Articles

Finding Meijer G function expressions

In Uncategorized on May 9, 2011 by nessgrh

Similarly to the Hypergeometric case, Meijer G functions should also be expressed in terms of named special functions when possible. The ideas for doing this are very similar to the Hypergeometric case, just more involved since there are many more indices. I will explain in this post why initially not much of an algorithm is needed, and will later expand on what can be done if we really need to.

So first of all the Meijer G function is defined as an inverse Mellin transform. The standard notation is fairly idiosyncratic. In Sympy we should probably nevertheless stick to it. However for this post let me use the following notation:

  • G\left({{a_m, b_n} \atop {c_p, d_q}} \middle| z \right) = \frac{1}{2 \pi \i} \int_L \frac{\prod_{k=1}^m \Gamma(1-a_k + s) \, \prod_{k=1}^n \Gamma(b_k - s)}{\prod_{k=1}^p \Gamma(1-c_k + s) \, \prod_{k=1}^n \Gamma(d_k - s)} z^s \mathrm{d}s

I won’t describe here what the contour L is or under what conditions the integral converges. One important thing is that L separates the poles of \Gamma(1-a_k + s) from those of \Gamma(b_k - s), so the G function is undefined if a_i - b_j \in \mathbb{Z}_{>0} for any i, j.

Slater’s Theorem

In many cases, a G function can be expressed in terms of Hypergeometric functions, due to the following:

  • Suppose m + q < n + p or m+q = n+p, |z| < 1. If b_i - b_j \notin \mathbb{Z} for all i \ne j, then G\left({{a_m, b_n} \atop {c_p, d_q}} \middle| z \right) = \sum_{h=1}^n (\text{complicated product of gamma functions}) \times {}_{m+q}F_{n+p-1} \left({1+b_h - (a_m, d_q)} \atop {1 + b_h -(b_n, c_p)*} \middle| (-1)^{p - m} z \right).
  • A similar theorem holds with inequalities reversed, and roles of a_m, b_n switched.

Beyond Slater’s Theorem

This reduces our problem to Hypergeometric functions unless two of the a_i (or b_j) differ by an integer. I will assume that this is not a common case. It can sometimes be treated by what is called the “paired index theorem”. Moreover there exists a theory of shift and inverse-shift operators for G functions. We can thus use much the same algorithm as for Hypergeometric functions. However most functions of interest are already expressible as Hypergeometric functions, so it should be possible to reduce to that case.

Later in the summer if use cases arise (or just if I get bored and then come up with use cases), I can implement the full algorithm for G functions as well.

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