On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients

  • T. H. Nguyen Department of Mathematics \& Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine
  • A. Vishnyakova Department of Mathematics \& Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine
Keywords: Laguerre-P\'olya class; entire functions of order zero; real-rooted polynomials; multiplier sequences; complex zero decreasing sequences

Abstract

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zero
with non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are
all equal and the odd-indexed ones are all equal:
$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$
We obtain necessary and sufficient conditions under which such functions
belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.

Author Biographies

T. H. Nguyen, Department of Mathematics \& Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine

Department of Mathematics \& Computer Sciences

V.N. Karazin Kharkiv National University

Kharkiv, Ukraine

A. Vishnyakova, Department of Mathematics \& Computer Sciences V.N. Karazin Kharkiv National University Kharkiv, Ukraine

Department of Mathematics \& Computer Sciences

V.N. Karazin Kharkiv National University

Kharkiv, Ukraine

References

A. Bohdanov, Determining bounds on the balues of barameters for a bunction '$varphi_{a}(z, m)=sum_{k=0}^infty frac{z^k}{a^{k^2}}(k!)^m,$ $m in (0,1), $ to belong to the Laguerre–P´olya class, Comput. Methods Funct. Theory, (2017), DOI:10.1007/ s40315-017-0210-6.

A. Bohdanov, A. Vishnyakova, On the conditions for entire functions related to the partial thetafunction to belong to the Laguerre–P´olya class, J. Math. Anal. Appl., 434 (2016), №2, 1740–1752. DOI:10.1016/j.jmaa.2015.09.084

T. Craven, G. Csordas, Complex zero decreasing sequences, Methods Appl. Anal., 2 (1995), 420–441.

G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of mathematics and its applications, Cambridge University Press, United Kingdom, Cambridge, 2004.

I.I. Hirschman, D.V. Widder, The convolution transform, Princeton University Press, Princeton, New Jersey, 1955.

J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc., 25 (1923), 325–332.

O. Katkova, T. Lobova and A. Vishnyakova, On power series having sections with only real zeros, Comput. Methods Funct. Theory, 3 (2003), №2, 425–441.

O. Katkova, T. Lobova, A. Vishnyakova, On entire functions having Taylor sections with only real zeros, J. Math. Phys., Anal., Geom., 11, (2004), №4, 449–469.

V.P. Kostov, About a partial theta function, C. R. Acad. Bulgare Sci., 66, (2013), 629–634.

V.P. Kostov, On the zeros of a partial theta function, Bull. Sci. Math., 137 (2013), №8, 1018–1030.

V.P. Kostov, On the spectrum of a partial theta function, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), №05, 925–933.

V.P. Kostov, Asymptotics of the spectrum of partial theta function, Revista Matematica Complutense, 27 (2014), №2, 677–684.

V.P. Kostov, A property of a partial theta function, C. R. Acad. Bulgare Sci., 67 (2014), 1319–1326.

V.P. Kostov, Asymptotic expansions of zeros of a partial theta function, C. R. Acad. Bulgare Sci., 68 (2015), 419–426.

V.P. Kostov, On the double zeros of a partial theta function, Bulletin des Sciences Mathamatiques, 140 (2016), №4, 98–111.

V.P. Kostov, On a partial theta function and its spectrum, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 146 (2016), №3, 609–623.

V.P. Kostov, The closest to 0 spectral number of the partial theta function, C. R. Acad. Bulgare Sci., 69, (2016), 1105–1112.

V.P. Kostov, B. Shapiro, Hardy-Petrovitch-Hutchinson’s problem and partial theta function, Duke Math. J., 162 (2013), №5, 825–861.

B.Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Mono., 5, Amer. Math. Soc., Providence, RI, 1964; revised ed. 1980.

T.H. Nguyen, A. Vishnyakova, On the entire functions from the Laguerre–P´olya class having the decreasing second quotients of Taylor coefficients, Journal of Mathematical Analysis and Applications, 465 (2018), №1, 348–359. https://doi.org/10.1016/j.jmaa.2018.05.018.

T.H. Nguyen, A. Vishnyakova, On the necessary condition for entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–P´olya class, Journal of Mathematical Analysis and Applications, 480 (2019), №2. https://doi.org/10.1016/j.jmaa.2019.123433.

T.H. Nguyen, A. Vishnyakova, On the closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–P´olya I class, Results in Mathematics, 75 (2020), №115. https: //doi.org/10.1007/s00025-020-01245-w.

T.H. Nguyen, A. Vishnyakova, On the entire functions from the Laguerre–P´olya I class having the increasing second quotients of Taylor coefficients, Journal of Mathematical Analysis and Applications, 498 (2021), №1. https://doi.org/10.1016/j.jmaa.2021.124955.

T.H. Nguyen, A. Vishnyakova, On the number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients. https://arxiv.org/abs/2101.11757

T.H. Nguyen, On the conditions for a special entire function related to the partial theta-function and the Euler function to belong to the Laguerre–P´olya class, Computational Methods and Function Theory, (2021). https://doi.org/10.1007/s40315-021-00361-0

N. Obreschkov, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963.

G. P´olya, Collected Papers, V.II Location of Zeros, (R.P.Boas ed.) MIT Press, Cambridge, MA, 1974.

G. P´olya, J. Schur, ¨ Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., 144 (1914), 89–113.

A.D. Sokal, The leading root of the partial theta function, Advances in Mathematics, 229 (2012), №5, 2063–2621.

S. O. Warnaar, Partial theta functions. https://www.researchgate.net/publication/327791878_Partial_theta_functions

Published
2021-12-26
How to Cite
Nguyen, T. H., & Vishnyakova, A. (2021). On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients. Matematychni Studii, 56(2), 149-161. https://doi.org/10.30970/ms.56.2.149-161
Section
Articles