Asymptotic distributions of the zeros of a family of hypergeometric polynomials

Zhou, Jian-Rong; Srivastava, H.; Wang, Zhi-Gang

doi:10.1090/S0002-9939-2011-11117-1

Asymptotic distributions of the zeros of a family of hypergeometric polynomials
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by Jian-Rong Zhou, H. M. Srivastava and Zhi-Gang Wang PDF

Proc. Amer. Math. Soc. 140 (2012), 2333-2346 Request permission

Abstract:

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, \[ \;_2F_1(-n, a; -n+b;z),\] where $n$ is a nonnegative integer. Owing to the connection between the classical Jacobi polynomials and the Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez-Finkelshtein, Martínez-González and Orive. Numerical evidence and graphical illustrations of the clustering of the 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
P. B. Borwein and Weiyu Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), no. 1, 85–106. MR 1323965, DOI 10.1007/BF01294340
Kung-Yu Chen and H. M. Srivastava, A new result for hypergeometric polynomials, Proc. Amer. Math. Soc. 133 (2005), no. 11, 3295–3302. MR 2161152, DOI 10.1090/S0002-9939-05-07895-0
Diego Dominici, Kathy Driver, and Kerstin Jordaan, Polynomial solutions of differential-difference equations, J. Approx. Theory 163 (2011), no. 1, 41–48. MR 2741218, 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
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
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 and A. Martínez-Finkelshtein, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, J. Anal. Math. 94 (2004), 195–234. MR 2124460, DOI 10.1007/BF02789047
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
A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, Zeros of Jacobi polynomials with varying non-classical parameters, Special functions (Hong Kong, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 98–113. MR 1805976
A. Martínez-Finkelshtein and R. Orive, Riemann-Hilbert analysis of Jacobi polynomials orthogonal on a single contour, J. Approx. Theory 134 (2005), no. 2, 137–170. MR 2142296, DOI 10.1016/j.jat.2005.02.004
P. Martínez-González and A. Zarzo, Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials, J. Comput. Appl. Math. 233 (2010), no. 6, 1577–1583. MR 2559348, DOI 10.1016/j.cam.2009.02.088
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
H. M. Srivastava, Jian-Rong Zhou, and Zhi-Gang Wang, Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials, Math. Comp. 80 (2011), no. 275, 1769–1784. MR 2785478, DOI 10.1090/S0025-5718-2011-02409-9
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