Evaluation of MeijerG function returns `Arguments are not consistent` error

Question

The following input

N[MeijerG[{{0}, {1}}, {{-1, 0, 0}, {}}, 1]]

causes Mathematica to complain that MeijerG[{{0},{1}},{{-1,0,0},{}},1] does not exist. Arguments are not consistent.

What is the problem and is there a way around it?

Answer 1

The definition of the Meijer G-function on wikipedia states that

The definition holds under the following assumptions:

• $0 ≤ m ≤ q$ and $0 ≤ n ≤ p$, where $m$, $n$, $p$ and $q$ are integer numbers,
• $a_k − b_j ≠ 1, 2, 3,...$ for $k = 1, 2, ..., n$ and $j = 1, 2, ..., m$, which implies that no pole of any $Γ(bj − s), j = 1, 2, ..., m$, coincides with any pole of any $Γ(1 − ak + s), k = 1, 2, ..., n$
• z ≠ 0

In particular your problem arises from the second condition. From the mathematica documentation we see that $a_k$ and $b_j$, for $1 \le k \le n$ and $1 \le j \le m$, are the elements in the first and third input list, respectively.

In your case, you passed $a_1 = 0$ and $b_1=-1, b_2=0, b_3=0$. The error is given because $a_1 - b_1 = 1$, value for which the function is not defined.

You can confirm that this is indeed the problem by simply slightly changing $b_1$, for example making it $b_1 = -1.1$, which evaluates without error and gives you the numerical output:

In[5]:= N@MeijerG[
  {{0}, {1}},
  {{-1.1, 0, 0}, {}},
  1
  ]

Out[5]= -11.0637 - 6.97741*10^-14 I