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A closed-form expression for second-order recurrences

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Author(s): E. Haji-Esmaili | S. H. Ghaderi

Journal: Acta Mathematica Universitatis Comenianae
ISSN 0862-9544

Volume: LXXX;
Issue: 2;
Start page: 251;
Date: 2011;
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Keywords: Second-order recurrence | characteristic polynomial | closed-form solution.

ABSTRACT
This paper introduces a closed-form expression for the second-order recurrence relation $a_{n}=c_1 a_{n-1}+c_2 a_{n-2}$, in which $c_1$ and $c_2$ are fixed constants and the value of two arbitrary terms $a_{n-p}$ and $a_{n-q}$ are known where $p$ and $q$ are positive integers and $p>q$. This expression is $$ a_{n}=frac{S(p+1)}{S(p-q+1)}a_{n-q}-(-c_2)^{(p-q)}frac{S(q+1)}{S(p-q+1)}a_{n-p}$$where$$S(n)=sum_{i=0}^{[n/2]} binom{n-i}{i}c_1^{n-i}c_2^{i}. $$
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