Abstract:We apply the binomial transforms to Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, and generating functions of these transforms are found by recurrence relations. Finally, we illustrate the relations between these transforms by deriving new formulas.
In this study, we define the binomial transform of the generalized fourth order Pell sequenceand as special cases, the binomial transform of the fourth order Pell and fourth order Pell-Lucassequences will be introduced. We investigate their properties in details.
In this study, we define the binomial transform of the generalized fourth order Pell sequenceand as special cases, the binomial transform of the fourth order Pell and fourth order Pell-Lucassequences will be introduced. We investigate their properties in details.
In this paper, we define the binomial transform of the generalized fifth order Pell sequence and as special cases, the binomial transform of the fifth order Pell and fifth order Pell-Lucas sequences will be introduced. We investigate their properties in details. We present Binet ’s formulas, generating functions, Simson formulas, recurrence properties, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.
“…The characteristic equation of the Padovan Sequence is 𝑥𝑥 3 − 𝑥𝑥 − 1 = 0. The real root of this characteristic equation is known as plastic ratio which was defined in 1924 by G´erard Cordonnier, [5].…”
At this work, we give a method for constructing the Perrin and Padovan sequences and obtain the De Moivre-type identity for Padovan numbers. Also, we define a Padovan sequence with new initial conditions and find some identities between all of these auxiliary sequences. Furthermore, we give quadratic approximations for these sequences.
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