Advertisement

Results in Mathematics

, 74:52 | Cite as

On Zeros of Some Entire Functions

Article
  • 13 Downloads

Abstract

Let
$$\begin{aligned} A_{q}^{(\alpha )}(a;z)=\sum _{k=0}^{\infty }\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}}, \end{aligned}$$
where \(\alpha >0,~0<q<1.\) In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire function \(A_{q}^{(\alpha )}(a;z)\) are all real and established some results on the zeros of \(A_{q}^{(\alpha )}(a;z)\) which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that \(A_{q}^{(\alpha )}(q^l;z),~l\ge 2\) has only infinitely many negative zeros that gives a partial answer to Zhang’s question. In addition, we establish some results on zeros of certain entire functions involving the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials.

Keywords

zeros of entire functions Pólya frequence sequence Vitali’s theorem Hurwitz’s theorem Rogers–Szegő polynomials Stieltjes–Wigert polynomials 

Mathematics Subject Classification

30C15 33D15 30C10 

Notes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11801451).

References

  1. 1.
    Ahlfors, L.: Complex Analysis, 3rd edn. McGraw-Hill, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.M.: On the generating functions of totally positive sequences. Proc. Nat. Acad. Sci. USA 37, 303–307 (1951)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  4. 4.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  5. 5.
    Boas, R.P.: Entire Functions. Academic Press, Cambridge (1954)zbMATHGoogle Scholar
  6. 6.
    Carnicer, J., Peña, J.M., Pinkus, A.: On some zero-increasing operators. Acta Math. Hung. 94, 173–190 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dimitrov, D.K., Peña, J.M.: Almost strict total positivity and a class of Hurwitz polynomials. J. Approx. Theory 132, 212–223 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Driver, K., Jordaan, K., Martinínez-Finkelshtein, A.: Pólya frequency sequences and real zeros of some \(_{3}F_{2}\) polynomials. J. Math. Anal. Appl. 332, 1045–1055 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  10. 10.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  11. 11.
    Ismail, M.E.H., Zhang, R.: \(q\)-Bessel functions and Rogers–Ramanujan type identities. Proc. Am. Math. Soc. 146(9), 3633–3646 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu, Z.-G.: On the \(q\)-partial differential equations and \(q\)-series, The Legacy of Srinivasa Ramanujan, 213–250, Ramanujan Math. Soc. Lect. Notes Ser., 20, Ramanujan Math. Soc., Mysore (2013)Google Scholar
  13. 13.
    Katkova, O.M., Lobova, T., Vishnyakova, A.: On power series having sections with only real zeros. Comput. Methods Funct. Theory 3, 425–441 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pólya, G., Szegő, G.: Problems and Theorems in Analysis II. Springer, New York (1976)CrossRefGoogle Scholar
  15. 15.
    Rogers, L.J.: On a three-fold symmetry in the elements of Heine’s series. Proc. Lond. Math. Soc. 24, 171–179 (1893)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Szegő, G.: Ein Betrag zur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. 19, 242–252 (1926)zbMATHGoogle Scholar
  17. 17.
    Titchmarsh, E.C.: The Theory of Functions, Corrected, 2nd edn. Oxford University Press, Oxford (1964)Google Scholar
  18. 18.
    Zhang, R.: Zeros of Ramanujan type entire functions. Proc. Am. Math. Soc. 145(1), 241–250 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaPeople’s Republic of China

Personalised recommendations