1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hermite identity help

  1. Jun 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Let a and b be integers and m an integer >1 Evaluate

    [b/m] + [(b+a)/m]+ [(b+2a)/m]+ [(b+3a)/m]+ [(b+4a)/m]+ [(b+5a)/m]+.....+ [(b+(m-1)a)/m]

    2. Relevant equations



    3. The attempt at a solution
    i tried to use hermite identity.

    [x] + [x + 1/n] + [x + 2/n] +...+ [x + (n-1)/n] = [nx]

    assuming x = b/m and 1/n = a/m.. but a/m is not an integer so i cant use it. I m stuck what to do?
     
  2. jcsd
  3. Jun 17, 2013 #2

    lurflurf

    User Avatar
    Homework Helper

    Funny you should mention that, just last week I read about it in pp 90-94 of Concrete Mathematics by Ronald L. Graham , Donald E. Knuth, and Oren Patashnik. Several interesting things are the dependence of the result on gcd(a,m) ,the fact that

    $$\sum_{0 \le k < m} \left[ \frac{b+k \, a}{m}\right] = \sum_{0 \le k < a} \left[ \frac{b+k \, m}{a}\right], \, \, \, \, \, \, \, \, \, \mathop{integers \, \, \, a,m>0} $$

    the closely related integrals

    $$\frac{1}{m} \int_0^m (a \, x+b) \! \mathop{dx}=\frac{1}{a} \int_0^a (m \, x+b) \! \mathop{dx}=\frac{a \, m}{2}+b$$

    The book uses several special cases to deduce the general one. Give it another try. If you have trouble describe the methods you tried and those you have know.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Hermite identity help
Loading...