In this paper, we investigate even-order linear difference equations
and their criticality. However, we restrict our attention only to several
special cases of the general Sturm–Liouville equation. We wish to
investigate on such cases a possible converse of a known theorem. This
theorem holds for second-order equations as an equivalence; however, only
one implication is known for even-order equations. First, we show the
converse in a sense for one term equations. Later, we show an upper bound
on criticality for equations with nonnegative coefficients as well.
Finally, we extend the criticality of the second-order linear self-adjoint
equation for the class of equations with interlacing indices. In this way,
we can obtain concrete examples aiding us with our investigation.
This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation
y′′+λfalse/t2y=0$$ {y}^{\prime \prime }+\lambda /{t}^2y=0 $$. Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm.
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