Calculation of some determinants using the s-shifted factorial

Details Calculation of some determinants using the s-shifted factorial Calculation of some determinants using the s-shifted factorial

2004-01-05

arxiv.org/abs/math-ph/0401006v2

J. Phys. A: Math. Gen. 37 (2004) 5737-5762

Physics

Mathematical Physics

25 pages; added section 5 for some examples of applications

Scientific paper

10.1088/0305-4470/37/22/003

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.

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