  We examine a fractional version of the discrete Nonlinear Schr\"{o}dinger
(dnls) equation, where the usual discrete laplacian is replaced by a fractional
discrete laplacian. This leads to the replacement of the usual nearest-neighbor
interaction to a long-range intersite coupling that decreases asymptotically as
a power-law. For the linear case, we compute both, the spectrum of plane waves
and the mean square displacement of an initially localized excitation in closed
form, in terms of regularized hypergeometric functions, as a function of the
fractional exponent. In the nonlinear case, we compute numerically the
low-lying nonlinear modes of the system and their stability, as a function of
the fractional exponent of the discrete laplacian. The selftrapping transition
threshold of an initially localized excitation shifts to lower values as the
exponent is decreased and, for a fixed exponent and zero nonlinearity, the
trapped fraction remains greater than zero.
