Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials
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- by H. M. Srivastava, Jian-Rong Zhou and Zhi-Gang Wang PDF
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Abstract:
The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Clausenian hypergeometric $\;_3F_2$ functions and polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Clausenian hypergeometric polynomials: \[ \;_3F_2(-n, \tau n+a, b;\tau n+c, -n+d;z),\] where $n$ is a nonnegative integer. Some families of the hypergeometric $_3F_2$ functions, which are connected (by means of a hypergeometric reduction formula) with the Gauss hypergeometric polynomials of the form \[ \;_2F_1(-n,kn+l+1;kn+l+2;z),\] are also investigated. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0).References
- W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
- Kathryn Boggs and Peter Duren, Zeros of hypergeometric functions, Comput. Methods Funct. Theory 1 (2001), no. 1, [On table of contents: 2002], 275–287. MR 1931616, DOI 10.1007/BF03320990
- D. Dominici, K. Driver and K. Jordaan, Polynomial solutions of differential-difference equations, J. Approx. Theory (2009) [doi:10.1016/j.jat.2009.05.010].
- Kathy Driver and Peter Duren, Zeros of the hypergeometric polynomials $F(-n,b;2b;z)$, Indag. Math. (N.S.) 11 (2000), no. 1, 43–51. MR 1809661, DOI 10.1016/S0019-3577(00)88572-9
- K. Driver and K. Jordaan, Zeros of ${}_3F_2\left (\smallmatrix -n,b,c\\ d,e\endsmallmatrix ;z\right )$ polynomials, Numer. Algorithms 30 (2002), no. 3-4, 323–333. MR 1927508, DOI 10.1023/A:1020126822435
- K. Driver and K. Jordaan, Asymptotic zero distribution of a class of $_3F_2$ hypergeometric functions, Indag. Math. (N.S.) 14 (2003), no. 3-4, 319–327. MR 2083078, DOI 10.1016/S0019-3577(03)90049-8
- K. Driver and K. Jordaan, Separation theorems for the zeros of certain hypergeometric polynomials, J. Comput. Appl. Math. 199 (2007), no. 1, 48–55. MR 2267530, DOI 10.1016/j.cam.2005.05.039
- K. Driver, K. Jordaan, and A. Martínez-Finkelshtein, Pólya frequency sequences and real zeros of some $_3F_2$ polynomials, J. Math. Anal. Appl. 332 (2007), no. 2, 1045–1055. MR 2324318, DOI 10.1016/j.jmaa.2006.10.080
- Kathy Driver, Kerstin Jordaan, and Norbert Mbuyi, Interlacing of zeros of linear combinations of classical orthogonal polynomials from different sequences, Appl. Numer. Math. 59 (2009), no. 10, 2424–2429. MR 2553144, DOI 10.1016/j.apnum.2009.04.007
- Kathy Driver and Manfred Möller, Zeros of the hypergeometric polynomials $F(-n,b;\ -2n;z)$, J. Approx. Theory 110 (2001), no. 1, 74–87. MR 1826086, DOI 10.1006/jath.2000.3548
- Peter L. Duren and Bertrand J. Guillou, Asymptotic properties of zeros of hypergeometric polynomials, J. Approx. Theory 111 (2001), no. 2, 329–343. MR 1849553, DOI 10.1006/jath.2001.3580
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
- A. B. J. Kuijlaars, A. Martinez-Finkelshtein, and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Electron. Trans. Numer. Anal. 19 (2005), 1–17. MR 2149265
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
- P. Martínez-González and A. Zarzo, Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials, J. Comput. Appl. Math. 233 (2010), 1577–1583.
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 3, Gordon and Breach Science Publishers, New York, 1990. More special functions; Translated from the Russian by G. G. Gould. MR 1054647
- H. M. Srivastava and Per W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1985. MR 834385
- H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1984. MR 750112
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- Nico M. Temme, Large parameter cases of the Gauss hypergeometric function, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 441–462. MR 1985714, DOI 10.1016/S0377-0427(02)00627-1
- Jian-Rong Zhou and Yu-Qiu Zhao, An infinite asymptotic expansion for the extreme zeros of the Pollaczek polynomials, Stud. Appl. Math. 118 (2007), no. 3, 255–279. MR 2305779, DOI 10.1111/j.1467-9590.2007.00373.x
Additional Information
- H. M. Srivastava
- Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
- Email: harimsri@math.uvic.ca
- Jian-Rong Zhou
- Affiliation: Department of Mathematics, Foshan University, Foshan 528000, Guangdong, People’s Republic of China
- Email: zhoujianrong2008@yahoo.com.cn
- Zhi-Gang Wang
- Affiliation: School of Mathematics and Computing Science, Changsha University of Science and Technology, Yuntang Campus, Changsha 410114, Hunan, People’s Republic of China
- Email: wangmath@163.com
- Received by editor(s): December 1, 2009
- Received by editor(s) in revised form: January 7, 2010
- Published electronically: February 11, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 80 (2011), 1769-1784
- MSC (2010): Primary 33C05, 33C20; Secondary 30C15, 33C45
- DOI: https://doi.org/10.1090/S0025-5718-2011-02409-9
- MathSciNet review: 2785478