## On generating functions of hausdorff moment sequences

Publication
, Journal Article

Liu, JG; Pego, RL

Published in: Transactions of the American Mathematical Society

January 1, 2016

The class of generating functions for completely monotone sequences (moments of finite positive measures on [0, 1]) has an elegant characterization as the class of Pick functions analytic and positive on (−∞, 1). We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [0, 1]. Also we provide a simple analytic proof that for any real p and r with p > 0, the Fuss-Catalan or Raney numbers (Formula Presented) are the moments of a probability distribution on some interval [0, τ] if and only if p ≥ 1 and p ≥ r ≥ 0. The same statement holds for the binomial coefficients (Formula Presented).

### Duke Scholars

## Published In

Transactions of the American Mathematical Society

## DOI

## ISSN

0002-9947

## Publication Date

January 1, 2016

## Volume

368

## Issue

12

## Start / End Page

8499 / 8518

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics

### Citation

APA

Chicago

ICMJE

MLA

NLM

Liu, J. G., & Pego, R. L. (2016). On generating functions of hausdorff moment sequences.

*Transactions of the American Mathematical Society*,*368*(12), 8499–8518. https://doi.org/10.1090/tran/6618Liu, J. G., and R. L. Pego. “On generating functions of hausdorff moment sequences.”

*Transactions of the American Mathematical Society*368, no. 12 (January 1, 2016): 8499–8518. https://doi.org/10.1090/tran/6618.Liu JG, Pego RL. On generating functions of hausdorff moment sequences. Transactions of the American Mathematical Society. 2016 Jan 1;368(12):8499–518.

Liu, J. G., and R. L. Pego. “On generating functions of hausdorff moment sequences.”

*Transactions of the American Mathematical Society*, vol. 368, no. 12, Jan. 2016, pp. 8499–518.*Scopus*, doi:10.1090/tran/6618.Liu JG, Pego RL. On generating functions of hausdorff moment sequences. Transactions of the American Mathematical Society. 2016 Jan 1;368(12):8499–8518.

## Published In

Transactions of the American Mathematical Society

## DOI

## ISSN

0002-9947

## Publication Date

January 1, 2016

## Volume

368

## Issue

12

## Start / End Page

8499 / 8518

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics