Geometric Interpretation of Perrin and Padovan Numbers

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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.

Contributed by: Marcello Artioli and Giuseppe Dattoli (September 2016)
Open content licensed under CC BY-NC-SA


Snapshots


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 (Wolfram MathWorld).

[2] E. W. Weisstein. "Padovan Sequence" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PadovanSequence.html (Wolfram MathWorld).

[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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