Is there a (possibly hypergeometric-type) explicit evaluation of the continued fraction $$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$ Even the special case $d=0$, $a=1$ would be interesting. Note that $$\atanh(z)=\dfrac{z}{1-\dfrac{1^2z^2}{3-\dfrac{2^2z^2}{5-\ddots}}}$$ but that does not seem to help.

Is there a (possibly hypergeometric-type) explicit evaluation of the continued fraction $$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$ Even the special case $d=0$, $a=1$ would be interesting. Note that $$\tanh^{-1}(z)=\dfrac{z}{1-\dfrac{1^2z^2}{3-\dfrac{2^2z^2}{5-\ddots}}}$$ but that does not seem to help.