Symmetric moment problems and a conjecture of Valent

TL;DR

This paper investigates the order and type of certain indeterminate moment problems linked to birth and death processes, confirming a conjecture about their asymptotic behavior and relating the type to multi-zeta values.

Contribution

It proves the asymptotic value of the type for these moment problems and establishes estimates for order and type based on polynomial growth conditions of orthogonal polynomials.

Findings

Order of the moment problem is 1/p, confirming Romanov's result.

Type of the moment problem is within a specific interval related to p.

Type asymptotically matches the conjectured value as p approaches infinity.

Abstract

In 1998 G. Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes having polynomial birth and death rates of degree p\ge 3. Romanov recently proved that the order is 1/p as conjectured, see \cite{Ro}. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval [\pi/(p\sin(\pi/p)),\pi/(p\sin(\pi/p)\cos(\pi/p))], which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as p\to\infty. The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials P_n and those of the second kind Q_n satisfy P_{2n}^2(0)\sim c_1n^{-1/\b} and Q_{2n-1}^2(0)\sim c_2 n^{-1/\a}, where 0<\a,\b<1 can be different, and c_1,c_2 are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of \a,\b. Here \alpha_n\sim \beta_n means that \alpha_n/\beta_n\to 1. This also leads to a new proof of Romanov's Theorem that the order is 1/p.