On 8/3/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Now B(n) is already known to be periodic mod p^t for n >= 0; with the extension, the periodicity appears to extend to all n. Using the fact that S(n, n-k) (extended) is polynomial in n of degree 2*k, B(n) can be extended to a continuous function of a continuous variable; p-adically this function is almost-periodic.
A nontrivial correction this time: although each S(n, n-k) is polynomial, n of them are added together --- so the inference that B(x) might be a continuous function of p-adic x falls through. In fact if it were, we should have the p-adically convergent Mahler (Newton?) expansion B(x) = \sum_{n >= 0} \Delta^n B(0) * (x_C_n), where \Delta denotes forward difference, and x_C_n binomial coefficient [don't rely on this last for negative n with Maple, btw!] However, the n-th differences \Delta^n B(0) turn out also to be apparently periodic mod p, and therefore cannot approach zero. So there is no such continuous p-adic B(x) after all. Incidentally, \Delta^n B(0) = (B - 1)^n umbrally = 1,0,1,1,4,11,41,162,715, ... turns out to be well-known [OEIS A000296]; simply (B + 1)^n = B(n+1) of course; and (B + 2)^n = 1,3,10,37,151,674,3263, ... also appears as [OEIS A005493]. The connection with Bell numbers is not explicitly mentioned in OEIS --- are you there, NJAS? --- but as its server is at present failing to respond, I cannot check whether this might be implicit in the algorithms given. Since by an extension of Touchard's recurrence we have B(p^(kn)) = (B+k)^n mod p, there is an obvious motive for studying the numbers (B+k)^n for any integer k. Maybe I'll get around to discussing these generalised Bell numbers some other time ... You have been warned! Fred Lunnon