Mat. Ezheg. 15 (1974), p. 169–202. – ISSN 0321–2270. Fernández, Lidia ; Pérez, Teresa E. ; Piñar, Miguel A.: Classical orthogonal polynomials in two variables: a matrix approach. En: Numer. Algorithms 39 (2005), Nr. 1-3, p. 131–142. – ISSN 1017–1398. Garza, Luis ; Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-coherent pairs on the unit circle. En: Abstr. Appl. Anal. 2013 (2013), p. 8. – Id/No 307974. – ISSN 1085–3375. Hahn, Wolfgang: Über die Jacobischen Polynome und zwei verwandte Polynomklassen. En: Mathematische Zeitschrift 39 (1935), p. 634–638. Hermite, M. ; des sciences (France), Académie: Sur Un Nouveau Développement en Série Des Fonctions. Imprimerie de Gauthier-Villars, 1864. Iserles, Arieh ; Koch, P. E. ; Nørsett, Syvert P. ; Sanz-Serna, J. M.: On polynomials orthogonal with respect to certain Sobolev inner products. En: J. Approx. Theory 65 (1991), Nr. 2, p. 151–175. – ISSN 0021–9045. Jacobi, C.G.J.: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. 1859 (1859), Nr. 56, p. 149–165. de Jesus, M. N. ; Marcellán, Francisco. ; Petronilho, J. ; Pinzón-Cortés, N. C.: (M,N)-coherent pairs of order (m, n) and Sobolev orthogonal polynomials. En: J. Comput. Appl. Math. 256 (2014), p. 16–35. – ISSN 0377–0427. Karlin, S. ; McGregor, J. On some stochastic models in genetics. Stochastic Models Med. Biol., Proc. Sympos. Univ. Wisconsin 1963, 245-279. 1964. Karlin, S. ; McGregor, J.: Linear Growth Models with Many Types and Multidimensional Hahn Polynomials. En: Askey, Richard A. (Ed.): Theory and Application of Special Functions. Academic Press, 1975. – ISBN 978–0–12–064850–4, p. 261–288. Koornwinder, Tom: Two-Variable Analogues of the Classical Orthogonal Polynomials. En: Askey, Richard A. (Ed.): Theory and Application of Special Functions. Academic Press, 1975. – ISBN 978–0–12–064850–4, p. 435–495. Kowalski, M. A.: Orthogonality and recursion formulas for polynomials in n variables. En: SIAM J. Math. Anal. 13 (1982), p. 316–323. – ISSN 0036–1410. Kowalski, M. A.: The recursion formulas for orthogonal polynomials in n variables. En: SIAM J. Math. Anal. 13 (1982), p. 309–315. – ISSN 0036–1410. Krall, H. L. ; Frink, Orrin: A new class of orthogonal polynomials: The Bessel polynomials. En: Trans. Am. Math. Soc. 65 (1949), p. 100–115. – ISSN 0002–9947. Krall, H. L. ; Sheffer, I. M.: Orthogonal polynomials in two variables. En: Ann. Mat. Pura Appl. (4) 76 (1967), p. 325–376. – ISSN 0373–3114. Legendre, Adrien M.: Recherches sur l’attraction des spheroides homogenes. En: Mémoires de mathématique et de physique : prés. á l’Acad´emie Royale des Sciences, par divers savans, et lûs dans ses assemblées 1785 (2007), p. 411 – 434. Marcellán, Francisco ; Marriaga, Misael E. ; Pérez, Teresa E. ; Piñar, Miguel A.: Coherent pairs of bivariate orthogonal polynomials. En: J. Approx. Theory 245 (2019), p. 40–63. – ISSN 0021–9045. Marcellán, Francisco ; Pérez, Teresa E. ; Piñar, Miguel A.: Orthogonal polynomials on weighted Sobolev spaces: The semiclassical case. En: Ann. Numer. Math. 2 (1995), Nr. 1-4, p. 93–122. – ISSN 1021–2655. Marcellán, Francisco ; Petronilho, J.: Orthogonal polynomials and coherent pairs: The classical case. En: Indag. Math., New Ser. 6 (1995), Nr. 3, p. 287–307. – ISSN 0019–3577. Marcellán, Francisco ; Petronilho, José C. ; Pérez, Teresa E. ; Piñar, Miguel A.: What is beyond coherent pairs of orthogonal polynomials? En: J. Comput. Appl. Math. 65 (1995), Nr. 1-3, p. 267–277. – ISSN 0377–0427. Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-q-coherent pairs. En: Numer. Algorithms 60 (2012), Nr. 2, p. 223–239. – ISSN 1017–1398. Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-Dω-coherent pairs. En: J. Difference Equ. Appl. 19 (2013), Nr. 11, p. 1828–1848. – ISSN 1023–6198. Meijer, H. G.: Coherent pairs and zeros of Sobolev-type orthogonal polynomials. En: Indag. Math., New Ser. 4 (1993), Nr. 2, p. 163–176. – ISSN 0019–3577. Meijer, H. G.: Determination of all coherent pairs. En: J. Approx. Theory 89 (1997), Nr. 3, p. 321–343. – ISSN 0021–9045. Proriol, Joseph: Sur une famille de polynômes à deux variables orthogonaux dans un triangle. En: C. R. Acad. Sci., Paris 245 (1957), p. 2459–2461. – ISSN 0001–4036. Stieltjes, T.-J.: Recherches sur les fractions continues. En: Annales de la Faculté des sciences de Toulouse : Mathématiques 8 (1894), Nr. 4, p. J1–J122. Suetin, P. K.: Anal. Methods Spec. Funct.. Vol. 3: Orthogonal polynomials in two variables. Transl. from the 1988 Russian original by E. V. Pankratiev . Amsterdam: Gordon and Breach Science Publishers, 1999. – ISBN 90–5699–167–1. Szegö, Gábor: Colloq. Publ., Am. Math. Soc.. Vol. 23: Orthogonal polynomials. American Mathematical Society (AMS), Providence, RI, 1939 ISSN 0065–9258. Tchebychev, Pafnutii L.: Théorie des mécanismes connus sous le nom de parallélogrammes. Imprimerie de l’Académie impériale des sciences, 1853. Tchebyshev, Pafnutii L.: Sur le développement des fonctions à une seule variable. En: Bull. Acad. Sci. St. Petersb 1 (1859), Nr. 193-200, p. 124. Wolfram Research, Inc. Mathematica, Version 12.1. Champaign, Illinois, (2020). Zernike, F. ; Brinkman, H. C.: Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome. En: Proc. Akad. Wet. Amsterdam 38 (1935), p. 161–170. – ISSN 0370–0348. |
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Atribución-NoComercial 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Pinzón Cortés, Natalia Camila4a612934f85871565b47483201ed4e44Cortés Garzón, Juan Estebane239e2a6e623e40849eeeae7c0cc37f62023-07-07T20:15:55Z2023-07-07T20:15:55Z2023-06https://repositorio.unal.edu.co/handle/unal/84168Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/En este trabajo nos centraremos en la obtención de pares x_k-coherentes de polinomios ortogonales en varias variables a partir de un sistema de polinomios ortogonales escogido inicialmente, concepto introducido en [28] de 2019 por Francisco Marcellán, Misael Marriaga, Teresa Pérez y Miguel Piñar, mediante el uso de programación en el software Wolfram Mathematica. (Texto tomado de la fuente)In this work we will focus on obtaining xk-coherent pairs of orthogonal polynomials in several variables from an initially chosen system of orthogonal polynomials, a concept introduced in [28] of 2019 by Francisco Marcellán, Misael Marriaga, Teresa Pérez and Miguel Piñar, by using programming in Wolfram Mathematica software.MaestríaMagíster en Ciencias - Matemáticasvii, 46 páginasapplication/pdfspaUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasBogotá,ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - MatemáticasFunciones ortogonalesFunctions, orthogonalSeries, orthogonalSeries ortogonalesPolinomios OrtogonalesVarias VariablesPares CoherentesProgramaciónPares coherentes generalizados de polinomios ortogonales en dos variablesGeneralized Coherent Pairs of Orthogonal Polynomials in Two VariablesTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMÁlvarez-Nodarse, Renato: Monogr. Semin. Mat. García Galdeano. Vol. 26: Polinomios hipergeométricos clásicos y q-polinomios. Zaragoza: Prensas Universitarias de Zaragoza, 2003. – ISBN 84–7733–637–7.Appell, P. ; Kampé de Fériet, J.: Fonctions hypergéométriques et hypersphériques. Polynômes d’Hermite. 