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A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices

Author

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  • Hongyan Xu

    (College of Arts and Sciences, Suqian University, Suqian 223800, China
    School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China
    School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China
    These authors contributed equally to this work.)

  • Guangsheng Chen

    (College of Mathematics and Computer Science, Guangxi Science & Technology Normal University, Laibin 546199, China)

  • Hari Mohan Srivastava

    (Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
    Department of Mathematics and Informatics, Azerbaijan University, AZ197 Baku, Azerbaijan
    Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy)

  • Hong Li

    (School of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China)

  • Zuxing Xuan

    (Department of General Education, Beijing Union University, Beijing 100101, China)

  • Yongqin Cui

    (Department of Informatics and Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, China)

Abstract

In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with smaller indices if these Dirichlet series have different growth indices; (ii) the q 1 -type of the Hadamard–Kong product series is equal to zero if p Dirichlet series are of q j -regular growth, and q 1 < q 2 < ⋯ < q p , q j ∈ N + , j = 1 , 2 , … , p . The second purpose is to reveal the properties of the growth in the Hadamard–Kong product series of two Dirichlet series—when one Dirichlet series is of finite order, the other is of logarithmic order, and two Dirichlet series are of finite logarithmic order—and obtain the growth relationships between the Hadamard–Kong product series and two Dirchlet series concerning the order, the logarithmic order, logarithmic type, etc. Finally, some examples are given to show that our results are best possible.

Suggested Citation

  • Hongyan Xu & Guangsheng Chen & Hari Mohan Srivastava & Hong Li & Zuxing Xuan & Yongqin Cui, 2022. "A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices," Mathematics, MDPI, vol. 10(13), pages 1-26, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2220-:d:847316
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    Cited by:

    1. Hari Mohan Srivastava, 2022. "Higher Transcendental Functions and Their Multi-Disciplinary Applications," Mathematics, MDPI, vol. 10(24), pages 1-3, December.

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