IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i16p3500-d1216374.html
   My bibliography  Save this article

Fractal Divergences of Generalized Jacobi Polynomials

Author

Listed:
  • Răzvan-Cornel Sfetcu

    (Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania)

  • Vasile Preda

    (Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania
    “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania
    “Costin C. Kiriţescu” National Institute of Economic Research, Calea 13 Septembrie 13, 050711 Bucharest, Romania)

Abstract

The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are convergent and the last is divergent.

Suggested Citation

  • Răzvan-Cornel Sfetcu & Vasile Preda, 2023. "Fractal Divergences of Generalized Jacobi Polynomials," Mathematics, MDPI, vol. 11(16), pages 1-12, August.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3500-:d:1216374
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/16/3500/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/16/3500/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Kaizoji, Taisei, 2006. "An interacting-agent model of financial markets from the viewpoint of nonextensive statistical mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 109-113.
    2. Iulia-Elena Hirica & Cristina-Liliana Pripoae & Gabriel-Teodor Pripoae & Vasile Preda, 2022. "Lie Symmetries of the Nonlinear Fokker-Planck Equation Based on Weighted Kaniadakis Entropy," Mathematics, MDPI, vol. 10(15), pages 1-22, August.
    3. Furuichi, Shigeru & Mitroi, Flavia-Corina, 2012. "Mathematical inequalities for some divergences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 388-400.
    4. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2008. "Scaling in the distribution of intertrade durations of Chinese stocks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5818-5825.
    5. Chen, Yanguang, 2020. "Equivalent relation between normalized spatial entropy and fractal dimension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    6. Y. Bar-Yam, 2004. "Multiscale Complexity/Entropy," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 47-63.
    7. Hasumi, Tomohiro, 2009. "Hypocenter interval statistics between successive earthquakes in the two-dimensional Burridge–Knopoff model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 477-482.
    8. Darooneh, Amir H. & Dadashinia, Cyruse, 2008. "Analysis of the spatial and temporal distributions between successive earthquakes: Nonextensive statistical mechanics viewpoint," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(14), pages 3647-3654.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Răzvan-Cornel Sfetcu & Vasile Preda, 2024. "Order Properties Concerning Tsallis Residual Entropy," Mathematics, MDPI, vol. 12(3), pages 1-16, January.
    2. Ferreira, D.S.R. & Ribeiro, J. & Oliveira, P.S.L. & Pimenta, A.R. & Freitas, R.P. & Dutra, R.S. & Papa, A.R.R. & Mendes, J.F.F., 2022. "Spatiotemporal analysis of earthquake occurrence in synthetic and worldwide data," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    3. Martins, Francisco Leonardo Bezerra & do Nascimento, José Cláudio, 2022. "Power law dynamics in genealogical graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    4. Răzvan-Cornel Sfetcu & Sorina-Cezarina Sfetcu & Vasile Preda, 2021. "Ordering Awad–Varma Entropy and Applications to Some Stochastic Models," Mathematics, MDPI, vol. 9(3), pages 1-15, January.
    5. Zhao, Pan & Pan, Jian & Yue, Qin & Zhang, Jinbo, 2021. "Pricing of financial derivatives based on the Tsallis statistical theory," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    6. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2009. "Detrended fluctuation analysis of intertrade durations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 433-440.
    7. Can Yilmaz Altinigne & Harun Ozkan & Veli Can Kupeli & Zehra Cataltepe, 2019. "An Empirical Study on Arrival Rates of Limit Orders and Order Cancellation Rates in Borsa Istanbul," Papers 1909.08308, arXiv.org.
    8. Niu, Hongli & Wang, Jun, 2017. "Return volatility duration analysis of NYMEX energy futures and spot," Energy, Elsevier, vol. 140(P1), pages 837-849.
    9. Quanbo Zha & Gang Kou & Hengjie Zhang & Haiming Liang & Xia Chen & Cong-Cong Li & Yucheng Dong, 2020. "Opinion dynamics in finance and business: a literature review and research opportunities," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 6(1), pages 1-22, December.
    10. Hasumi, Tomohiro, 2009. "Hypocenter interval statistics between successive earthquakes in the two-dimensional Burridge–Knopoff model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 477-482.
    11. Popescu, P.G. & Preda, V. & Sluşanschi, E.I., 2014. "Bounds for Jeffreys–Tsallis and Jensen–Shannon–Tsallis divergences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 280-283.
    12. Niu, Hongli & Wang, Weiqing & Zhang, Junhuan, 2019. "Recurrence duration statistics and time-dependent intrinsic correlation analysis of trading volumes: A study of Chinese stock indices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 838-854.
    13. Kluza, Paweł A., 2020. "On Jensen–Rényi and Jeffreys–Rényi type f-divergences induced by convex functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 548(C).
    14. Ren, Fei & Gu, Gao-Feng & Zhou, Wei-Xing, 2009. "Scaling and memory in the return intervals of realized volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(22), pages 4787-4796.
    15. Yanqi Zhao & Yue Zhang & Ying Yang & Fan Li & Rongkun Dai & Jianlin Li & Mingshi Wang & Zhenhua Li, 2023. "The Impact of Land Use Structure Change on Utilization Performance in Henan Province, China," IJERPH, MDPI, vol. 20(5), pages 1-18, February.
    16. Song, Dong-Ming & Jiang, Zhi-Qiang & Zhou, Wei-Xing, 2009. "Statistical properties of world investment networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(12), pages 2450-2460.
    17. Guoxin Qiu & Kai Jia, 2018. "Extropy estimators with applications in testing uniformity," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 30(1), pages 182-196, January.
    18. Murakami, Ryo & Nakamura, Tomomichi & Kimura, Shin & Manabe, Masashi & Tanizawa, Toshihiro, 2015. "On possible origins of trends in financial market price changes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 420(C), pages 179-189.
    19. Gao-Feng Gu & Xiong Xiong & Hai-Chuan Xu & Wei Zhang & Yongjie Zhang & Wei Chen & Wei-Xing Zhou, 2021. "An empirical behavioral order-driven model with price limit rules," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 7(1), pages 1-24, December.
    20. Ni, Xiao-Hui & Jiang, Zhi-Qiang & Gu, Gao-Feng & Ren, Fei & Chen, Wei & Zhou, Wei-Xing, 2010. "Scaling and memory in the non-Poisson process of limit order cancelation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2751-2761.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3500-:d:1216374. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.