Abstract
We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein–Chen–Dilcher’s polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of iterated integrals of ultraspherical polynomials with parameters \((2\alpha +1)/2\), \(\alpha \in \mathbb {Z}_{+}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Here we state results for \(m=n\) but analogous statements hold for \(m=n+j\) with j a fixed integer.
The zeros of \(I_n(\widehat{P}_n)\) different from 0.
Hereafter, if \(\nu \) is a positive Borel measure with compact support in the complex plane, its logarithmic potential and energy are respectively \(V(\nu ,z):=\int \log \frac{1}{|z-x|}\, d\nu (x)\) and \(I(\nu ):=\int V(\nu ,z)\, d\nu (z)\).
The zeros of \(I_n(\widehat{P}_n^{(1/2)})\) different from 0.
References
Alfaro, M., Pérez, T.E., Piñar, M.A., Rezola, M.L.: Sobolev orthogonal polynomials: the discrete-continuous case. Methods Appl. Anal. 6, 593–616 (1999)
Bello Cruz, J.Y., Pijeira Cabrera, H., Márquez, C., UrbinaRomero, W.: Sobolev-Gegenbauer-type orthogonality and a hydrodynamical interpretation. Integral Transf. Spec. Funct. 22, 711–722 (2011)
Blatt, H.-P., Saff, E.B., Simkani, M.: Jentzsch–Szegő type theorems for the zeros of best approximates. J. Lond. Math. Soc. 38, 192–204 (1988)
Borwein, P.B., Chen, W., Dilcher, K.: Zeros of iterated integrals of polynomials. Can. J. Math. 47, 65–87 (1995)
García-Caballero, E.M., Pérez, T.E., Piñar, M.A.: Hermite interpolation and Sobolev orthogonality. Acta Appl. Math. 61, 87–99 (2000)
Khavinson, D., Pereira, R., Putinar, M., Saff, E. B., Shimorin, S.: Borcea’s variance conjectures on the critical points of polynomials. Notions of positivity and the geometry of polynomials, 283–309, Trends Math., Birkhäuser/Springer Basel AG, Basel (2011)
Cabrera, H.P., Cruz, J.Y.B., Romero, W.U.: On polar Legendre polynomials. Rock. Mount. J. Math. 40, 2025–2036 (2010)
Pijeira Cabrera, H., Rivero Castillo, D.: Iterated integrals ofJacobi polynomials. Bull. Malays. Math. Sci. Soc. (2019). https://doi.org/10.1007/s40840-019-00831-8
Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford Univ. Press, Oxford (2002)
Ransford, T.: Potential Theory in the Complex Plane. Cambridge Univ. Press, Cambridge (1995)
Rodríguez, J.M.: Zeros of Sobolev orthogonal polynomials via Muckenhoupt inequality with three measures. Acta Appl. Math. 142, 9–37 (2016)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer, New York (1997)
Sharapudinov, I.I.: Approximation properties of Fourier series of Sobolev orthogonal polynomials with Jacobi weight and discrete masses. Math. Notes 101, 718–734 (2017)
Sheil-Small, T.: Complex Polynomials. Cambridge Univ. Press, Cambridge (2002)
Szegő, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Publ. 23, 4th ed., Providence, RI (1975)
Widom, H.: Polynomials associated with measures in the complex plane. J. Math. Mech. 16, 997–1013 (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Manuel Bello Hernández: Research partially supported by Ministry of Economy and Competitiveness of Spain, under Grant MTM2014-54043-P. Héctor Pijeira-Cabrera: Research partially supported by Ministry of Science, Innovation and Universities of Spain, under Grant PGC2018-096504-B-C33.
Rights and permissions
About this article
Cite this article
Bello-Hernández, M., Pijeira-Cabrera, H. & Rivero-Castillo, D. Iterated Integrals and Borwein–Chen–Dilcher Polynomials. Mediterr. J. Math. 17, 148 (2020). https://doi.org/10.1007/s00009-020-01571-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-020-01571-x