<s>[INST] For what $a$ and $b$ are there explicit expressions for $I(a, b) =\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b} $? For what $a$ and $b$
are there explicit expressions for
$I(a, b)
=\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b}
$?
This is inspired by
the answer to
https://www.quora.com/What-is-displaystyle-int_-0-1-int_-0-1-frac-1-1-xy-3-mathrm-d-x-mathrm-d-y
where it is shown that
$I(1, 3)
=\dfrac34\left(\ln(3)+\dfrac{\pi\sqrt{3}}{9}\right)
$.
Here's what I've done
so far.
$\begin{array}\\
I(a, b)
&=\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b}\\
&=\int_0^1 \int_0^1 dx\,dy\sum_{n=0}^{\infty} (x^ay^b)^n\\
&=\sum_{n=0}^{\infty} \int_0^1 \int_0^1 dx\,dy(x^ay^b)^n\\
&=\sum_{n=0}^{\infty} \int_0^1 x^{an}dx \int_0^1 y^{bn}dy\\
&=\sum_{n=0}^{\infty} \dfrac{x^{an+1}}{an+1}\big|_0^1 \dfrac{y^{bn+1}}{bn+1}\big|_0^1\\
&=\sum_{n=0}^{\infty} \dfrac{1}{an+1} \dfrac{1}{bn+1}\\
&=ab\sum_{n=0}^{\infty} \dfrac{1}{abn+b} \dfrac{1}{abn+a}\\
\\
I(a.a)
&=\sum_{n=0}^{\infty} \dfrac{1}{(an+1)^2}\\
&=\dfrac1{a^2}\sum_{n=0}^{\infty} \dfrac{1}{(n+1/a)^2}\\
&=\dfrac1{a^2}\psi^{(1)}(1/a)\\
\text{If } a \ne b\\
I(a, b)
&=\dfrac{ab}{a-b}\sum_{n=0}^{\infty} \left(\dfrac{1}{abn+b} -\dfrac{1}{abn+a}\right)\\
&=\dfrac{ab}{a-b}\sum_{n=0}^{\infty} \int_0^1 \left(x^{abn+b-1} -x^{abn+a-1}\right)dx\\
&=\dfrac{ab}{a-b} \int_0^1 \left(\sum_{n=0}^{\infty}x^{abn+b-1} -\sum_{n=0}^{\infty}x^{abn+a-1}\right)dx\\
&=\dfrac{ab}{a-b} \int_0^1 \left(x^{b-1}\sum_{n=0}^{\infty}x^{abn} -x^{a-1}\sum_{n=0}^{\infty}x^{abn}\right)dx\\
&=\dfrac{ab}{a-b} \int_0^1 \left(x^{b-1}\dfrac1{1-x^{ab}} -x^{a-1}\dfrac1{1-x^{ab}}\right)dx\\
&=\dfrac{ab}{a-b} \int_0^1 \left(\dfrac{x^{b-1} -x^{a-1}}{1-x^{ab}}\right)dx\\
&=\dfrac{ab}{a-b} \int_0^1 x^{b-1}\left(\dfrac{1 -x^{a-b}}{1-x^{ab}}\right)dx\\
\end{array}
$
But,
in general,
I can't go further.
Special cases can be handled.
For example,
if $b=1$
and $a$ is a positive integer,
we can show that
$I(a, 1)
=\dfrac{a}{(a-1)^2} \left(\ln(a)-\int_0^1 \left(\dfrac{\sum_{k=0}^{a-3}(a-2-k)x^k}{\sum_{k=0}^{a-1}x^k}\right)dx\right)
$.
This gives,
as above,
$I(3, 1)
=\dfrac34\left(\ln(3)-\int_0^1 \left(\dfrac{1}{1+x+x^2}\right)dx\right)
$
and completing the square
gives the answer above.
Also
$I(2, 1)
=2\ln(2)
$.
Similarly,
$\begin{array}\\
I(4, 1)
&=\dfrac49\left(\ln(4)-\int_0^1 \left(\dfrac{2+x}{1+x+x^2+x^3}\right)dx\right)\\
&=\dfrac49\left(\ln(4)- \dfrac{3 π + \ln(4)}{8}\right)
\quad\text{(According to Wolfy)}\\
&=\dfrac49\left(\dfrac{7\ln(4)-3 π}{8}\right)\\
\end{array}
$
We could get
$I(5, 1)$
since Wolfy gives
$$\dfrac1{1+x+x^2+x^3+x^4}
=\dfrac{-2 x + \sqrt{5} - 1}{\sqrt{5} (2 x^2 - (\sqrt{5}-1) x + 2)} + \dfrac{2 x + \sqrt{5} + 1}{\sqrt{5} (2 x^2 + (\sqrt{5}+1) x + 2)}
$$
but I'm not going to bother
to work it out.
 [/INST] The following is trivial due to expansion of digamma (formula 14 of this page) $$\sum _{n=0}^{\infty } \frac{1}{(a+n) (b+n)}=\frac{\psi ^{(0)}(a)-\psi ^{(0)}(b)}{a-b}$$ Thus OP's integral equals to $$I(a,b)=\sum _{n=0}^{\infty } \frac{a b}{(a b n+b) (a b n+a)}=\frac{\psi ^{(0)}\left(\frac{1}{b}\right)-\psi ^{(0)}\left(\frac{1}{a}\right)}{a-b}$$ Due to Gauss's digamma theorem (formula 11 of link above) for all $a,b\in \mathbb Q$ the integral can be expressed in terms of log and trig functions (even square roots, when $a,b$ are small). For instance $$I(3,5)=\frac{1}{2} \pi  \sqrt{\frac{1}{6} \left(-\sqrt{3+\frac{6}{\sqrt{5}}}+\frac{3}{\sqrt{5}}+2\right)}+\frac{1}{8} \log \left(\frac{3125}{729}\right)+\frac{1}{4} \sqrt{5} \coth ^{-1}\left(\sqrt{5}\right)$$
</s>