Abstract
The purpose of this paper is to introduce several finite sets of orthogonal polynomials in two variables, and investigate some general properties of them such as recurrence relations, generating functions, differential equations, and Rodrigues type representations.
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References
Aktaş, R.: Representations for parameter derivatives of some Koornwinder polynomials in two variables. J. Egypt. Math. Soc. 24(4), 555–561 (2016)
Aktaş, R.: A note on parameter derivatives of the Jacobi polynomials on the triangle. Appl. Math. Comp. 247, 368–372 (2014)
Aktaş, R., Altın, A., Taşdelen, F.: A note on a family of two-variable polynomials. J. Comput. Appl. Math. 235, 4825–4833 (2011)
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 155, 2nd edn. Cambridge University Press, Cambridge (2014)
Fernández, L., Pérez, T.E., Piñar, M.A.: On Koornwinder classical orthogonal polynomials in two variables. J. Comput. Appl. Math. 236, 3817–3826 (2012)
Fernández, L., Pérez, T.E., Piñar, M.A.: Classical orthogonal polynomials in two variables: a matrix approach. Numer. Algorithms 39, 131–142 (2005)
Koepf, W., Masjed-Jamei, M.: Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials. Proc. Am. Math. Soc. 135(11), 3599–3606 (2007)
Koornwinder, T.H.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R. (ed.) Theory and Application of Special Functions, pp. 435–495. Academic Press, New York (1975)
Marcellán, F., Marriaga, M., Pérez, T.E., Piñar, M.A.: On bivariate classical orthogonal polynomials. Appl. Math. Comput. 325, 340–357 (2018)
Marcellán, F., Marriaga, M., Pérez, T.E., Piñar, M.A.: Matrix Pearson equations satisfied by Koornwinder weights in two variables. Acta Appl. Math. 153, 81–100 (2018)
Marriaga, M., Pérez, T.E., Piñar, M.A.: Three term relations for a class of bivariate orthogonal polynomials. Mediterr. J. Math. 14(54), 26 (2017)
Masjed-Jamei, M.: Classical orthogonal polynomials with weight function \(\left( \left( ax+b\right) ^{2}+\left( cx+d\right) ^{2}\right) ^{-p}\exp \left( q\arctan \left( \frac{ax+b}{cx+d}\right) \right) \); \(-\infty <x<\infty \) and a generalization of T and F distributions. Integr. Trans. Spec. Funct. 15(2), 137–153 (2004)
Masjed-Jamei, M.: Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation. Integr. Trans. Spec. Funct. 13(2), 169–190 (2002)
Masjed-Jamei, M., Soleyman, F., Area, I., Nieto, J.J.: Two finite q-Sturm liouville problems and their orthogonal polynomial solutions. Filomat 32(1), 231–244 (2018)
Milovanovic, G., Öztürk, G., Aktaş, R.: Properties of some of two-variable orthogonal polynomials. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00750-8
Soleyman, F., Masjed-Jamei, M., Area, I.: A finite class of q-orthogonal polynomials corresponding to inverse gamma distribution. Anal. Math. Phys. 7, 479–492 (2017)
Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, New York (1984)
Szegö, G.: Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, 4th edn. American Mathematical Society, Providence (1975)
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The work of the third author has been supported by the Alexander von Humboldt Foundation under the Grant number: Ref 3.4-IRN-1128637-GF-E.
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Communicated by Davod Khojasteh Salkuyeh.
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Güldoğan, E., Aktaş, R. & Masjed-Jamei, M. On Finite Classes of Two-Variable Orthogonal Polynomials. Bull. Iran. Math. Soc. 46, 1163–1194 (2020). https://doi.org/10.1007/s41980-019-00319-y
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DOI: https://doi.org/10.1007/s41980-019-00319-y