On the domain of convergence and poles of complex J-fractions

Consider the infinite J-fraction [GRAPHICS] where a(n) is an element of C\{0}, b(n) is an element of C. Under very general conditions on the coefficients {a(n)}, {b(n)}, we prove that this continued fraction converges to a meromorphic function in C/R. Such conditions hold, in particular, if lim(n) zeta(a(n))=lim(n) zeta(b(n))=0 and Sigma n greater than or equal to 0 (1/\a(n)\) = infinity (or Sigma n greater than or equal to 0 (\b(n)\/\a(n)a(n+1)\) = infinity). 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. (C) 1998 Academic Press.