[Axiom-developer] [Guessing formulas for sequences]

[kratt6]

Changes 
http://page.axiom-developer.org/zope/mathaction/GuessingFormulasForSequences/diff
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??changed:
-\Rate, Doron Zeilberger's program 'GuessRat' and the relevant parts of Bruno
'Rate', Doron Zeilberger's program 'GuessRat' and the relevant parts of Bruno

--removed:
-

??changed:
-  $f(n)-f(n-1)$, and
    $f(n)-f(n-1)$, and

??changed:
-    Guessing formulas for sequences of rational numbers
-
-      For example, if we suspect that a sequence of integers or rationals like
  Guessing formulas for sequences of rational numbers

    For example, if we suspect that a sequence of integers or rationals like

??changed:
-  guessPade(n, [1, 1, 2, 3, 5], n+->n)\$GuessInteger
      guessPade(n, [1, 1, 2, 3, 5], n+->n)\$GuessInteger

??changed:
-    Guessing formulas for sequences of rational functions
-
-      Most of the previous section still applies, the only change being that 
we now
  Guessing formulas for sequences of rational functions

    Most of the previous section still applies, the only change being that we 
now

??changed:
-  guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat], 
-        [guessSum, guessProduct])\$GuessPolynomial
      guess(n, [1, 1+q, 1+q+q^2, 1+q+q^2+q^3], n+->n, [guessRat], 
            [guessSum, guessProduct])\$GuessPolynomial

??changed:
-Sequence (4)::
Sequence (4) ::

??changed:
-    Some remarks
-    
-      - All of the presented guessing algorithms are insensitive to the shift
-        operator. I.e., if a formula for the sequence $(s_1,s_2,\dots,s_m)$ 
can be
-        guessed, then the corresponding formula for $(s_2,s_3,\dots,s_{m+1})$ 
will
-        be found, too.
-        
-      - Except of the class of functions covered by 'guessExpRat', all are
-        closed under addition. I.e., if formulas for $(s_1,s_2,\dots,s_m)$ and
-        $(t_1,t_2,\dots,t_m)$ can be guessed, then also for
-        $(s_1+t_1,s_2+t_2,\dots,s_m+t_m)$. However, the class of functions of
-        type (5) is not even closed under addition of a constant! On the other
-        hand, all classes are closed under multiplication.
-        
-      - Note that the class of functions covered by 'guessPRec', i.e., the
-        class of $D$-finite functions, is closed under the operator $\Delta_n$.
-        Thus, it does not make to try to guess a function for some sequence $s$
-        with::
-
-[9 more lines...]
  Some remarks
    
    - All of the presented guessing algorithms are insensitive to the shift
      operator. I.e., if a formula for the sequence $(s_1,s_2,\dots,s_m)$ can be
      guessed, then the corresponding formula for $(s_2,s_3,\dots,s_{m+1})$ will
      be found, too.
        
    - Except of the class of functions covered by 'guessExpRat', all are
      closed under addition. I.e., if formulas for $(s_1,s_2,\dots,s_m)$ and
      $(t_1,t_2,\dots,t_m)$ can be guessed, then also for
      $(s_1+t_1,s_2+t_2,\dots,s_m+t_m)$. However, the class of functions of
      type (5) is not even closed under addition of a constant! On the other
      hand, all classes are closed under multiplication.
        
    - Note that the class of functions covered by 'guessPRec', i.e., the
      class of $D$-finite functions, is closed under the operator $\Delta_n$.
      Thus, it does not make sense to try to guess a function for some 
      sequence $s$ with::

        guess(n, s, n+->n, [guessPRec], [guessSum]).

    - The situation is very different, if the operator 'guessProduct' is 
specified, too. The 
      class of functions covered by::

        guess(n, s, n+->n, [guessPRec], [guessSum, guessProduct])

    - is bigger than the class of functions covered by::

        guess(n, s, n+->n, [guessPRec], [guessProduct])!

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forwarded from http://page.axiom-developer.org/zope/mathaction/address@hidden