On infinite Jacobi matrices with a trace class resolvent

Abstract Let { P n ( x ) } be an orthonormal polynomial sequence and denote by { w n ( x ) } the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for { P n ( x ) } is determinate and denote by J the corresponding Jacobi matrix operator on l 2 . We show that if J is positive definite and J − 1 belongs to the trace class then the series on the right-hand side of the defining equation F ( z ) ≔ 1 − z ∑ n = 0 ∞ w n ( 0 ) P n ( z ) converges locally uniformly on ℂ and it holds true that F ( z ) = ∏ n = 1 ∞ ( 1 − z ∕ λ n ) where { λ n ; n = 1 , 2 , 3 , … } = s p e c J . Furthermore, the Al-Salam–Carlitz II polynomials are treated as an example of orthogonal polynomials to which this theorem can be applied.