zbMATH — the first resource for mathematics

Hardy-Petrovitch-Hutchinson’s problem and partial theta function. (English) Zbl 1302.30008
A polynomial \(p(x)=a_0+a_1x+a_2x^2+\ldots +a_nx^n\) with positive coefficients is called section-hyperbolic if, for all \(i=1,2,3,\ldots ,n\), each of its section \(a_0+a_1x+a_2x^2+\ldots +a_i x^i\) has only real roots. The paper deals with the class \(\Delta\) of all section-hyperbolic polynomials. The authors give the conclusive answer to the following question:

Problem (Hardy-Petrovitch-Hutchinson-Ostrovskii). For a given positive integer \(i\), find or estimate
\[ m_i=\inf_{p\in \Delta} \frac{a_i^2}{a_{i-1}a_{i+1}}. \]

Using an inductive procedure, they construct an entire function \(p_{\infty }=1+x+\sum_{n=2}^{\infty } A_n x^n\) such that, for any positive integer \( i,\) one has \( m_i=\frac{A_i^2}{A_{i-1}A_{i+1}}\), that is \(p_{\infty }\) minimizes all \(m_i\) simultaneously. The authors prove that the limit \(m_{\infty}=\lim_{i \to \infty} m_i\) exists and coincides with \(1/\tilde {q}\), where \(\tilde{q} >0\) is the maximal positive number for which the partial theta function \(\Psi(q,u)=\sum_{j=0}^{\infty }q^{j(j+1)/2}u^j\) has only real roots as a function of \(u\). They state that this number \(\tilde{q} \) is the unique solution of the equation \(\frac{1}{x}-\frac{2}{1-x}=\sum_{j=1}^{\infty} \frac{x^j}{1-x^{j+1}}\) beloning to the interval \((0,1)\).

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals
30D20 Entire functions of one complex variable, general theory
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
26C05 Real polynomials: analytic properties, etc.
Full Text: DOI Euclid arXiv
[1] G. E. Andrews, An introduction to Ramanujan’s “lost” notebook , Amer. Math. Monthly 86 (1979), 89-108. · Zbl 0401.01003 · doi:10.2307/2321943
[2] G. E. Andrews, Ramanujan’s “lost” notebook, I: Partial theta functions , Adv. Math. 41 (1981), 137-172. · Zbl 0477.33001 · doi:10.1016/0001-8708(81)90013-X
[3] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II. Springer, New York, 2009. · Zbl 1180.11001
[4] G. E. Andrews and S. O. Warnaar, The product of partial theta functions , Adv. in Appl. Math. 39 (2007), 116-120. · Zbl 1120.33014 · doi:10.1016/j.aam.2005.12.003
[5] B. C. Berndt and B. Kim, Asymptotic expansions of certain partial theta functions , Proc. Amer. Math. Soc. 139 (2011), 3779-3788. · Zbl 1272.11057 · doi:10.1090/S0002-9939-2011-11062-1
[6] K. Bringmann, A. Folsom, and R. C. Rhoades, Partial theta functions and mock modular forms as \(q\)-hypergeometric series , Ramanujan’s 125th birthday volume, Ramanujan J. 29 (2012), 295-310. · Zbl 1283.11077 · doi:10.1007/s11139-012-9370-1
[7] T. Craven and G. Csordas, “Composition theorems, multiplier sequences and complex zero decreasing sequences” in Value Distribution Theory and Related Topics, Adv. Complex Anal. Appl. 3 , Kluwer, Boston, 2004, 131-166. · Zbl 1101.26015 · doi:10.1007/1-4020-7951-6_6
[8] I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants , Birkhäuser, Boston, 1994. · Zbl 0827.14036
[9] D. Handelman, Arguments of zeros of highly log concave polynomials , [math.CA]. 1009.6022v1 · arxiv.org
[10] G. H. Hardy, On the zeros of a class of integral functions, Messenger of Mathematics 34 (1904), 97-101. · JFM 35.0416.05
[11] J. I. Hutchinson, On a remarkable class of entire functions , Trans. Amer. Math. Soc. 25 (1923), 325-332. · JFM 49.0217.02 · doi:10.1090/S0002-9947-1923-1501248-1
[12] O. M. Katkova, T. Lobova, and A. M. Vishnyakova, On power series having sections with only real zeros , Comput. Methods Funct. Theory 3 (2003), 425-441. · Zbl 1058.30009 · doi:10.1007/BF03321047
[13] O. M. Katkova, T. Lobova, and A. M. Vishnyakova, On entire functions having Taylor sections with only real zeros , Mat. Fiz. Anal. Geom. 11 (2004), 449-469. · Zbl 1078.30022
[14] V. P. Kostov, On the zeros of a partial theta function , to appear in Bull. Sci. Math. · Zbl 1278.26019
[15] D. C. Kurtz, A sufficient condition for all roots of a polynomial to be real , Amer. Math. Monthly 99 (1992), 259-263. · Zbl 0761.26011 · doi:10.2307/2325063
[16] B. Ja. Levin, Distribution of Zeros of Entire Functions , Transl. Math. Monogr. 5 , Amer. Math. Soc., Providence, 1964; rev. ed. 1980. · Zbl 0152.06703
[17] E. Laguerre, Sur quelques points de la théorie des équations numériques , Acta Math. 4 (1884), 97-120. · JFM 16.0069.03 · doi:10.1007/BF02418414
[18] C. Niculescu, A new look at Newton’s inequalities , J. Inequal. Pure Appl. Math. 1 (2000), no. 2, art. ID 17. · Zbl 0972.26010
[19] I. V. Ostrovskii, On zero distribution of sections and tails of power series , Israel Math. Conf. Proc. 15 (2001), 297-310. · Zbl 0994.30002
[20] M. Passare, J. M. Rojas, and B. Shapiro, New multiplier sequences via discriminant amoebae , Moscow Math. J. 11 (2011), 547-560. · Zbl 1290.12001
[21] M. Petrovitch, Une classe remarquable de séries entières , Atti del IV Congresso Internationale dei Matematici, Rome, Ser. 1, 2 (1908), 36-43. · JFM 40.0460.01
[22] G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen , J. Reine Angew. Math. 144 (1914), 89-113. · JFM 45.0176.01
[23] G. Pólya and G. Szegö, Problems and Theorems in Analysis , Vol. 1, Springer, Heidelberg, 1976. · Zbl 0338.00001
[24] S. Ramanujan, The Lost Notebook and Other Unpublished Papers: Mathematical Works of Srinivasa Ramanujan , Narosa, New Delhi, 1988. · Zbl 0639.01023
[25] A. Sokal, The leading root of the partial theta function , Adv. Math. 229 (2012), 2603-2621. · Zbl 1239.05018 · doi:10.1016/j.aim.2012.01.012
[26] S. O. Warnaar, Partial theta functions, I: Beyond the lost notebook , Proc. London Math. Soc. 87 (2003), 363-395. · Zbl 1089.05009 · doi:10.1112/S002461150201403X
[27] N. Zheltukhina, On sections and tails of power series , Ph.D. diss., Bilkent University, Ankara, Turkey, 2002. · Zbl 1014.30021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.