This paper isolates and studies a class of Markov chains with a
special quasi-triangular form of the transition matrix [so-called Δm,n(Δ′m,n)-matrix]. Many discrete stochastic processes encountered in
applications (queues, inventories and dams) have transition matrices
which are special cases of a Δm,n(Δ′m,n)-matrix. Necessary and
sufficient conditions for the ergodicity of a Markov chain with
transition Δm,n(Δ′m,n)-matrix are determined in the article in two
equivalent versions. According to the first version, these conditions are
expressed in terms of certain restrictions imposed on the generating
functions Ai(x) of the elements of the i-th row of the transition matrix,
i=0,1,2,…; in the other version they are connected with the
characterization of the roots of a certain associated function in the unit
circle of the complex plane. Results obtained in the article generalize,
complement, and refine similar results existing in the literature.