Expressing $G_{m,m+1}^{m+1,0}\left(x\middle| \begin{array}{c}1,\cdots,1 \\0,0,\cdots,0\\\end{array}\right)$ as a power series.

Question

I have this family of MeijerG functions: $$ G_{m,m+1}^{m+1,0}\left(x\left| \begin{array}{c} 1,\cdots,1 \\ 0,0,\cdots,0 \\ \end{array} \right.\right) $$ which I'd like to express in terms of a power series in $x$, eventually generalized hyper-geometric functions.

Does anyone knows how this can be done for this specific case?

Thanks.

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Here are the first 3 elements of the family in Mathematica syntaxe:

MeijerG[{{}, {1}}, {{0, 0}, {}}, x]
MeijerG[{{}, {1, 1}}, {{0, 0, 0 }, {}}, x]
MeijerG[{{}, {1, 1, 1}}, {{0, 0, 0, 0}, {}}, x]

Answer 1

I assume you mean the expansion around zero. We have $$G_{m, m + 1}^{m + 1, 0} {\left( x \middle| {\mathbf 1 \atop \mathbf 0} \right)} = \frac 1 {2 \pi i} \int_\gamma \Gamma(s) s^{-m} x^{-s} ds.$$ The sum of the residues over the simple poles at the negative integers gives a hypergeometric series: $$\operatorname*{Res}_{s = -k} \Gamma(s) s^{-m} x^{-s} = (-k)^{-m} x^k \,[(s + k)^{-1}] \,\Gamma(s) = (-1)^m \frac {(-x)^k} {k^m k!}, \\ \operatorname*{Res}_{s = 0} \Gamma(s) s^{-m} x^{-s} = [s^m] \,\Gamma(s + 1) e^{-s \ln x} = \sum_{k = 0}^m \frac {\Gamma^{(k)}(1)} {k!} \frac {(-\ln x)^{m - k}} {(m - k)!}, \\ G_{m, m + 1}^{m + 1, 0} {\left( x \middle| {\mathbf 1 \atop \mathbf 0} \right)} = \operatorname*{Res}_{s = 0} \Gamma(s) s^{-m} x^{-s} - (-1)^m x \,{_{m + 1}\hspace{-2px}F_{m + 1}}(\mathbf 1; \mathbf 2; -x).$$