Geometric Interpretation of Perrin and Padovan Numbers

On the left of each graphic is a three-dimensional plot of a lacunary Legendre polynomial in two variables and . On the right is a two-dimensional plot of the surface cut by a plane perpendicular to the axis. Also shown is the trace of a plane perpendicular to the axis. These lacunary Legendre polynomials are defined in the Details.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
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DETAILS

The Padovan and Perrin numbers play an important role in combinatorics [1, 2] and in applied research [3].
They can be calculated using the generating functions:
,
.
Their explicit form can be derived using a procedure involving multivariable forms of Hermite [4] and Legendre polynomials, discussed in [5].
Consider the polynomials defined by the generating function:
.
By the integral representation method discussed in [5]:
where are three-variable Hermite–Kampé de Fériet polynomials [4] defined by
,
.
Using the property of the two-variable Hermite–Kampé de Fériet polynomials when the first parameter is null,
,
allows the definition of the two-variable polynomials (call them lacunary Legendre polynomials),
and of the associated numbers
.
The - plots of this family of polynomials are shown in the Snapshots, reproducing the first and the tenth numbers of the series, as well as a case not generating any number .
The Padovan and Perrin numbers are finally expressible in terms of as
,
.
References
[1] E. W. Weisstein. "Perrin Sequence" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PerrinSequence.html.
[2] E. W. Weisstein. "Padovan Sequence" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PadovanSequence.html.
[3] J. Ablinger and J. Blümlein, "Harmonic Sums, Polylogarithms, Special Numbers, and Their Generalizations," in Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions (C. Schneider and J. Blümlein, eds.), Vienna: Springer Vienna, 2013. doi:10.1007/978-3-7091-1616-6_ 1.
[4] M. Artioli and G. Dattoli. "Geometric Properties of Generalized Hermite Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometricPropertiesOfGeneralizedHermitePolynomials
[5] M. Artioli and G. Dattoli. "Geometry of Two-Variable Legendre Polynomials" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/GeometryOfTwoVariableLegendrePolynomials.
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