tailieunhanh - Báo cáo toán học: "Determinant expressions for q-harmonic congruences and degenerate Bernoulli numbers"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Determinant expressions for q-harmonic congruences and degenerate Bernoulli numbers. | Determinant expressions for q-harmonic congruences and degenerate Bernoulli numbers Karl Dilcher Department of Mathematics and Statistics Dalhousie University Halifax NS B3H 3J5 Canada dilcher@ Submitted Feb 16 2008 Accepted Apr 17 2008 Published Apr 28 2008 Mathematics Subject Classification 11B65 11B68 Abstract The generalized harmonic numbers H k P 1 j k satisfy the well-known congruence Htt 0 mod p for all primes p 3 and integers k 1. We derive TT k q-analogs of this congruence for two different q-analogs of the sum Hn . The results can be written in terms of certain determinants of binomial coefficients which have interesting properties in their own right. Furthermore it is shown that one of the classes of determinants is closely related to degenerate Bernoulli numbers and new properties of these numbers are obtained as a consequence. 1 Introduction The harmonic numbers Hn are defined by H X 1 n 0 j j i J where by convention H0 0. These numbers have been studied extensively see . 10 p. 272 ff. and they have important applications in combinatorics number theory and the analysis of algorithms. The harmonic numbers have also been generalized in various different ways for a recent summary of generalizations see 5 . In this paper we will be concerned with the generalized harmonic numbers defined by JU 1 n 0 n z y j k Supported in part by the Natural Sciences and Engineering Research Council of Canada THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R63 1 where we restrict our attention to positive integer parameters k. Obviously Hn1 Hn and limn 1 ll n k for k 2 where k is the Riemann zeta function. See . 10 for some further properties. Many special functions sequences and identities have interesting and meaningful q-analogs. For a general discussion of q-series q-analogs and their importance in combinatorics analysis number theory and other areas see . 2 Ch. 10-12 . A q-analog of Hn is given by the q-harmonic numbers 1 Hn q j 2 j- n 0 q 1 .