1926.Area, Iván: Hypergeometric multivariate orthogonal polynomials. En: Orthogonal polynomials. Proceedings of the 2nd AIMS-Volkswagen Stiftung workshop on introduction to orthogonal polynomials and applications, Douala, Cameroon, October 5–12, 2018. Cham: Birkhäuser, 2020. – ISBN 978–3–030–36743–5; 978–3–030–36746–6; 978–3–030– 36744–2, p. 165–193.Area, Iván. ; Godoy, E. ; Ronveaux, A. ; Zarzo, A.: Bivariate second-order linear partial differential equations and orthogonal polynomial solutions. En: J. Math. Anal.Appl. 387 (2012), Nr. 2, p. 1188–1208. – ISSN 0022–247X.Barrio, Roberto ; Peña, Juan M. ; Sauer, Tomas: Three term recurrence for the evaluation of multivariate orthogonal polynomials. En: Journal of Approximation Theory 162 (2010), Nr. 2, p. 407–420. – ISSN 0021–9045.Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, 1978.Delgado, Antonia M.: Ortogonalidad no estándar: Problemas Directos e Inversos, Doctoral Dissertation, Universidad Carlos III de Madrid. In Spanish, Tesis de Grado, 2006.Delgado, Antonia M. ; Geronimo, Jeffrey S. ; Iliev, Plamen ; Marcellán, Francisco: Two variable orthogonal polynomials and structured matrices. En: SIAM J. Matrix Anal. Appl. 28 (2006), Nr. 1, p. 118–147. – ISSN 0895–4798.Delgado, Antonia M. ; Marcellán, Francisco: Companion linear functionals and Sobolev inner products: a case study. En: Methods Appl. Anal. 11 (2004), Nr. 2, p. 237–266.Delgado, Antonia M. ; Marcellán, Francisco: On an extension of symmetric coherent pairs of orthogonal polynomials. En: J. Comput. Appl. Math. 178 (2005), Nr. 1-2, p. 155–168.Dunkl, Charles F. ; Xu, Yuan: Orthogonal polynomials of several variables. Cambridge: Cambridge University Press, 2001 (Encycl. Math. Appl.). ISSN 0953–4806.Engelis, G. K.: On some two-dimensional analogues of the classical orthogonal polynomials. En: Latv. Mat. Ezheg. 15 (1974), p. 169–202. – ISSN 0321–2270.Fernández, Lidia ; Pérez, Teresa E. ; Piñar, Miguel A.: Classical orthogonal polynomials in two variables: a matrix approach. En: Numer. Algorithms 39 (2005), Nr. 1-3, p. 131–142. – ISSN 1017–1398.Garza, Luis ; Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-coherent pairs on the unit circle. En: Abstr. Appl. Anal. 2013 (2013), p. 8. – Id/No 307974. – ISSN 1085–3375.Hahn, Wolfgang: Über die Jacobischen Polynome und zwei verwandte Polynomklassen. En: Mathematische Zeitschrift 39 (1935), p. 634–638.Hermite, M. ; des sciences (France), Académie: Sur Un Nouveau Développement en Série Des Fonctions. Imprimerie de Gauthier-Villars, 1864.Iserles, Arieh ; Koch, P. E. ; Nørsett, Syvert P. ; Sanz-Serna, J. M.: On polynomials orthogonal with respect to certain Sobolev inner products. En: J. Approx. Theory 65 (1991), Nr. 2, p. 151–175. – ISSN 0021–9045.Jacobi, C.G.J.: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. 1859 (1859), Nr. 56, p. 149–165.de Jesus, M. N. ; Marcellán, Francisco. ; Petronilho, J. ; Pinzón-Cortés, N. C.: (M,N)-coherent pairs of order (m, n) and Sobolev orthogonal polynomials. En: J. Comput. Appl. Math. 256 (2014), p. 16–35. – ISSN 0377–0427.Karlin, S. ; McGregor, J. On some stochastic models in genetics. Stochastic Models Med. Biol., Proc. Sympos. Univ. Wisconsin 1963, 245-279. 