1

As everyone know Meijer $G$ function is a vary general function that includes most of the known special functions; is it possible to consider Lauricella Functions as a special case of $G$?

Alessio Bocci
  • 187

1 Answers1

3

No. Meijer $G$-function is equivalent to generalized hypergeometric function $_pF_q(\ldots;z)$ in one variable. Lauricella functions are generalizations of $_2F_1(\ldots;z)$ to many variables $z_1,\ldots,z_n$.

As you see, the directions of generalizations are different and concern the number of parameters in the former case and the number of arguments in the latter. The "intersection" of the two constructions is given by the Gauss hypergeometric series $_2F_1$.

Start wearing purple
  • 53,234
  • 13
  • 164
  • 223
  • Why are Meijer $G$-functions equivalent to generalized hypergeometric functions? Their Wikipedia article seems to indicate that the latter are a specific case of the former ones. Even though it doesn't explicitly say it. – Giovanni De Gaetano Nov 12 '16 at 10:18
  • @GiovanniDeGaetano No, Wiki also says they are equivalent provided the parameters are sufficiently generic. See the first formula in https://en.wikipedia.org/wiki/Meijer_G-function#Relationship_between_the_G-function_and_the_generalized_hypergeometric_function And also http://dlmf.nist.gov/16.17.E2 – Start wearing purple Nov 12 '16 at 10:24
  • Thanks for the prompt reply. This is actually the formula that may me doubt about the statement in the first place. Quoting from Wikipedia: "If the integral converges when evaluated along the second path introduced above, and if no confluent poles appear among the Γ(bj − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as...". I don't really see why is this a request on the genericity of the parameters, but, even if this is the case, this doesn't imply the statement "Meijer G-function is equivalent to generalized hypergeometric function pFq(…;z) in one variable". – Giovanni De Gaetano Nov 12 '16 at 10:29
  • Could you please provide some details or a detailed reference? – Giovanni De Gaetano Nov 12 '16 at 10:29
  • @GiovanniDeGaetano I will definitely go into the details by lack of motivation. The meaning of genericity is explained in more details just below in the same Wiki section: "$p<q$ or $p=q$ and $|z|<1$ and $b_j$ distinct mod $\mathbb Z$" and, for another generalized hypergeometric representation slightly below "$p>q$ or $p=q$ and $|z|>1$ and $a_j$ distinct mod $\mathbb Z$". – Start wearing purple Nov 12 '16 at 11:10
  • @GiovanniDeGaetano I think OP is looking for a simple answer about the relation Meijer/Lauricella. And the answer is that these are generalizations of Gauss $_2F_1$ in different directions. – Start wearing purple Nov 12 '16 at 11:12
  • I suspect a multivariate version of Meijer $G$ (see e.g. Agrawal for the bivariate case) is the desired generalization of Lauricella. But, as you have mentioned, there does not seem to be any nontrivial way to relate univariate Meijer and Lauricella except in the ${}_2 F_1$ case. – J. M. ain't a mathematician Feb 28 '18 at 14:35