prod1[NN_,k_,i_]:=Product[k*i-c,{c,0,NN-1}]

d[NN_,k_,l_]:=Sum[(((-1))^i)*prod1[NN,k,i]*(l!)/(i!)/((l-i)!),{i,1,l}]

c2[NN1_,NN2_,l_,m_,t_,nu1_,nu2_,s_,k_]:=((((-1))^(NN1+NN2+t))*(E^(s/k))*(nu1^l)*(nu2^m)*d[NN1,k,l]*d[NN2,k,m])/(t!(NN1+NN2-1-t)!*l!*m!*(s^(l+m-t/k))*(k^(1+t/k)))

crossmoment[NN1_,NN2_,nu1_,nu2_,s_,k_]:=Sum[c2[NN1,NN2,l,m,t,nu1,nu2,s,k]*Gamma[l+m-t/k,s/k],{l,1,NN1},{m,1,NN2},{t,0,NN1+NN2-1}]

NN1=1;NN2=1;nu1=1;nu2=2;s=5;k=1/5;

result=crossmoment[NN1,NN2,nu1,nu2,s,k];

FullSimplify[result]

N[result,10]