Operator monotone functions, Jacobi operators and orthogonal polynomials

We reveal a connection between operator monotone functions and orthogonal polynomials. Especially, we express an operator monotone function with a Jacobi operator, and show that it is a limit of rational operator monotone functions. Further we prove that the ‘principal inverse’ of an orthogonal polynomial is operator monotone and hence it has a holomorphic extension to the open upper half plane, namely a Pick function (or Nevanlinna function).