Cesaro′s Theorems for Complex Sequences

The classical Cesaro′s theorems apply to ratios of real-valued sequences like ƒn/gn where gn → ∞ monotonically, or ƒn → 0, and gn → 0 monotonically. Here such results are generalized to complex-valued sequences and, at the same time, the restriction of monotonicity on gn is removed in the real case. Such a condition is replaced with a more general property that does not require the ordering of the range. An application to the asymptotics of linear difference equations is presented.