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Thread: Problem with a proof

  1. #1
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    Problem with a proof

    I'm having alot of trouble with the proof below. The parts I don't understand will be bolded, but I have written the whole proof so you can also follow the argument.

    Let $\displaystyle r,s$ be non negative intgers. If $\displaystyle r>s$, then

    a) $\displaystyle \int^1_0\int^1_0 \frac{x^ry^s}{1-xy}dxdy$ is a rational number whose denominator is a divisor of $\displaystyle d_r^3$

    b) $\displaystyle \int^1_0\int^1_0 -\frac{\log(xy)}{1-xy}x^ry^sdxdy$ is a rational number whose denominator is a divisor of $\displaystyle d_r^3$

    Where $\displaystyle d_r$ is the lowest common multiple of $\displaystyle 1,2, \ldots r$.

    The proof is given as follows:

    Let $\displaystyle \sigma$ be any non-negative number. Consider the integral

    $\displaystyle \int^1_0\int^1_0 \frac{x^{r+\sigma}y^{s+\sigma}}{1-xy}dxdy$ (1)

    Develop $\displaystyle (1-xy)^{-1}$ into a geometrical series and perform the double integration. Then we obtain

    $\displaystyle \sum^{\infty}_{k=0} \frac{1}{(k+r+\sigma + 1)(k + s + \sigma +1)}$ (2)

    Assume that $\displaystyle r>s$. Then we can write this sum as

    $\displaystyle \sum^{\infty}_{k=0} \frac{1}{r-s}\{\frac{1}{k+s+\sigma +1}-\frac{1}{k+r+\sigma+1}\} = \frac{1}{r-s}\{\frac{1}{s+\sigma +1}+ \cdots + \frac{1}{r+\sigma}\} $

    If we put $\displaystyle \sigma = 0$ then assertion a) follows immediately. If we differentiate with respect to $\displaystyle \sigma$ and put $\displaystyle \sigma = 0$, then integral (1) changes into

    $\displaystyle \int^1_0\int^1_0 \frac{\log(xy)}{1-xy}x^ry^sdxdy$

    and summation (2) becomes

    $\displaystyle \frac{-1}{r-s}\{\frac{1}{(s+1)^2} + \ldots + \frac{1}{r^2}\}$

    and assertion (b) now follows straight away.

    Thank you thank you thank you to anyone who can explain those bolded sentences, I will be forever indebted to you!

    Havelock (formerly HTale)




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  2. #2
    MHF Contributor

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    Quote Originally Posted by Havelock View Post

    Let $\displaystyle r,s$ be non negative intgers. If $\displaystyle r>s$, then

    a) $\displaystyle \int^1_0\int^1_0 \frac{x^ry^s}{1-xy}dxdy$ is a rational number whose denominator is a divisor of $\displaystyle \color{red}d_r^3$.

    here $\displaystyle d_r^3$ should be $\displaystyle d_r^2.$



    Let $\displaystyle \sigma$ be any non-negative number. Consider the integral

    $\displaystyle \int^1_0\int^1_0 \frac{x^{r+\sigma}y^{s+\sigma}}{1-xy}dxdy$ (1)


    Develop $\displaystyle (1-xy)^{-1}$ into a geometrical series and perform the double integration. Then we obtain

    $\displaystyle \sum^{\infty}_{k=0} \frac{1}{(k+r+\sigma + 1)(k + s + \sigma +1)}$ (2)
    we have $\displaystyle \frac{1}{1-xy}=\sum_{k=0}^{\infty}x^ky^k.$ put this in (1) and integrate the sum term by term to get (2).


    Assume that $\displaystyle r>s$. Then we can write this sum as


    $\displaystyle \sum^{\infty}_{k=0} \frac{1}{r-s}\{\frac{1}{k+s+\sigma +1}-\frac{1}{k+r+\sigma+1}\} = \frac{1}{r-s}\{\frac{1}{s+\sigma +1}+ \cdots + \frac{1}{r+\sigma}\} $

    If we put $\displaystyle \sigma = 0$ then assertion a) follows immediately.
    if we put $\displaystyle \sigma=0,$ we'll get: $\displaystyle \int^1_0\int^1_0 \frac{x^ry^s}{1-xy}dxdy=\frac{1}{r-s}(\frac{1}{s+1} + \cdots + \frac{1}{r})=\frac{a}{(r-s)b},$ where $\displaystyle b=\text{lcm}\{s+1, \cdots, r \}.$ now since $\displaystyle \{s+1, \cdots, r \} \subseteq \{1, \cdots, r \},$ and $\displaystyle d_r=\text{lcm}\{1,2, \cdots, r \},$

    we have $\displaystyle b \mid d_r.$ also since $\displaystyle r-s \leq r,$ we have $\displaystyle r - s \mid d_r.$ thus $\displaystyle (r-s)b \mid d_r^2.$



    If we differentiate with respect to $\displaystyle \sigma$ and put $\displaystyle \sigma = 0$, then integral (1) changes into

    $\displaystyle \int^1_0\int^1_0 \frac{\log(xy)}{1-xy}x^ry^sdxdy$

    and summation (2) becomes

    $\displaystyle \frac{-1}{r-s}\{\frac{1}{(s+1)^2} + \ldots + \frac{1}{r^2}\}$

    and assertion (b) now follows straight away.

    exactly the same argument as above but this time $\displaystyle \text{lcm}\{(s+1)^2 , \cdots , r^2 \} \mid d_r^2$ and $\displaystyle r-s \mid d_r.$ thus the denominator will divide $\displaystyle d_r^3.$
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