On the solutions of second order difference equations with variable coefficients

Kadyrov, Shirali; Shynarbek, Nurdaulet; Orynbassar, Alibek

doi:10.47974/JIM-2093

Open Access Research Article

On the solutions of second order difference equations with variable coefficients

Shirali Kadyrovsh.kadyrov@newuu.uzDepartment of General EducationNew Uzbekistan UniversityTashkent, UzbekistanView full profile → , ^*Nurdaulet ShynarbekCorresponding authornurdaulet.shynarbek@sdu.edu.kzPedagogy of Natural Sciences1/1 Abylay KhanSuleyman Demirel UniversityKaskelen, KazakhstanView full profile → , Alibek Orynbassaralibek.orynbassar@sdu.edu.kzPedagogy of Natural Sciences1/1 Abylay KhanSuleyman Demirel UniversityKaskelen, KazakhstanView full profile →

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pp. 1517–1527Vol. 28Issue 4June 2025DOI: 10.47974/JIM-2093XML

Received:: 07 Feb 2024
Published Online:: 02 Jun 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2093
Pages:: 1517–1527

Abstract

In this article, we explore solutions to second-order linear difference equations featuring variable coefficients. By imposing mild conditions, we present closed-form solutions by utilizing finite continued fraction representations. The proof of our results relies on elementary techniques, specifically involving factoring a quadratic shift operator. As a consequential application, we unveil two novel generalized continued fraction formulas for the mathematical constant π2.

Keywords

Difference equations Continued fractions Non-homogeneous equations Closed form solutions Apéry-like series π

Subject Classifications

39A0611J70

References

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