REMARKS ON HYPERGEOMETRIC CAUCHY NUMBERS.

For a positive integer N, hypergeometric Cauchy numbers cN,n are defined by ... where ₂F₁(a,b;c;z) is the Gauss hypergeometric function. When N = 1, cn = c1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers (see (4), (6) and (3) in the text). Hypergeometric numbers can be recognized as one of the most natural extensions of the classical Cauchy numbers in terms of determinants (see Section 2), though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In addition, there are some relations between the hypergeometric Cauchy numbers and the classical Cauchy numbers. In this paper, we give the determinant expressions of hypergeometric Cauchy numbers and their generalizations, and show some interesting expressions of hypergeometric Cauchy numbers. As applications, we can get the inversion relations such that hypergeometric Cauchy numbers as cN,n/n! and the numbers N/(N + n) are interchanged in terms of determinants of the so-called Hassenberg matrices. [ABSTRACT FROM AUTHOR]