@MISC{_b-adicnumbers, author = {}, title = {b-ADIC NUMBERS IN PASCAL'S TRIANGLE MODULO b}, year = {} }

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Abstract

h31 Form the product k 1 '(n- kY by using this formula to expand k l and (n- k) J '. Sum both sides and we get (8-3) S(iJ;n}- ^t^^^jt^^tBAmAn- k), r=0 ' d s=0 k = 0 which brings in a convolution of Bernoulli polynomials. Since the Bernoulli polynomials may be expressed in terms of Bernoulli numbers by the further formula n (8.4) BAx) = X) {l) xn " " B ^ m = Q it would be possible to secure a convolution of the Bernoulli numbers. However, the author has not reduced this to any interesting or useful formula that appears to offer any advantages over those wev have derived here or those in [9]. We leave this as a project for the reader. It is also possible to obtain a mixed formula by proceeding first as in [9] to get 7(i,j;n) = £(-1) ' ( ^ J '~l> i+r> v = 0 ' k = 0 apply one of our Stirling number expansions to the inner sum and get, e.g., J (8.5) S(i,j;n) = £ ('V ( ^""'"'E (l X ± 1} klS(i +r> k)> V=0 fc=0 but the writer sees no remarkable advantages to be gained. i + r