Lower incomplete gamma function format in series representation and R 
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*As known that the lower incomplete gamma function can be written as $\gamma(a,x) = x^{a}e^{-x}\sum_k^\infty{{x^{k}}\over a^{k+1}}.$ What is the format for  $\sum_j^\infty{\gamma(v/p-j,rx^{p})} $ in series representation? (where $v$, $p$ and $r$ are parameters) 

*$\gamma(a,x)$ in R is zipfR::Igamma(a, x, lower=TRUE, log=FALSE). If $\sum_j^\infty{\gamma(v/p-j,rx^{p})} $, what is the $a$ and $b$ should be in Igamma()?
 A: If your question is about the generalized incomplete gamma function that you mention in an earlier question, then note that the path you try to follow here has been done before (I have now added this also to my previous answer to your previous question):
Miller, Allen R., and Ira S. Moskowitz. "On certain generalized incomplete gamma functions." Journal of computational and applied mathematics 91.2 (1998): 179-190
They eventually express the function in terms of a Kampé de Fériet function (a two-variable generalization of the generalized hypergeometric series)

$$\begin{array}{rl} \Gamma \left( -v, \frac{1}{x};\frac{z^2}{4} \right) = &  \Gamma\left( - v , \frac{1}{x} \right) \Gamma(1+c) \left( \frac{2}{z} \right)^v I_v(z) \\ & - \frac{z^2}{4} \frac{x^{1+v}}{1+v} e^{-1/x}   F \mathstrut_{2:0;0}^{0:2;1}
    \left[
      \begin{array}{ r r r  }
        \dfrac{\phantom{2,2+v}}{\phantom{2,2+v}}  : & 
          1,1+v  ; &
          1 ;  \\ 
       2, 2+v : &
          \dfrac{\phantom{1,1+v}}{\phantom{1,1+v}}  ; &
          \dfrac{\phantom{1}}{\phantom{1}}  ; &
      \end{array} 
      -x \dfrac{z^2}{4}, \dfrac{z^2}{4}
    \right] \end{array}$$
      where now $v \neq -1, -2, -3, ...$

and for non negative integers

$$\begin{array}{rl} \Gamma \left( n, x; z \right) = &
   \frac{(-z)^n}{n!} \lbrace \sum_{k=1}^n \frac{(-n)_k}{z^k} \Gamma(k,x) + _0F_1[-;n+1;z]\Gamma(0,x)  \\& - \frac{z}{n+1} \frac{e^{-x}}{x}   F
 \mathstrut_{2:0;0}^{0:2;1}
\left[
  \begin{array}{ r   r r  }
\dfrac{\phantom{2,n+2}}{\phantom{2,2+v}}  : & 
  1,1  ; &
  1 ;  \\ 
   2, n+2 \;: &
  \dfrac{\phantom{1,1}}{\phantom{1,1}}  ; &
  \dfrac{\phantom{1}}{\phantom{1}}  ; &
  \end{array} 
  -x^{-1} z, z
\right]  \rbrace \end{array}$$

