Duke Mathematical Journal

Hardy–Petrovitch–Hutchinson’s problem and partial theta function

Vladimir Petrov Kostov and Boris Shapiro

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Abstract

In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E. Laguerre on uniform limits of sequences of real-rooted polynomials and by an interesting result of G. H. Hardy. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities, one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J. I. Hutchinson has shown that an entire function p(x)=a0+a1x++anxn+ with strictly positive coefficients has the property that all of its finite segments aixi+ai+1xi+1++ajxj have only real roots if and only if ai2/ai1ai+14 for i=1,2, . In the present paper, we give sharp lower bounds on the ratios ai2/ai1ai+1 (i=1,2,) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when i equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre–Pólya class L−PI. We also explain the relation between Newton’s and Hutchinson’s inequalities and the logarithmic image of the set of all real-rooted polynomials with positive coefficients.

Article information

Source
Duke Math. J., Volume 162, Number 5 (2013), 825-861.

Dates
First available in Project Euclid: 29 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1364562912

Digital Object Identifier
doi:10.1215/00127094-2087264

Mathematical Reviews number (MathSciNet)
MR3047467

Zentralblatt MATH identifier
1302.30008

Subjects
Primary: 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Secondary: 12D10: Polynomials: location of zeros (algebraic theorems) {For the analytic theory, see 26C10, 30C15} 26C05: Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx]

Citation

Kostov, Vladimir Petrov; Shapiro, Boris. Hardy–Petrovitch–Hutchinson’s problem and partial theta function. Duke Math. J. 162 (2013), no. 5, 825--861. doi:10.1215/00127094-2087264. https://projecteuclid.org/euclid.dmj/1364562912


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References

  • [1] G. E. Andrews, An introduction to Ramanujan’s “lost” notebook, Amer. Math. Monthly 86 (1979), 89–108.
  • [2] G. E. Andrews, Ramanujan’s “lost” notebook, I: Partial theta functions, Adv. Math. 41 (1981), 137–172.
  • [3] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook, Part II. Springer, New York, 2009.
  • [4] G. E. Andrews and S. O. Warnaar, The product of partial theta functions, Adv. in Appl. Math. 39 (2007), 116–120.
  • [5] B. C. Berndt and B. Kim, Asymptotic expansions of certain partial theta functions, Proc. Amer. Math. Soc. 139 (2011), 3779–3788.
  • [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.
  • [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.
  • [8] I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
  • [9] D. Handelman, Arguments of zeros of highly log concave polynomials, arXiv:1009.6022v1 [math.CA].
  • [10] G. H. Hardy, On the zeros of a class of integral functions, Messenger of Mathematics 34 (1904), 97–101.
  • [11] J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc. 25 (1923), 325–332.
  • [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.
  • [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.
  • [14] V. P. Kostov, On the zeros of a partial theta function, to appear in Bull. Sci. Math.
  • [15] D. C. Kurtz, A sufficient condition for all roots of a polynomial to be real, Amer. Math. Monthly 99 (1992), 259–263.
  • [16] B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monogr. 5, Amer. Math. Soc., Providence, 1964; rev. ed. 1980.
  • [17] E. Laguerre, Sur quelques points de la théorie des équations numériques, Acta Math. 4 (1884), 97–120.
  • [18] C. Niculescu, A new look at Newton’s inequalities, J. Inequal. Pure Appl. Math. 1 (2000), no. 2, art. ID 17.
  • [19] I. V. Ostrovskii, On zero distribution of sections and tails of power series, Israel Math. Conf. Proc. 15 (2001), 297–310.
  • [20] M. Passare, J. M. Rojas, and B. Shapiro, New multiplier sequences via discriminant amoebae, Moscow Math. J. 11 (2011), 547–560.
  • [21] M. Petrovitch, Une classe remarquable de séries entières, Atti del IV Congresso Internationale dei Matematici, Rome, Ser. 1, 2 (1908), 36–43.
  • [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.
  • [23] G. Pólya and G. Szegö, Problems and Theorems in Analysis, Vol. 1, Springer, Heidelberg, 1976.
  • [24] S. Ramanujan, The Lost Notebook and Other Unpublished Papers: Mathematical Works of Srinivasa Ramanujan, Narosa, New Delhi, 1988.
  • [25] A. Sokal, The leading root of the partial theta function, Adv. Math. 229 (2012), 2603–2621.
  • [26] S. O. Warnaar, Partial theta functions, I: Beyond the lost notebook, Proc. London Math. Soc. 87 (2003), 363–395.
  • [27] N. Zheltukhina, On sections and tails of power series, Ph.D. diss., Bilkent University, Ankara, Turkey, 2002.