"Notes on van der Poorten's proof of the Hadamard Quotient Theorem: Part I" Robert Rumely (~1985?) Hairy stuff! And presumably still unpublished. The journal shelves groan under a mass of mediocrity, while worthwhile gems languish as yellowing typescripts. Not doubt 'twas ever thus. Did Part II ever come to fruition, I wonder? A search for "dominant root method" yielded numerous hits, including Umberto Zannier (2000) "A proof of Pisot’s d-th root conjecture" Annals of Mathematics, 151 (2000), 375–383 http://arxiv.org/pdf/math/0010024.pdf citing further references. WFL On 7/11/13, Victor Miller <victorsmiller@gmail.com> wrote:
Here's a paper by Robert Rumely giving a proof of a result of Alf van der Poorten which characterizes quotients.
Victor
http://www.math.uga.edu/~rr/HadamardQuotient1.pdf
On Wed, Jul 10, 2013 at 5:53 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Difficult to know what at level to answer this question ...
All these sequences satisfy homogeneous linear recurrences with constant coefficients, about which there is a considerable body of classical theory.
Try a web search on "linear recurring sequence" or "linear feedback shift-register sequence"; or see http://en.wikipedia.org/wiki/Recurrence_relation
The term-by-term sum and product of such LFSR sequences are also LFSR sequences, but not in general the quotient. BIll Gosper recently asked for an algorithm to detect the quotient of LFSR sequences. I have been waiting for sombody like Neil Sloane or Simon Plouffe to wade in, but in vain ... maybe there's a trade secret to protect?
Or maybe they just can't think of anything intelligent to say about it. Like me.
WFL
On 7/10/13, Stuart Anderson <stuart.errol.anderson@gmail.com> wrote:
If Fibonacci had used more realistic assumptions (ie his rabbits dying) he would have produced the Padovan (or Perrin?) sequence.
http://matheminutes.blogspot.com.au/2012/02/killing-fibonaccis-rabbits.html
Another example of this sort of thing; summing A000079 (powers of 2), and subtracting previous terms 2 generations back gives A003945
Is there a general form for such sequences?
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