Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Asymptotic distributions of the zeros of certain classes of hypergeometric functions and polynomials


Authors: H. M. Srivastava, Jian-Rong Zhou and Zhi-Gang Wang
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
Published electronically: February 11, 2011
MathSciNet review: 2785478
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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:

$\displaystyle \;_3F_2(-n, \tau n+a, b;\tau n+c, -n+d;z),$

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

$\displaystyle \;_2F_1(-n,kn+l+1;kn+l+2;z),$

are also investigated. Numerical evidence and graphical illustrations of the clustering of zeros on certain curves are generated by Mathematica (Version 4.0).


References [Enhancements On Off] (What's this?)

  • 1. W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Cambridge University Press, Cambridge, London and New York, 1935; Reprinted by Stechert-Hafner Service Agency, New York and London, 1964. MR 0185155 (32:2625)
  • 2. K. Boggs and P. Duren, Zeros of hypergeometric functions, Comput. Methods Funct. Theory 1 (2001), 275-287. MR 1931616 (2003k:33003)
  • 3. 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].
  • 4. K. Driver and P. Duren, Zeros of the hypergeometric polynomials $ F(-n,b;2b;z)$, Indag. Math. (New Ser.) 11 (2000), 43-51. MR 1809661 (2002d:33006)
  • 5. K. Driver and K. Jordaan, Zeros of $ _3F_2(-n,b,c;d,e;z)$ polynomials, Numer. Algorithms 30 (2002), 323-333. MR 1927508 (2003j:33018)
  • 6. K. Driver and K. Jordaan, Asymptotic zero distribution of $ _3F_2$ polynomials, Indag. Math. $ ($New Ser.$ )$ 14 (2003), 319-327. MR 2083078 (2005f:33004)
  • 7. K. Driver and K. Jordaan, Separation theorems for the zeros of certain hypergeometric polynomials, J. Comput. Appl. Math. 199 (2007), 48-55. MR 2267530 (2007m:33009)
  • 8. K. Driver and K. Jordaan, Pólya frequency sequences and real zeros of some $ _3F_2$ polynomials, J. Math. Anal. Appl. 332 (2007), 1045-1055. MR 2324318 (2009m:33013)
  • 9. K. Driver, K. Jordaan and N. Mbuyi, Interlacing of zeros of linear combinations of classical orthogonal polynomials from different sequences, Appl. Numer. Math. 59 (2009), 2424-2429. MR 2553144
  • 10. K. Driver and M. Möller, Zeros of the hypergeometric polynomials $ F(-n,b;-2n;z)$, J. Approx. Theory 110 (2001), 74-87. MR 1826086 (2002c:33001)
  • 11. P. Duren and B. J. Guillou, Asymptotic Properties of zeros of hypergeometric polynomials, J. Approx. Theory 111 (2001), 329-343. MR 1849553 (2002f:33011)
  • 12. E. Hille, Analytic Function Theory, Vol. II, Chelsea Publishing Company, Bronx, New York, 1973. MR 0201608 (34:1490)
  • 13. A. B. J. Kuijlaars, A. Martínez-Finkelshtein and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Electron. Trans. Numer. Anal. 19 (2005), 1-17. MR 2149265 (2006e:33010)
  • 14. M. Marden, Geometry of Polynomials, American Mathematical Society, Providence, Rhode Island, 1996. MR 0225972 (37:1562)
  • 15. 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.
  • 16. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 3 (Moscow, Nauka, 1986 (in Russian); English translation, Gordon and Breach, New York, 1990); Errata in Math. Comput. 65 (1996), 1380-1384. MR 1054647 (91c:33001)
  • 17. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985. MR 834385 (87f:33015)
  • 18. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984. MR 750112 (85m:33016)
  • 19. G. Szegö, Orthogonal Polynomials, Fourth edition, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, Rhode Island, 1975. MR 0372517 (51:8724)
  • 20. N. M. Temme, Large parameter cases of the Gauss hypergeometric function, J. Comput. Appl. Math. 153 (2003), 441-462. MR 1985714 (2004f:33006)
  • 21. J.-R. Zhou and Y.-Q. Zhao, An infinite asymptotic expansion for the extreme zeros of the Pollaczek polynomials, Stud. Appl. Math. 118 (2007), 255-279. MR 2305779 (2008e:33025)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 33C05, 33C20, 30C15, 33C45

Retrieve articles in all journals with MSC (2010): 33C05, 33C20, 30C15, 33C45


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

DOI: https://doi.org/10.1090/S0025-5718-2011-02409-9
Keywords: Generalized hypergeometric functions, Clausenian hypergeometric function, Gauss hypergeometric polynomials, asymptotic distribution of zeros, zeros of $_{3}F_{2}(-n, \tau n+a, b;\tau n+c, -n+d;z)$, zeros of $_{3}F_{2}(-n,a,b;c,d;z)$, Jacobi polynomials, Rice polynomials, Pasternack polynomials, hypergeometric reduction formulas, Euler-Mascheroni constant, Vitali’s theorem, Hurwitz’s theorem, Eneström-Kakeya theorem, \textit{Mathematica} (Version 4.0).
Received by editor(s): December 1, 2009
Received by editor(s) in revised form: January 7, 2010
Published electronically: February 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society