Skip to main content
Log in

Wiman–Valiron Theory for a Polynomial Series Based on the Askey–Wilson Operator

  • Published:
Constructive Approximation Aims and scope

Abstract

We establish a Wiman–Valiron theory of a polynomial series based on the Askey–Wilson operator \({\mathcal {D}}_q\), where \(q\in (0,1)\). For an entire function f of log-order smaller than 2, this theory includes (i) an estimate which shows that f behaves locally like a polynomial consisting of the terms near the maximal term of its Askey–Wilson series expansion, and (ii) an estimate of \({\mathcal {D}}_q^n f\) compared to f. We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey–Wilson operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The logarithmic density of a set \(E\subseteq [1,\infty )\) is defined by

    $$\begin{aligned}\mathrm {logdens}\,E:=\limsup _{r\rightarrow \infty }{\frac{\mathrm {logmeas}\,(E\cap [1,r])}{\ln r}}=\limsup _{r\rightarrow \infty }{\frac{1}{\ln r}\int _{E\cap [1,r]}{\frac{1}{x}\,dm}}, \end{aligned}$$

    where m is the Lebesgue measure on \({\mathbb {R}}\).

References

  1. Askey, R., Wilson, J.A.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54(319), (1985), iv+55. MR 783216 (87a:05023)

  2. Bergweiler, W., Ishizaki, K., Yanagihara, N.: Growth of meromorphic solutions of some functional equations I. Aequ. Math. 63, 140–151 (2002)

    Article  MathSciNet  Google Scholar 

  3. Cheng, K.H.: Wiman–Valiron theory for a polynomial series based on the Wilson operator. J. Approx. Theory 239, 174–209 (2019)

    Article  MathSciNet  Google Scholar 

  4. Cheng, K.H., Chiang, Y.M.: Nevanlinna theory of the Wilson divided-difference operator. Ann. Acad. Sci. Fenn. Math. 42, 175–209 (2017)

    Article  MathSciNet  Google Scholar 

  5. Chiang, Y.M., Feng, S.J.: On the growth of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)

    Article  MathSciNet  Google Scholar 

  6. Chiang, Y.M., Feng, S.J.: Nevanlinna theory of the Askey–Wilson divided difference operator. Adv. Math. 329, 217–272 (2018)

    Article  MathSciNet  Google Scholar 

  7. Clunie, J.: The determination of an integral function of finite order by its Taylor series. J. Lond. Math. Soc. 28, 58–66 (1953)

    Article  MathSciNet  Google Scholar 

  8. Clunie, J.: On the determination of an integral function from its Taylor series. J. Lond. Math. Soc. 30, 32–42 (1955)

    Article  MathSciNet  Google Scholar 

  9. Cooper, S.: The Askey–Wilson operator and the \( _6\phi _5\) summation formula, South East Asian. J. Math. Math. Sci. 1(1), 71–82 (2002)

    MathSciNet  Google Scholar 

  10. Fenton, P.C.: Some results of Wiman–Valiron type for integral functions of finite lower order. Ann. Math. 2(183), 237–252 (1976)

    Article  MathSciNet  Google Scholar 

  11. Fenton, P.C.: A note on the Wiman–Valiron method. Proc. Edinb. Math. Soc. (2) 37(1), 53–55 (1994)

    Article  MathSciNet  Google Scholar 

  12. Fenton, P.C.: A glance at Wiman–Valiron theory. Contemp. Math. 382, 131–139 (2005)

    Article  MathSciNet  Google Scholar 

  13. Gelfond, A.O.: Calculus of Finite Differences. State Publishing Office for Physical and Mathematical Literature, Moscow (1971) (in Russian)

  14. Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Hayman, W.K.: The local growth of power series: a survey of the Wiman–Valiron method. Can. Math. Bull. 17(3), 317–358 (1974)

    Article  MathSciNet  Google Scholar 

  16. Ishizaki, K., Yanagihara, N.: Wiman–Valiron method for difference equations. Nagoya Math. J. 175, 75–102 (2004)

    Article  MathSciNet  Google Scholar 

  17. Ismail, M.E.H.: The Askey–Wilson operator and summation theorems. Contemp. Math. 190, 171–178 (1995)

    Article  MathSciNet  Google Scholar 

  18. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  19. Ismail, M.E.H., Stanton, D.: \(q\)-Taylor theorems, polynomials expansions, and interpolation of entire functions. J. Approx. Theory 123–1, 125–146 (2003)

    Article  MathSciNet  Google Scholar 

  20. Juneja, O.P., Kapoor, G.P., Bajpai, S.K.: On the \((p, q)\)-order and lower \((p, q)\)-order of an entire function. J. Reine Angew. Math. 282, 53–67 (1976)

    MathSciNet  MATH  Google Scholar 

  21. Juneja, O.P., Kapoor, G.P., Bajpai, S.K.: On the \((p, q)\)-type and lower \((p, q)\)-type of an entire function. J. Reine Angew. Math. 290, 180–190 (1977)

    MathSciNet  MATH  Google Scholar 

  22. Kövari, T.: On the theorems of G. Pólya and P. Turan. J. Anal. Math. 6, 323–332 (1958)

    Article  Google Scholar 

  23. Kövari, T.: On the Borel exceptional values of lacunary integral functions. J. Anal. Math. 9, 71–109 (1961)

    Article  MathSciNet  Google Scholar 

  24. Saxer, W.: Über die Picardschen Ausnahmewerte sukzessiver Derivierten. Math. Z. 17, 206–227 (1923)

    Article  MathSciNet  Google Scholar 

  25. Valiron, G.: Lectures on the general theory of integral functions (reprinted), Chelsea (1949)

  26. Wiman, A.: Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen Taylorschen Reihe. Acta Math. 37, 305–326 (1914)

    Article  MathSciNet  Google Scholar 

  27. Wiman, A.: Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Betrage bei gegebenem Argumente der Funktion. Acta Math. 41, 1–28 (1916)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee for her/his helpful and constructive comments and bringing Fenton’s work [12] to their attention.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yik-Man Chiang.

Additional information

Communicated by Mourad Ismail.

Dedicated to the memories of Jim Clunie and Walter Hayman.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Both authors were partially supported by GRF No. 16306315 from the Research Grant Council of Hong Kong. The first author was also partially supported by the PDFS (No. PDFS2021-6S04), and the second author was also partially supported by GRF No. 600609 from the Research Grant Council of Hong Kong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, K.H., Chiang, YM. Wiman–Valiron Theory for a Polynomial Series Based on the Askey–Wilson Operator. Constr Approx 54, 259–294 (2021). https://doi.org/10.1007/s00365-021-09528-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-021-09528-3

Keywords

Mathematics Subject Classification

Navigation