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Approximate closed-form formulas for the zeros of the Bessel polynomials. (English) Zbl 1255.33003

Summary: We find approximate expressions \(\tilde{x}(k, n, a)\) and \(\tilde{y}(k, n, a)\) for the real and imaginary parts of the \(k\)-th zero \(z_k = x_k + iy_k\) of the Bessel polynomial \(y_n(x; a)\). To obtain these closed-form formulas, we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas, and, then, a fit to the real and imaginary parts as functions of \(k, n\) and \(a\) is obtained. It is shown that the resulting complex numbers \(\tilde{x}(k, n, a) + i\tilde{y}(k, n, a)\) are \(O(1/n^2)\)-convergent to \(z_k\) for fixed \(k\).

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:

[1] H. L. Krall and O. Frink, “A new class of orthogonal polynomials: the Bessel polynomials,” Transactions of the American Mathematical Society, vol. 65, pp. 100-115, 1949. · Zbl 0031.29701 · doi:10.2307/1990516
[2] E. Grosswald, Bessel Polynomials, vol. 698 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978. · Zbl 0416.33008
[3] H. M. Srivastava, “Some orthogonal polynomials representing the energy spectral functions for a family of isotropic turbulence fields,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 64, no. 6, pp. 255-257, 1984. · Zbl 0509.33008 · doi:10.1002/zamm.19840640612
[4] J. L. López and N. M. Temme, “Large degree asymptotics of generalized Bessel polynomials,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 30-42, 2011. · Zbl 1213.33011 · doi:10.1016/j.jmaa.2010.10.030
[5] Ö. E\ugecio\uglu, “Bessel polynomials and the partial sums of the exponential series,” SIAM Journal on Discrete Mathematics, vol. 24, no. 4, pp. 1753-1762, 2010. · Zbl 1238.05007 · doi:10.1137/090760337
[6] C. Berg and C. Vignat, “Linearization coefficients of Bessel polynomials and properties of Student t-distributions,” Constructive Approximation, vol. 27, no. 1, pp. 15-32, 2008. · Zbl 1132.33308 · doi:10.1007/s00365-006-0643-6
[7] L. Pasquini, “Accurate computation of the zeros of the generalized Bessel polynomials,” Numerische Mathematik, vol. 86, no. 3, pp. 507-538, 2000. · Zbl 0965.65045 · doi:10.1007/s002110000166
[8] A. J. Carpenter, “Asymptotics for the zeros of the generalized Bessel polynomials,” Numerische Mathematik, vol. 62, no. 4, pp. 465-482, 1992. · Zbl 0782.33003 · doi:10.1007/BF01396239
[9] F. W. J. Olver, “The asymptotic expansion of Bessel functions of large order,” Philosophical Transactions of the Royal Society of London Series A, vol. 247, pp. 328-368, 1954. · Zbl 0070.30801 · doi:10.1098/rsta.1954.0021
[10] H.-J. Runckel, “Zero-free parabolic regions for polynomials with complex coefficients,” Proceedings of the American Mathematical Society, vol. 88, no. 2, pp. 299-304, 1983. · Zbl 0521.30005 · doi:10.2307/2044720
[11] M. G. de Bruin, E. B. Saff, and R. S. Varga, “On the zeros of generalized Besselpolynomials I, II,” Indagationes Mathematicae, vol. 84, pp. 1-25, 1981. · Zbl 0467.33004 · doi:10.1016/1385-7258(81)90013-5
[12] F. Gálvez and J. S. Dehesa, “Some open problems of generalised Bessel polynomials,” Journal of Physics A, vol. 17, no. 14, pp. 2759-2766, 1984. · Zbl 0583.33007 · doi:10.1088/0305-4470/17/14/019
[13] E. Hendriksen and H. van Rossum, “Electrostatic interpretation of zeros,” in Orthogonal Polynomials and Their Applications, M. Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia, and J. Vinuesa, Eds., vol. 1329 of Lecture Notes in Mathematics, pp. 241-250, Springer, Berlin, Germany, 1988. · Zbl 0685.33009 · doi:10.1007/BFb0083363
[14] G. Valent and W. Van Assche, “The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: additional material,” Journal of Computational and Applied Mathematics, vol. 65, no. 1-3, pp. 419-447, 1995. · Zbl 0856.33002 · doi:10.1016/0377-0427(95)00128-X
[15] G. Szeg\Ho, Orthogonal Polynomials, Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1975.
[16] F. Marcellán, A. Martínez-Finkelshtein, and P. Martinez-Gonzalez, “Electrostatic models for zeros of polynomials: old, new, and some open problems,” Journal of Computational and Applied Mathematics, vol. 207, pp. 258-272, 2007. · Zbl 1131.30002 · doi:10.1016/j.cam.2006.10.020
[17] R. G. Campos, “Perturbed zeros of classical orthogonal polynomials,” Boletin de la Sociedad Matematica Mexicana, vol. 5, no. 1, pp. 143-153, 1999. · Zbl 0958.33005
[18] R. G. Campos, “Solving singular nonlinear two-point boundary value problems,” Boletin de la Sociedad Matematica Mexicana, vol. 3, no. 2, pp. 279-297, 1997. · Zbl 0895.41003
[19] R. G. Campos and L. A. Avila, “Some properties of orthogonal polynomials satisfying fourth order differential equations,” Glasgow Mathematical Journal, vol. 37, no. 1, pp. 105-113, 1995. · Zbl 0820.33005 · doi:10.1017/S0017089500030445
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