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<p>Abstract</p> <p>In this article we study the difference equation</p> <p><display-formula><m:math name="1687-1847-2011-28-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo class="MathClass-bin">+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo class="MathClass-rel">=</m:mo> <m:mfrac> <m:mrow> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>l</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mi>b</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-bin">-</m:mo> <m:mi>c</m:mi> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo class="MathClass-punc">,</m:mo> <m:mspace width="2.77695pt" class="tmspace"/> <m:mi>n</m:mi> <m:mspace width="2.77695pt" class="tmspace"/> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mn>1</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mspace width="2.77695pt" class="tmspace"/> <m:mo class="MathClass-op">&#8230;</m:mo> <m:mo class="MathClass-punc">,</m:mo> </m:mrow> </m:math> </display-formula></p> <p>where the initial conditions <it>x</it>_-<it>r</it>, <it>x</it>_{-<it>r</it>+1}, <it>x</it>_{<it>-r</it>+2},..., <it>x</it>₀are arbitrary positive real numbers, <it>r </it>= max{<it>l</it>, <it>k</it>, <it>p</it>, <it>q</it>} is nonnegative integer and <it>a</it>, <it>b</it>, <it>c </it>are positive constants: Also, we study some special cases of this equation.</p>