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Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

### Abstract

We investigate the asymptotic behavior of the Selberg-like integral as *N* → ∞ for different scalings of the parameters *a* and *b* with *N*. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting *N* channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.

© 2010 American Institute of Physics

Received 02 April 2010
Accepted 19 October 2010
Published online 21 December 2010

Acknowledgments:
The authors are grateful to M. Novaes for fruitful discussions on the factorization conjecture and to C. Krattenthaler for his very interesting comments about hypergeometric functions. This paper is partially supported by the ANR project PhysComb, ANR-08-BLAN- 0243-04.

Article outline:

I. INTRODUCTION
II. SOME SELBERG-LIKE INTEGRALS
A. Schur functions
B. Selberg-like integrals with a power sum in the integrand
III. INVERSE BINOMIAL TRANSFORM
A. Inverse binomial transform and Newton's interpolation formula
B. An example of generalized (inverse) binomial transform
C. Leading coefficients
IV. ASYMPTOTIC BEHAVIOR OF
A. Convergence
B. Computation of the limit
V. SOME SPECIAL CASES RELATED TO COMBINATORICS
A. Simplest cases
B. Central binomial coefficients and the specialization
C. Catalan triangle and the specialization
D. Symmetric Dyck paths counted by number of peaks and the specialization
VI. CONCLUSION