The best tool for trying to deal with such integrals is Fredéric Chyzak's MGfun package (available as part of the Algolib library).
For your example, you should get a system of differential equations (for the integrand) for $a,b,c$ and $x$; you can leave $\nu$ as a parameter, or get a (mixed) difference equation for it. Then using this package, you can try to do elimination, which will give you a new system for the answer.
Note that this method is a vast generalization of using the Meijer G-function as an intermediary (since MeijerG is the most general function whose series expansion / asymptotic expansion at 0 has coefficients satisfy a recurrence of order 1).
Maple finds the integral under consideration for concrete values of $n$ and $k$, producing huge outputs. For example, $$restart; with(orthopoly):VectorCalculus:-int(L(2, alpha, x)^3*L(3, alpha, x)*x^{alpha-delta}*exp(-x), x = 0 .. infinity) assuming\, alpha > delta-1 $$ gives $$-1/48\,\Gamma \left( 4+\alpha-\delta \right) \left( {\delta}^{6}+12
\,\alpha\,{\delta}^{4}-27\,{\delta}^{5}+30\,{\alpha}^{2}{\delta}^{2}-
208\,\alpha\,{\delta}^{3}+319\,{\delta}^{4}+8\,{\alpha}^{3}-246\,{
\alpha}^{2}\delta+1422\,\alpha\,{\delta}^{2}-2081\,{\delta}^{3}+532\,{
\alpha}^{2}-4466\,\alpha\,\delta+7828\,{\delta}^{2}+5380\,\alpha-15976
\,\delta+13736 \right) +$$ $$1/48\,\Gamma \left( \alpha-\delta+1 \right)
\left( \alpha+3 \right) \left( {\alpha}^{2}{\delta}^{6}-3\,\alpha\,{
\delta}^{7}+3\,{\delta}^{8}+12\,{\alpha}^{3}{\delta}^{4}-63\,{\alpha}^
{2}{\delta}^{5}+120\,\alpha\,{\delta}^{6}-66\,{\delta}^{7}+30\,{\alpha
}^{4}{\delta}^{2}-298\,{\alpha}^{3}{\delta}^{3}+1069\,{\alpha}^{2}{
\delta}^{4}-1464\,\alpha\,{\delta}^{5}+698\,{\delta}^{6}+8\,{\alpha}^{
5}-270\,{\alpha}^{4}\delta+2274\,{\alpha}^{3}{\delta}^{2}-7067\,{
\alpha}^{2}{\delta}^{3}+9342\,\alpha\,{\delta}^{4}-4356\,{\delta}^{5}+
556\,{\alpha}^{4}-6356\,{\alpha}^{3}\delta+23458\,{\alpha}^{2}{\delta}
^{2}-34181\,\alpha\,{\delta}^{3}+17171\,{\delta}^{4}+6176\,{\alpha}^{3
}-38038\,{\alpha}^{2}\delta+72258\,\alpha\,{\delta}^{2}-42898\,{\delta
}^{3}+23980\,{\alpha}^{2}-81016\,\alpha\,\delta+65264\,{\delta}^{2}+
36800\,\alpha-54248\,\delta+18448 \right)
.$$
Best Answer
I think that http://arxiv.org/pdf/math-ph/0409066v1 (Multivariate Orthogonal Polynomials (symbolically) page 15, has the representation you are looking for [whether it will help you compute your sum, I am not sure, but maybe their Maple package will do the thinking for you?