Signal, Image and Video Processing

, Volume 7, Issue 1, pp 173–188 | Cite as

Tartaglia-Pascal’s triangle: a historical perspective with applications

  • A. FarinaEmail author
  • S. Giompapa
  • A. Graziano
  • A. Liburdi
  • M. Ravanelli
  • F. Zirilli
Original Paper


The aim of this paper is to provide a historical perspective of Tartaglia-Pascal’s triangle with its relations to physics, finance, and statistical signal processing. We start by introducing Tartaglia’s triangle and its numerous properties. We then consider its relationship with a number of topics: the Newton binomial, probability theory (in particular with the Gaussian probability density function, pdf), the Fibonacci sequence, the heat equation, the Schrödinger equation, the Black–Scholes equation of mathematical finance and stochastic filtering theory. Thus, the main contribution of this paper is to present a systematic review of the triangle properties, its connection to statistical theory, and its numerous applications. The paper has mostly a scientific-educational character and is addressed to a wide circle of readers. Sections 7 and 8 are more technical; thus, they may be of interest to more expert readers.


Tartaglia-Pascal triangle Yang Hui triangle Newton binomial Fibonacci sequence Probability theory Heat equation Schrödinger equation Black–Scholes equation Fokker-Planck equation Kushner equation Stochastic filtering 


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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • A. Farina
    • 1
    Email author
  • S. Giompapa
    • 2
  • A. Graziano
    • 2
  • A. Liburdi
    • 2
  • M. Ravanelli
    • 2
  • F. Zirilli
    • 3
  1. 1.Engineering, Large Systems Business UnitSELEX Sistemi IntegratiRomaItaly
  2. 2.Analysis of Integrated Systems Unit, SELEX Sistemi IntegratiRomaItaly
  3. 3.CERI—Centro di Ricerca “Previsione, Prevenzione Controllo dei Rischi Geologici”Università di Roma “La Sapienza”, Palazzo Doria PamphiljValmontone, RomaItaly

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