| Title:
|
Zeros of a certain class of Gauss hypergeometric polynomials (English) |
| Author:
|
Abathun, Addisalem |
| Author:
|
Bøgvad, Rikard |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
1021-1031 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that as $n\to \infty $, the zeros of the polynomial $$ _{2}{F}_{1}\left [ \begin {matrix} -n, \alpha n+1\\ \alpha n+2 \end {matrix} ; z\right ] $$ cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al.\ (1999--2001) to the case of a complex parameter $\alpha $ and partially proves a conjecture made by the authors in an earlier work. (English) |
| Keyword:
|
asymptotic zero-distribution |
| Keyword:
|
hypergeometric polynomial |
| Keyword:
|
saddle point method |
| MSC:
|
30C15 |
| MSC:
|
33C05 |
| idZBL:
|
Zbl 07031694 |
| idMR:
|
MR3881893 |
| DOI:
|
10.21136/CMJ.2018.0055-17 |
| . |
| Date available:
|
2018-12-07T06:20:09Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147518 |
| . |
| Reference:
|
[1] Abathun, A., Bøgvad, R.: Asymptotic distribution of zeros of a certain class of hypergeometric polynomials.Comput. Methods Funct. Theory 16 (2016), 167-185. Zbl 1339.33009, MR 3503349, 10.1007/s40315-015-0131-1 |
| Reference:
|
[2] Andrews, G. E., Askey, R., Roy, R.: Special Functions.Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, Cambridge (1999). Zbl 0920.33001, MR 1688958, 10.1017/CBO9781107325937 |
| Reference:
|
[3] Bleistein, N.: Mathematical Methods for Wave Phenomena.Computer Science and Applied Mathematics, Academic Press, Orlando (1984). Zbl 0554.35002, MR 0755514, 10.1016/B978-0-08-091695-8.50001-7 |
| Reference:
|
[4] Boggs, K., Duren, P.: Zeros of hypergeometric functions.Comput. Methods Funct. Theory 1 (2001), 275-287. Zbl 1009.33004, MR 1931616, 10.1007/BF03320990 |
| Reference:
|
[5] Borcea, J., Bøgvad, R., Shapiro, B.: Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions.Publ. Res. Inst. Math. Sci. 45 (2009), 525-568 corrigendum ibid. 48 2012 229-233. Zbl 1182.30008, MR 2510511, 10.2977/prims/1241553129 |
| Reference:
|
[6] Bruijn, N. G. de: Asymptotic Methods in Analysis.Bibliotheca Mathematica 4, North-Holland Publishing, Amsterdam (1961). Zbl 0109.03502, MR 0177247 |
| Reference:
|
[7] Driver, K., Duren, P.: Asymptotic zero distribution of hypergeometric polynomials.Numer. Algorithms 21 (1999), 147-156. Zbl 0935.33004, MR 1725722, doi.org/10.1023/A:1019197027156 |
| Reference:
|
[8] Duren, P. L., Guillou, B. J.: Asymptotic properties of zeros of hypergeometric polynomials.J. Approximation Theory 111 (2001), 329-343. Zbl 0983.33008, MR 1849553, 10.1006/jath.2001.3580 |
| . |
