  For an indeterminate moment problem we denote the orthonormal polynomials by
P_n. We study the relation between the growth of the function
P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the
sequence (P_n(z)). Under certain assumptions on the recurrence coefficients
from the three term recurrence relation
zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P
is of order \alpha with 0<\alpha<1, if and only if the sequence (P_n(z)) is
absolutely summable to any power greater than 2\alpha. Furthermore, the order
\alpha is equal to the exponent of convergence of the sequence (b_n).
  Similar results are obtained for logarithmic order and for more general types
of slow growth. To prove these results we introduce a concept of an order
function and its dual.
  We also relate the order of P with the order of certain entire functions
defined in terms of the moments or the leading coefficient of P_n
