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 threevariable Hermite–Kampé de Fériet polynomials [4] defined by
,
.
Using the property of the twovariable Hermite–Kampé de Fériet polynomials
when the first parameter is null,
,
allows the definition of the twovariable 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
,
.
[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:10.1007/9783709116166_ 1.