Metadata

slug: karp-prilepkina-2015-meijer-zero-count
bibkey: karpprilepkina2016gammaratio
doi: 10.1007/s40315-015-0128-9
triage: theorem
motivation_gids:
  - D5/S3/Analytic/MeijerZeroCount/Result.result
url: https://arxiv.org/abs/1501.05388

The Karp–Prilepkina Meijer zero-count conjecture

Problem

Karp and Prilepkina, Completely monotonic gamma ratio and infinitely divisible H-function of Fox, arXiv:1501.05388v2, Section 2, Conjecture 1:

If aᵢ ≥ 0, i = 1, …, p and Σᵢ₌₁ᵖ (bᵢ − aᵢ) > 0 then # zeros of G^{p,0}_{p,p}(x | b/a) on (0, 1) ≤ # zeros of v(t) on (0, 1).

Here $p \geq 1$ , $a_{i}, b_{i}$ are real,

$v (t) = k = 1 \sum p (t^{a_{k}} - t^{b_{k}}), ψ = i = 1 \sum p (b_{i} - a_{i}),$

and the contour-integral definition is

$G_{p, p}^{p, 0} (x b a) = \frac{1}{2 πi} \int_{L} \frac{\prod _{i = 1}^{p} Γ ( a _{i} + s )}{\prod _{i = 1}^{p} Γ ( b _{i} + s )} x^{- s} d s .$

The contour separates the numerator poles in the usual Mellin–Barnes representation. The zero counts below count distinct zeros.

Formal reading

All Lean names below have prefix D5.S3.Analytic.MeijerZeroCount.

Source object	Lean name	Correspondence
$R (s) = \prod_{i} Γ (a_{i} + s) / \prod_{i} Γ (b_{i} + s)$	`Defs.gammaQuot`	Complex Gamma quotient of the real parameter vectors.
$G_{p, p}^{p, 0} (x ∣ b / a)$	`Defs.meijerG`	`mellinInv 1 (gammaQuot a b) x`, with normalization $1/ (2 π)$ in the real integral.
$v (t) = \sum_{k} (t^{a_{k}} - t^{b_{k}})$	`Defs.control`	Real powers of real arguments; the comparison uses $0 < t < 1$ .
$p \geq 1$ , $a_{i} \geq 0$ , $ψ > 0$	`Defs.claim`	The formal claim restricts to $ψ > 1$ . It quantifies over natural $p$ ; the case $p = 0$ cannot satisfy $ψ > 1$ .
Number of zeros on $(0, 1)$	`Set.encard` of each displayed zero set	Extended natural cardinality, including infinite zero sets.
$a = (n + c, n + c)$ , $b = (c, 2 n + d + c)$	`Result.family`	All natural $n, d \geq 2$ and real $c > 0$ ; $p = 2$ and $ψ = d$ .
Refutation of the comparison	`Result.result`	The unconditional theorem has type `¬ Defs.claim`.

The formal claim restricts to $ψ > 1$ , where the contour may be taken as the vertical line $ℜ s = 1$ . Since $a_{i} \geq 0$ , this line is to the right of every numerator Gamma pole. The Gamma quotient decays as $O (∣ℑ s ∣^{- ψ})$ on this line, so the integral is absolutely convergent. A counterexample in this restricted range is also a counterexample to the source statement with $ψ > 0$ .

Motivation

The settling declaration is D5/S3/Analytic/MeijerZeroCount/Result.result. The source note is D5/L/Asymptotics/karpprilepkina2016gammaratio. The conjecture compares oscillation of an inverse Mellin transform with a finite power sum built from the same parameter vectors.

Gap

Status: Refuted. The family $a = (n + c, n + c)$ , $b = (c, 2 n + d + c)$ with $n, d \geq 2$ and $c > 0$ has $n$ interior zeros of $G$ and exactly one of $v$ . The excess $n - 1$ is unbounded even for fixed $p = 2$ and $ψ = d = 2$ .

Route

Put $N = n + d - 1$ . Gamma recurrence gives

$R (s) = \frac{( s + c ) _{n}}{( s + c + n ) _{n + d}} = k = 0 \sum N \frac{A _{k}}{s + c + n + k}, A_{k} = \frac{( - 1 ) ^{n + k} ( n + k )!}{( k ! ) ^{2} ( N - k )!} .$

The partial fractions follow by polynomial interpolation at the distinct denominator roots. Define $W (x) = x^{n} (1 - x)^{N}$ and, on $(0, 1)$ ,

$f (x) = \frac{( - 1 ) ^{n}}{N !} x^{n + c} W^{(n)} (x) = k = 0 \sum N A_{k} x^{n + c + k},$

extending $f$ by zero outside that interval. Its Mellin transform is $R (s)$ for $ℜ s > 0$ . On the inversion line, $∣ R (1 + i t) ∣ \leq (1 + t^{2})^{- 1}$ ; continuity in the interior and Mellin inversion give $G = f$ there.

