Analogues of the binomial coefficient theorems of Gauss and Jacobi

作者：Al Shaghay Abdullah^*; Dilcher Karl

来源：International Journal of Number Theory, 2016, 12(8): 2125-2145.

DOI：10.1142/S179304211650127X

• Binomial coefficients
• factorials
• congruences
• Jacobi sums
• p-adic gamma function
• Catalan numbers

The theorems of Gauss and Jacobi that give modulo p evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo p(2). In this paper, we further extend these results to congruences modulo p(3). In the process, we prove congruences to arbitrarily high powers of p for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's p-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

• 出版日期2016-12