On the approximation of analytic functions by infinite series of fractional Ruscheweyh derivatives bases

Mohra Zayed; Gamal Hassan; Mohra Zayed; Gamal Hassan

doi:10.3934/math.2024422

AIMS Mathematics

2024, Volume 9, Issue 4: 8712-8731. doi: 10.3934/math.2024422

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Research article

On the approximation of analytic functions by infinite series of fractional Ruscheweyh derivatives bases

Mohra Zayed ^{1
,
,},
Gamal Hassan ²

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Mathematics Department, Faculty of Science, University of Assiut, Assiut 71516, Egypt

Received: 03 November 2023 Revised: 04 December 2023 Accepted: 11 December 2023 Published: 29 February 2024
MSC : 30E10, 30D10, 41A10, 26A33

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the $ \mathbb{T}_{\rho} $-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results.
- complex conformable derivative,
- Ruscheweyh derivative,
- bases of polynomials,
- effectiveness,
- order and type,
- Fréchet space
Citation: Mohra Zayed, Gamal Hassan. On the approximation of analytic functions by infinite series of fractional Ruscheweyh derivatives bases[J]. AIMS Mathematics, 2024, 9(4): 8712-8731. doi: 10.3934/math.2024422

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Abstract

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the $ \mathbb{T}_{\rho} $-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results.

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