|MSC:||39A11||Stability and asymptotics of difference equations; oscillatory and periodic solutions, etc.|
|41A60||Asymptotic approximations, asymptotic expansions (steepest descent, etc.)|
The difference equation in the title is solved by means of
functions, which can be represented as composed functions of the
exponential function and a function being odd with respect to one
or two arbitrary parameters. In the case $\beta=1/4$, $\gamma=3/4$
there is given a conjecture concerning a solution of a new type.
A second conjecture concerns the existence of asymptotically 3-periodic solutions.
Though the difference equation is of second order, we point out singular cases where
three initial values can be prescribed.