$$\Large{\text{Goal:}}$$ One goal is to find better ways of expressing:
$$\sum_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b)}$$
$$\Large{\text{Special Case:}}$$ Here are some closed form special cases using the Regularized Lower aand upper Incomplete Gamma function and Lower Incomplete gamma function
$$γ(a,z)=\Gamma(a)-\Gamma(a,z)\\\text P(a,z)=\frac{γ(a,z)}{\Gamma(a)}\\Q(a,z)=\frac{\Gamma(a,z)}{\Gamma(a)}\\\text P(a,z)+Q(a,z)=1\\\sum_{n=1}^\infty \frac{γ(n+a,bn)}{n! n^{a+1}}\mathop=^{0\le b\le1}\frac{b^{a+1}}{a+1}\implies\sum_{n=1}^\infty \frac{\Gamma(n+a,bn)}{n! n^{a+1}}= -\frac{b^{a+1}}{a+1}+\sum_{n=1}^\infty \frac{(n+1)_{a-1}}{ n^{a+1}} =? $$
The $\sum\limits_{n=1}^\infty \frac{(n+1)_{a-1}}{ n^{a+1}}$ seems to be a sum of Riemann Zeta function values since you can try the following for $a\in\Bbb N$ to get a closed form of the simpler sum:
$$\sum_{n=1}^\infty \frac{\Gamma\left(n+4,\frac n2\right)}{n! n^{5}}=\frac{\pi^2}{6}+\frac{11\pi^4}{90}+6ζ(3)+6ζ(5)-\frac1{160}= 26.9781475874885318135350777164… $$
where appears the Pochhammer symbol $(A)_x$. The restriction on $b$ is loose since:
$$\sum_{n=1}^\infty \frac{γ\left(n+i,\frac n3\right)}{n!n^{i+1}}=\frac{3^{-1-i}}{1+i}= \frac{1-i}{6}(\cos(\ln(3))-i\sin(\ln(3)))=-0.072624103140189550674560360495022601835... - 0.2242349107490594690170820750267034298651... i$$
Conjectures:
$$\sum_{n=0}^\infty \frac{Q(n+1,n)}{n^3}\mathop=^? ζ(3)-\frac13$$
$$\sum_{n=0}^\infty \frac{Q(n+1,n)}{n^2}\mathop=^?\frac{\pi^2}6-\frac12$$
Please correct me and give me feedback!
$$\Large{\text{Mini Goal:}}$$
Using the above notation, I have noticed a closed form for the following $γ(a,z)=\int_0^z t^{a-1} e^{-t} dt$ series:
$$\sum\limits_{n=1}^\infty \frac{γ(n+a,bn)n^k}{(n+c)!}\\\\ \sum_{n=1}^\infty \frac{γ\left(n+2,\frac n5\right)n^3}{n!}=\frac{3982173}{10485760}\\ \sum_{n=1}^\infty \frac{γ\left(n+1,\frac n5\right)}{n!} =\frac1{32}\\ \sum_{n=1}^\infty \frac{γ\left(n,\frac n4\right)}{(n-3)!}=\frac{1717}{2916}\\ \sum_{n=1}^\infty \frac{γ\left(n,\frac n2 \right)}{n!}=\ln(2)\\ \sum_{n=1}^\infty \frac{γ\left(n,\frac n2 \right)}{n^3 n!}=\frac{119}{288} \\ \sum_{n=1}^\infty \frac{γ\left(n+2,\frac n3 \right)}{n^2 (n-2)!} =\ln\left(\frac23\right)+\frac{89}{192}$$
What is a closed form for:$\sum\limits_{n=1}^\infty \frac{γ(n+a,bn)n^k}{(n+c)!}$? The easiest cases would be those with $a,-c\in\Bbb N \text{ or } 0,\frac1b\in\Bbb N,k\in\Bbb Z$, but these are just a suggestion if there is no closed form outside of these restrictions.