1964.Karlin, S. ; McGregor, J.: Linear Growth Models with Many Types and Multidimensional Hahn Polynomials. En: Askey, Richard A. (Ed.): Theory and Application of Special Functions. Academic Press, 1975. – ISBN 978–0–12–064850–4, p. 261–288.Koornwinder, Tom: Two-Variable Analogues of the Classical Orthogonal Polynomials. En: Askey, Richard A. (Ed.): Theory and Application of Special Functions. Academic Press, 1975. – ISBN 978–0–12–064850–4, p. 435–495.Kowalski, M. A.: Orthogonality and recursion formulas for polynomials in n variables. En: SIAM J. Math. Anal. 13 (1982), p. 316–323. – ISSN 0036–1410.Kowalski, M. A.: The recursion formulas for orthogonal polynomials in n variables. En: SIAM J. Math. Anal. 13 (1982), p. 309–315. – ISSN 0036–1410.Krall, H. L. ; Frink, Orrin: A new class of orthogonal polynomials: The Bessel polynomials. En: Trans. Am. Math. Soc. 65 (1949), p. 100–115. – ISSN 0002–9947.Krall, H. L. ; Sheffer, I. M.: Orthogonal polynomials in two variables. En: Ann. Mat. Pura Appl. (4) 76 (1967), p. 325–376. – ISSN 0373–3114.Legendre, Adrien M.: Recherches sur l’attraction des spheroides homogenes. En: Mémoires de mathématique et de physique : prés. á l’Acad´emie Royale des Sciences, par divers savans, et lûs dans ses assemblées 1785 (2007), p. 411 – 434.Marcellán, Francisco ; Marriaga, Misael E. ; Pérez, Teresa E. ; Piñar, Miguel A.: Coherent pairs of bivariate orthogonal polynomials. En: J. Approx. Theory 245 (2019), p. 40–63. – ISSN 0021–9045.Marcellán, Francisco ; Pérez, Teresa E. ; Piñar, Miguel A.: Orthogonal polynomials on weighted Sobolev spaces: The semiclassical case. En: Ann. Numer. Math. 2 (1995), Nr. 1-4, p. 93–122. – ISSN 1021–2655.Marcellán, Francisco ; Petronilho, J.: Orthogonal polynomials and coherent pairs: The classical case. En: Indag. Math., New Ser. 6 (1995), Nr. 3, p. 287–307. – ISSN 0019–3577.Marcellán, Francisco ; Petronilho, José C. ; Pérez, Teresa E. ; Piñar, Miguel A.: What is beyond coherent pairs of orthogonal polynomials? En: J. Comput. Appl. Math. 65 (1995), Nr. 1-3, p. 267–277. – ISSN 0377–0427.Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-q-coherent pairs. En: Numer. Algorithms 60 (2012), Nr. 2, p. 223–239. – ISSN 1017–1398.Marcellán, Francisco ; Pinzón-Cortés, Natalia C.: (1, 1)-Dω-coherent pairs. En: J. Difference Equ. Appl. 19 (2013), Nr. 11, p. 1828–1848. – ISSN 1023–6198.Meijer, H. G.: Coherent pairs and zeros of Sobolev-type orthogonal polynomials. En: Indag. Math., New Ser. 4 (1993), Nr. 2, p. 163–176. – ISSN 0019–3577.Meijer, H. G.: Determination of all coherent pairs. En: J. Approx. Theory 89 (1997), Nr. 3, p. 321–343. – ISSN 0021–9045.Proriol, Joseph: Sur une famille de polynômes à deux variables orthogonaux dans un triangle. En: C. R. Acad. Sci., Paris 245 (1957), p. 2459–2461. – ISSN 0001–4036.Stieltjes, T.-J.: Recherches sur les fractions continues. En: Annales de la Faculté des sciences de Toulouse : Mathématiques 8 (1894), Nr. 4, p. J1–J122.Suetin, P. K.: Anal. Methods Spec. Funct.. Vol. 3: Orthogonal polynomials in two variables. Transl. from the 1988 Russian original by E. V. Pankratiev . Amsterdam: Gordon and Breach Science Publishers, 1999. – ISBN 90–5699–167–1.Szegö, Gábor: Colloq. Publ., Am. Math. Soc.. Vol. 23: Orthogonal polynomials. American Mathematical Society (AMS), Providence, RI, 1939 ISSN 0065–9258.Tchebychev, Pafnutii L.: Théorie des mécanismes connus sous le nom de parallélogrammes. Imprimerie de l’Académie impériale des sciences, 1853.Tchebyshev, Pafnutii L.: Sur le développement des fonctions à une seule variable. En: Bull. Acad. Sci. St. Petersb 1 (1859), Nr. 193-200, p. 124.Wolfram Research, Inc. Mathematica, Version 12.1. Champaign, Illinois, (2020).Zernike, F. ; Brinkman, H. C.: Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome. En: Proc. Akad. Wet. 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