On the Conditions for a Special Entire Function Related to the Partial Theta-Function and the q-Kummer Functions to Belong to the Laguerre–Pólya Class

Abstract

In this paper, we discuss the conditions for the function

$$\begin{aligned} F_a(z) =\sum _{k=0}^\infty \frac{z^k}{(a+1)(a^2+1) \cdots (a^k+1)},\quad a >1, \end{aligned}$$

to belong to the Laguerre–Pólya class, or to have only real zeros.

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Acknowledgements

This article was partially supported by the Akhiezer Foundation. The author is very grateful to her scientific advisor, A. Vishnyakova, for her patience, support and contribution to her research career. The author is also deeply grateful to the referees for valuable and attentive remarks.

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Correspondence to Thu Hien Nguyen.

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Thu Hien Nguyen was partially supported by the Akhiezer Foundation.

Communicated by Hendrik Laurberg Petersen.

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Nguyen, T.H. On the Conditions for a Special Entire Function Related to the Partial Theta-Function and the q-Kummer Functions to Belong to the Laguerre–Pólya Class. Comput. Methods Funct. Theory (2021). https://doi.org/10.1007/s40315-021-00361-0

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Keywords

  • Laguerre–Pólya class
  • Entire functions of order zero
  • Real-rooted polynomials
  • Multiplier sequences
  • Complex zero decreasing sequences

Mathematics Subject Classification

  • 30C15
  • 30D15
  • 30D35
  • 26C10