On the order of indeterminate moment problems

For an indeterminate moment problem we denote the orthonormal polynomials by Pn. We study the relation between the growth of the function P(z)=(∑n=0∞|Pn(z)|2)1/2 and summability properties of the sequence (Pn(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zPn(z)=bnPn+1(z)+anPn(z)+bn−1Pn−1(z), we show that the function P is of order α with 0<α<1, if and only if the sequence (Pn(z)) is absolutely summable to any power greater than 2α. Furthermore, the order α is equal to the exponent of convergence of the sequence (bn). 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 Pn.