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On the solutions of second order difference equations with variable coefficients

Shirali Kadyrov, Alibek Orynbassar

Abstract

In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are elementary and based on factoring a quadratic shift operator. As an application, we obtain two new generalized continued fraction formulas for the mathematical constant $π^2$.

On the solutions of second order difference equations with variable coefficients

Abstract

In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are elementary and based on factoring a quadratic shift operator. As an application, we obtain two new generalized continued fraction formulas for the mathematical constant .

Paper Structure

This paper contains 5 sections, 9 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}
  4. Generalized continued fraction representations of $\pi^2$
  5. Discussion and Conclusion

Key Result

Theorem 1.1

Assume that there exist sequences $(c_n)$ and $(d_n)$ of complex numbers satisfying for any $n \ge 1.$ Then, the solution to the second order equation eq:main with initial values $y_{-1}$ and $y_0$ is given by

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1