  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.

Classical Analysis and ODEs