On the Domain of Convergence and Poles of ComplexJ-Fractions Original Research Article

Consider the infiniteJ-fraction [formula] wherean∈C\{0},bn∈C. Under very general conditions on the coefficients {an}, {bn}, we prove that this continued fraction converges to a meromorphic function in C\R. Such conditions hold, in particular, if limn J(an)=limn J(bn)=0 and ∑n⩾0 (1/|an|)=∞ (or ∑n⩾0 (|bn|/|anan+1|)=∞). The poles are located in the point spectrum of the associated tridiagonal infinite matrix and their order determined in terms of the asymptotic behavior of the zeros of the denominators of the corresponding partial fractions.