bibkey: kalmykovkarp2018qgamma
authors: S. I. Kalmykov and D. B. Karp
year: 2018
title: Inequalities for series in q-shifted factorials and q-gamma functions
doi: 10.1016/j.jmaa.2017.11.041
url: https://arxiv.org/abs/1702.03788
claim: "Conjecture 1 extends Theorem 3 from positive integral alpha to all positive real alpha, including the coefficient positivity assertion under its remaining parameter restrictions."
strata_touched:
  - D5/S3/Analytic/QGammaTuran/Result
  - D5/S3/Analytic/QGammaTuran/FunctionResult
license: citation-only
triage: anchor

Reciprocal q-gamma Turanian coefficients

Verified locator

DOI: 10.1016/j.jmaa.2017.11.041

Source: https://arxiv.org/abs/1702.03788

Pinned source: https://arxiv.org/abs/1702.03788v1. Journal publication: Journal of Mathematical Analysis and Applications 460 (2018), 332–351. The page numbers below are the printed pages of the arXiv version.

Source statements

Conjecture 1, p. 12:

All conclusions of Theorem 3 are valid for all α > 0, while conditions on other parameters remain intact. In particular, µ → g(µ; x) is log-concave for each fixed 0 < x < Rg.

Theorem 3, p. 11:

For a real sequence {gₙ}ₙ≥₀ and fixed 0 < q < 1 define

$g(\mu ;x)=\sum _{n=0}^{\infty }\frac{g_n{x}^n}{{\Gamma }_q(\mu +n)}.$ (16)

Suppose {gₙ}∞ₙ₌₀ is doubly positive. Then (16) has a positive radius of convergence Rg and the generalized Turánian

${\Delta }_g(\alpha ,\beta ;x)=g(\mu +\alpha ;x)g(\mu +\beta ;x)-g(\mu ;x)g(\mu +\alpha +\beta ;x)$

is positive for 0 < x < Rg if α ∈ N, β > 0 and µ ≥ max(−β, −1). If, in addition, α ≤ β + 1, then Δg(α, β; x) has positive power series coefficients at all powers of x.

Definition (6), p. 4:

The q-gamma function is defined by

${\Gamma }_q(z)=(1-q{)}^{1-z}\frac{(q;q{)}_{\infty }}{(q^z;q{)}_{\infty }}$ (6)

for |q| < 1 and all complex z such that qᶻ⁺ᵏ ≠ 1 for k ∈ N₀.

The sequence definition on p. 3 reads:

A sequence f : N₀ → R₊ is PF₂ (Pólya frequency sub two) or doubly positive if it is nontrivial, log-concave, fₖ² ≥ fₖ₋₁fₖ₊₁, k ∈ N, and has no internal zeros. The last claim means that fN = 0 implies either fN+i = 0 for all i ∈ N₀ or fN−i = 0 for i = 0, . . . , N.

Mathematical correspondence

The reciprocal of (6), on real arguments and $0<q<1$, is

$$R_q(x)=\frac{{\prod }_{j\ge 0}(1-q^{x+j})}{\left({\prod }_{j\ge 0}(1-q^{j+1})\right)(1-q{)}^{1-x}}.$$

The numerator vanishes at $x\in -{\mathbb{N}}_0$, giving the reciprocal extension by zero. This is the object Result.rqgamma. The existing D5.S3.Analytic.ReciprocalGammaTuran.Defs.AdmissibleSeq expresses exactly the printed sequence assumptions, including nonnegativity from the codomain ${\mathbb{R}}_{+}$.

Result.qturanCoeff is the finite Cauchy coefficient of the Turanian. Result.claim states the coefficient clause for real $\alpha ,\beta >0$, $\mu \ge \max (-\beta ,-1)$ and $\alpha \le \beta +1$. Result.result refutes that clause. The function clause is separately refuted by FunctionResult.result; log-concavity restricted to nonnegative parameters $\mu$ remains OPEN. The bounded literature scope and computational evidence are stated in Problems/kalmykov-karp-2018-qgamma-turanian-coefficients.md.

Function clause

FunctionResult.qseries is the infinite sum in (16), and FunctionResult.radius q µ g is Mathlib’s scalar-series radius of (16) at the given $\mu$, with coefficients $g_n{R}_q(\mu +n)$. The claim requires $0<x<R_g$ for this radius. On p. 12 the source specifies:

Writing Rg > 0 for the radius of convergence in (16), we claim that the next conjecture is true.

For the unit sequence, put $y=x(1-q)$ and

$$H(a,y)=\sum _{n\ge 0}{y}^n(a{q}^n;q{)}_{\infty }.$$

Then $g(s;x)=H(q^s,y)/((q;q{)}_{\infty }(1-q{)}^{1-s})$. For $s\ge -1$ the terms after index zero are nonnegative and bounded by a geometric sequence when $0\le y<1$. At the example, absolute summability at $x=3/2$ proves $3/2\le R_g$, so $0<93/100<3/2\le R_g$. All four shifted series also converge absolutely at $x=93/100$. The function clause quoted from Theorem 3 above imposes no $\alpha \le \beta +1$ restriction. Its extension to positive real $\alpha$ is false at $q=(93/100{)}^{10}$, $g_n=1$, $\mu =-7/10$, $\alpha =3/10$, $\beta =7/10$ and $x=93/100$. The corresponding dossier is Problems/kalmykov-karp-2018-qgamma-turanian-function.md.

As a consequence in prose of FunctionResult.witness_negative, at the same $q,x$ and $g_n=1$ the function cannot be log-concave on an interval containing $[-7/10,3/10]$. The two convex combinations $-2/5=(7/10)(-7/10)+(3/10)(3/10)$ and $0=(3/10)(-7/10)+(7/10)(3/10)$ have respective right-endpoint weights $3/10$ and $7/10$. Log-concavity at both points would imply, after multiplication, $g(-2/5;x)g(0;x)\ge g(-7/10;x)g(3/10;x)$, contradicting the proved strict reverse inequality. The endpoint values are strictly positive by the normalized lower bounds and positive denominators. This consequence is not a separate Lean theorem. Log-concavity restricted to nonnegative parameters $\mu$ remains OPEN.