This paper deals with the bounded input bounded output (BIBO) stability criteria of discrete-time nonlinear systems in the form of \[ \begin{aligned} y(t) &+ \sum a_ i y(t-i)+ \sum a_{ij} y(t-i) y(y-j)+ \cdots+ \sum a_{i_ 1\dots i_ n} y(i-i_ 1) y(t-i_ n)= \\=u(t) &+ \sum b_ i u(t-i)+ \sum b_{ij} u(t-i) u(t-j) +\cdots+ \sum b_{i_ 1 \dots i_ n} u(t- i_ 1) u(t- i_ n)+\tag{1}\\ &+ \sum c_{i_ 1\dots i_ n; j_ 1\dots j_ m} y(t-i_ 1) \dots y(t-i_ n) u(t-j_ 1) u(t-j_ m)\end{aligned} \] which may be viewed as a finite Taylor series expansion of the following sufficiently smooth function \[ y(t)= f(y(t-1), \dots, y(t- k), u(t-1), \dots, u(t-l)). \tag{2} \] The paper gives several theorems on the sufficient conditions derived by using an equivalent \(\delta \varepsilon\)-series representation, whose coefficients are exponentially bounded. The result obtained may be viewed as an extension of the stability criteria for discrete linear time-invariant systems (i.e. a system is BIBO stable, if and only if its impulse response is absolutely summable). The theorems presented can be applied to systems identification and adaptive control.