Congruences and Recurrences for Bernoulli Numbers of Higher Order
نویسنده
چکیده
In particular, B^\0) = B^\ the Bernoulli number of order k, and BJp = Bn, the ordinary Bernoulli number. Note also that B^ = 0 for n > 0. The polynomials B^\z) and the numbers B^ were first defined and studied by Niels Norlund in the 1920s; later they were the subject of many papers by L. Carlitz and others. For the past twenty-five years not much has been done with them, although recently the writer found an application for B^ involving congruences for Stirling numbers (see [8]). For the writer, the higher-order Bernoulli polynomials and numbers are still of interest, and they are worthy of further investigation. Apparently, not much is known about the divisibility properties of B^ for general k. Carlitz [2] proved that if/? is prime and
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On a Multidimensional Volkenborn Integral and Higher Order Bernoulli Numbers
In particular, the values at x = 0 are called Bernoulli numbers of order k, that is, Bn (0) = Bn k) (see [1, 2, 4, 5, 9, 10, 14]). When k = 1, the polynomials or numbers are called ordinary. The polynomials Bn (x) and numbers Bn were first defined and studied by Norlund [9]. Also Carlitz [2] and others investigated their properties. Recently they have been studied by Adelberg [1], Howard [5], a...
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