3,510 research outputs found
Integral representation of some functions related to the Gamma function
We prove that the functions Phi(x)=[Gamma(x+1)]^{1/x}(1+1/x)^x/x and log
Phi(x) are Stieltjes transforms
Fibonacci numbers and orthogonal polynomials
We prove that the sequence of reciprocals of the
Fibonacci numbers is a moment sequence of a certain discrete probability, and
we identify the orthogonal polynomials as little -Jacobi polynomials with
. We prove that the corresponding kernel
polynomials have integer coefficients, and from this we deduce that the inverse
of the corresponding Hankel matrices have integer entries. We
prove analogous results for the Hilbert matrices.Comment: A note dated June 2007 has been added with some historical comments.
Some references have been added and complete
On powers of Stieltjes moment sequences, II
We consider the set of Stieltjes moment sequences, for which every positive
power is again a Stieltjes moment sequence, we and prove an integral
representation of the logarithm of the moment sequence in analogy to the
L\'evy-Khinchin representation. We use the result to construct product
convolution semigroups with moments of all orders and to calculate their Mellin
transforms. As an application we construct a positive generating function for
the orthonormal Hermite polynomials.Comment: preprint, 21 page
Closable Hankel operators and moment problems
In a paper from 2016 D. R. Yafaev considers Hankel operators associated with
Hamburger moment sequences q_n and claims that the corresponding Hankel form is
closable if and only if the moment sequence tends to 0. The claim is not
correct, since we prove closability for any indeterminate moment sequence but
also for certain determinate moment sequences corresponding to measures with
finite index of determinacy. It is also established that Yafaev's result holds
if the moments satisfy \root{2n}\of{q_{2n}}=o(n).Comment: 10 pages. The notation for the closure of an operator A is changed to
\overline{A} from \o A on pages 7,
On a fixed point in the metric space of normalized Hausdorff moment sequences
We show that the transformation (x_n)_{n\ge 1}\to (1/(1+x_1+...+x_n))_{n\ge
1} of the compact set of sequences (x_n)_{n\ge 1} of numbers from the unit
interval [0,1] has a unique fixed point, which is attractive. The fixed point
turns out to be a Hausdorff moment sequence studied in papers by Berg and
Dur\'an in 2008
A determinant characterization of moment sequences with finitely many mass-points
To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of
Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the
corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy
D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive
semi-definite, and in particular (s_n) is the sequence of moments of a discrete
measure concentrated in n_0 points on the real line. We stress that the
conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of
the Hankel matrices.Comment: 8 page
On the order of indeterminate moment problems
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_nComment: 45 pages. To appear in Adv. Mat
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