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Poly family

  1. Jun 8, 2012 #1
    [itex]x^2+2\\\\
    \frac{2}{3} x^3 + \frac{13}{3} x\\\\
    \frac{1}{3} x^4 + \frac{14}{3} x^2 + 2\\ \\
    \frac{2}{15} x^5 + \frac{10}{3} x^3 + \frac{83}{15} x\\ \\
    \frac{2}{45} x^6 + \frac{16}{9} x^4 + \frac{323}{45} x^2 + 2\\\\
    \dots[/itex]
    ?
     
  2. jcsd
  3. Jun 8, 2012 #2
    What is your question? We can't just guess.
     
  4. Jun 8, 2012 #3
    But please guess :-) how to continue?
     
  5. Jun 8, 2012 #4
    How are these polynomials supposed to be related?
     
  6. Jun 8, 2012 #5

    micromass

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    Apparently, the OP is not a native english speaker and can't describe the problem well.

    I guess that he encountered these polynomials while working and he is asking if somebody recognizes them or sees an easy pattern in them.
     
    Last edited: Jun 8, 2012
  7. Jun 10, 2012 #6
    Thank you Micromass
     
  8. Jun 10, 2012 #7

    haruspex

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    Clearly the constant term alternates between 0 and 2. At the other end, you can get the leading term from the preceding line by multiplying by 2x/n. So the leading coefficient is 2n-1/n! So a natural thing to try is:
    - multiply each line by n! (starting with n=2 in the first line) to form the poly sequence Pn(x)
    - form a new sequence from this according to Qn(x) = Pn(x) - 2x*Pn-1(x)
    The coefficients that result look a little friendlier. The highest prime that occurs in this sample is 19, a lot better than 83.
     
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