Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials

“…with initial conditions 01 1. QQ  Incomplete and biperiodic studies have been discussed by many authors [1,4,[9][10][11][12][13][14]19]. Depending on k any integer, incomplete recurrences of the k-Pell and k-Pell-Lucas sequence [3] are defined as follows:…”

Section: Introductionmentioning

confidence: 99%

“…The following studies are the continuation of these studies; for example, Catarino et al [3] worked with generalized Pell numbers. Similarly, later Kuloğlu et al [10] studied the polynomials of Vieta-Pell number sequences and brought a more up-to-date version to these studies as a whole.…”

mentioning

confidence: 99%

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

Kuloğlu¹,

Özkan²,

Shannon³

2023

NNTDM

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In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called ‘incomplete’ number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.

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“…Motivated by the above presentation, we are interested in applying Veita-Pell Polynomials (VPPs) to find the approximate solution for optimal control problems. As regards the properties of Veita-Pell family Functions, many authors had been applied them with their important results [10][11][12][13][14][15][16].A novel algorithm is suggested in the present work to solve OCPs approximately. The proposed method is utilized together with VPPs to parameterize the states functions.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution for Some Continuous Optimal Control Problems Using Veita-Pell Polynomials

Shihab,

Ouda,

Mitlif

2023

J. Al-Qadisiyah Comp. Sci. Math.

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In this paper, Veita-Pell polynomials (VPPs) are first presented with some interesting properties. These properties are utilized to construct a general explicit formula of their operational matrix of derivative. Then an appropriate direct technique is suggested for solving quadratic optimal control problems based on VPPs and the idea of state parameterization algorithm. The resulting performance index optimal value shows the proposed method is able to provide a good treatment with fast convergence. The effectiveness of the presented method is illustrated by solving three numerical examples. The obtained results show that as the number of basis functions VPPs increase the error in the solution by the present method will be decreased and it may exactly close with the analytical one. This is the main modification of the algorithm and this contribution in the using special basis functions in obtaining an approximate solution with minimum number of VPPs and satisfactory accuracy.

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Incomplete generalized Vieta–Pell and Vieta–Pell–Lucas polynomials

Abstract: In this paper, we introduce the incomplete Vieta–Pell and Vieta–Pell–Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

Cited by 7 publications

References 16 publications

ON THE (p,q)-NARAYANA n-DIMENSIONAL RECURRENCES

ON THE (p,q)-NARAYANA n-DIMENSIONAL RECURRENCES

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

Approximate Solution for Some Continuous Optimal Control Problems Using Veita-Pell Polynomials

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