bibkey: karpprilepkina2016gammaratio
authors: D. B. Karp and E. G. Prilepkina
year: 2016
title: Completely monotonic gamma ratio and infinitely divisible H-function of Fox
doi: 10.1007/s40315-015-0128-9
url: https://arxiv.org/abs/1501.05388
claim: "Conjecture 1 bounds the number of interior zeros of the Meijer G-function by that of the power-sum control function when all a_i are nonnegative and the parameter excess is positive."
strata_touched:
  - D5/S3/Analytic/MeijerZeroCount/Defs
  - D5/S3/Analytic/MeijerZeroCount/Result
license: citation-only
triage: anchor

Gamma quotients and the Meijer zero-count conjecture

Verified locator

DOI: 10.1007/s40315-015-0128-9

Source: https://arxiv.org/abs/1501.05388

Pinned version: https://arxiv.org/abs/1501.05388v2, Section 2, equations (9), (10), the definition of $G_{p,p}^{p,0}$ and Conjecture 1. Journal publication: Computational Methods and Function Theory 16 (2016), 135–153.

Mathematical correspondence

For real parameter vectors $a,b$ of length $p$, put

$$R(s)=\frac{{\prod }_{i=1}^p\Gamma (a_i+s)}{{\prod }_{i=1}^p\Gamma (b_i+s)},v(t)=\sum _{i=1}^p(t^{a_i}-t^{b_i}),\psi =\sum _{i=1}^p(b_i-a_i).$$

The source defines

$$G_{p,p}^{p,0}\left(x|\begin{array}{c}b\\ a\end{array}\right)=\frac{1}{2\pi i}{\int }_{L}R(s)x^{-s}ds.$$

Conjecture 1 asserts that if $a_i\ge 0$ and $\psi >0$, then the number of zeros of this $G$ on $(0,1)$ is at most the number of zeros of $v$ there. Defs.gammaQuot and Defs.control encode these two expressions. Defs.meijerG uses the inverse Mellin integral on $\Re s=1$. Defs.claim restricts the proposed comparison to $\psi >1$, where this vertical integral is absolutely convergent and lies to the right of the numerator poles. Refuting this restriction refutes the source’s universal assertion.

Result.family gives exactly $n$ distinct zeros of $G$ and one of $v$ for every integer $n,d\ge 2$ and real $c>0$, with $a=(n+c,n+c)$ and $b=(c,2n+d+c)$. Result.result proves the negation of Defs.claim.

The bounded literature scope and its limitations are stated in Problems/karp-prilepkina-2015-meijer-zero-count.md.