Repeated Rolle arguments use the endpoint factors of $W$ to produce at least $n$ distinct interior zeros of $W^{(n)}$ . Dividing out $(1 - x)^{d - 1}$ leaves a polynomial of degree at most $n$ , proving the matching upper bound. The nonvanishing factor $x^{n + c} (- 1)^{n} / N!$ preserves this zero set.

For the control function,

$v (x) = x^{c} (2 x^{n} - 1 - x^{2 n + d}) .$

The second factor is $(x - 1) h (x)$ , where $h (x) = \sum_{j = 0}^{n - 1} x^{j} - x^{n} \sum_{j = 0}^{n + d - 1} x^{j}$ . It is positive at zero and negative at one. Its normalization $h(x)/x^n=\sum_{j=0}^{n-1}x^{j-n} -\sum_{j=0}^{n+d-1}x^j $i ss t r i c tl y d ecre a s in g o n$ (0,1)$, which proves existence and uniqueness of its interior zero.

Falsifier

The explicit admissible vectors $a = (3, 3)$ and $b = (1, 7)$ have $ψ = 2$ . Their interior zero counts are two for $G$ and one for $v$ , contradicting the proposed inequality $2 \leq 1$ .

Evidence

D5/S3/Analytic/MeijerZeroCount/Result.result proves exactly ¬ Defs.claim, with no additional hypotheses. Its principal dependency is Result.family; both zero-count conjuncts are used in the contradiction. The analytic dependencies are Mellin.gammaQuot_family, Mellin.mellin_f and Mellin.meijerG_eq_f. The polynomial expansion and count are Rodrigues.f_eq_sum and Rodrigues.zeros_f; the control count is Control.zeros_control. The axiom closure of Result.result is {propext, Classical.choice, Quot.sound}.

The refutation route was derived with the assistance of the ChatGPT Pro model. The source proves that an interior zero of $G$ forces a zero of $v$ . Karp–Prilepkina 2017, on delta-neutral H-functions, settles the different Conjecture 2. No settlement of Conjecture 1 was found in the searched scope: the source and its identifier, the authors’ work on Gamma quotients and zero counts, the 2017 delta-neutral paper, Berg–Çetinkaya–Karp 2021, Karp’s publication list and the distinct Toeplitz-determinant conjecture discussed in arXiv:2610.18895. This bounded scope does not establish exhaustive literature coverage or priority.

Triage

PROVED. For all integers $n, d \geq 2$ and real $c > 0$ , with $a = (n + c, n + c)$ and $b = (c, 2 n + d + c)$ , $G_{2, 2}^{2, 0}$ has exactly $n$ distinct zeros on $(0, 1)$ and $v$ exactly one (Result.family). Conjecture 1 fails (Result.result), and the excess $n - 1$ is unbounded for fixed $p = 2$ , $ψ = 2$ .
PROVED. On $(0, 1)$ , $G (x) = x^{n + c} (- 1)^{n} W^{(n)} (x) / N!$ , with $W (x) = x^{n} (1 - x)^{n + d - 1}$ and $N = n + d - 1$ (Mellin.meijerG_eq_f, Rodrigues.f, Rodrigues.rodriguesPoly).
PROVED in prose. The interior zeros are simple. After removing $(1 - x)^{d - 1}$ from $W^{(n)}$ , the remaining polynomial has degree at most $n$ and $n$ distinct roots, hence degree $n$ and no repeated root. The other factors are nonzero on $(0, 1)$ . The shifted Jacobi Rodrigues identity gives $G(x)=[n!/(n+d-1)!]x^{n+c}(1-x)^{d-1} P_n^{(d-1,0)}(2x-1)$. The same family refutes the unit-scale case of the source’s Conjecture 3, because its Fox H-function specializes to this Meijer G-function when every scale is one. Simplicity, the Jacobi identity and this specialization are prose consequences, not separate Lean theorems.
COMPUTED. For $n = d = 2$ , $c = 1$ , $G (x) = x^{3} (1 - x) (10 x^{2} - 8 x + 1) /3$ on $(0, 1)$ , with zeros $(4 \pm 6) /10$ . Also $v (x) = x (2 x^{2} - 1 - x^{6})$ has its single interior zero at $(5 - 1) /2$ .
OPEN. Conjecture 1 on the remaining parameter range $0 < ψ \leq 1$ , a zero-count bound depending on further structure, and the Fox H Conjecture 3 in general remain outside this result.

ASSUMED-UNVERIFIED

Exhaustive literature coverage, citation-index completeness and priority are unverified. The bounded literature scope does not imply that no other settlement exists.

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