Indeterminate Stieltjes moment problems revisited

Abstract We consider a normalized indeterminate Hamburger moment sequence s which is supposed to be Stieltjes. We revisit old results of Chihara, Berg-Valent and Henrik L. Pedersen about determinacy/indeterminacy in the sense of Stieltjes for s and related to a constant $$\alpha (s)\le 0,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which can be defined using the corresponding orthogonal polynomials and those of the second kind. We recall how $$\alpha (s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be defined in terms of the entire functions B and D from the Nevanlinna matrix, and we find a similar expression involving the functions A and C . The constant $$\alpha (s)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> determines the Friedrichs extension of the Jacobi matrix when used as parameter in the Nevanlinna parametrization of the indeterminate moment problem. We prove how the zeros of the corresponding orthogonal polynomials converge to the mass-points of the Friedrichs solution of the moment problem. The graph of the function $$j(x)=D(x)/B(x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> shows that the supports of the N-extremal solutions are bounded below and the lower bound can be any number in the interval $$(-\infty ,\xi _1],$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$\xi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is the lower bound of the Friedrichs measure.