A generalization of the $ q $-Lidstone series

Maryam AL-Towailb; Maryam AL-Towailb

doi:10.3934/math.2022518

AIMS Mathematics

2022, Volume 7, Issue 5: 9339-9352. doi: 10.3934/math.2022518

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Research article

A generalization of the $ q $-Lidstone series

Maryam AL-Towailb ^,

Department of Computer Science and Engineering, King Saud University, Riyadh, KSA

Received: 27 September 2021 Revised: 09 February 2022 Accepted: 17 February 2022 Published: 11 March 2022
MSC : 05A30, 11B68, 30B10, 30E20, 39A13

In this paper, we study the existence of solutions for the general $ q $-Lidstone problem:

$ \begin{equation*} (D_{q^{-1}}^{r_n}f)(1) = a_n, \quad (D_{q^{-1}}^{s_n}f)(0) = b_n, \quad (n\in \mathbb{N}) \end{equation*} $

where $ (r_n)_n $ and $ (s_n)_n $ are two sequences of non-negative integers and $ (a_n)_n $ and $ (b_n)_n $ are two sequences of complex numbers. We define a $ q^{-1} $-standard set of polynomials and then we introduce a generalization of the $ q $-Lidstone expansion theorem.
- $ q $-difference equations,
- standard sets of polynomials,
- $ q $-Lidstone series
Citation: Maryam AL-Towailb. A generalization of the $ q $-Lidstone series[J]. AIMS Mathematics, 2022, 7(5): 9339-9352. doi: 10.3934/math.2022518

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Abstract

In this paper, we study the existence of solutions for the general $ q $-Lidstone problem:

$ \begin{equation*} (D_{q^{-1}}^{r_n}f)(1) = a_n, \quad (D_{q^{-1}}^{s_n}f)(0) = b_n, \quad (n\in \mathbb{N}) \end{equation*} $

where $ (r_n)_n $ and $ (s_n)_n $ are two sequences of non-negative integers and $ (a_n)_n $ and $ (b_n)_n $ are two sequences of complex numbers. We define a $ q^{-1} $-standard set of polynomials and then we introduce a generalization of the $ q $-Lidstone expansion theorem.

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