bibkey: kalmykovkarp2017pfq
authors: S. I. Kalmykov and D. B. Karp
year: 2017
title: Log-concavity and Turán-type inequalities for the generalized hypergeometric function
doi: 10.1007/s10476-017-0503-z
url: https://arxiv.org/abs/1611.06724
claim: "Conjectures 1 and 2 assert coefficient nonnegativity for shifted hypergeometric Turanians when mu is at least one."
strata_touched:
  - D5/S3/Analytic/HypergeometricShiftTuran/Defs
  - D5/S3/Analytic/HypergeometricShiftTuran/Result
  - D5/S3/Analytic/HypergeometricShiftTuran/TwoBlockResult
  - D5/S3/Analytic/HypergeometricShiftTuran/FunctionResult
license: citation-only
triage: anchor

Shifted hypergeometric Turanian coefficients

Verified locator

DOI: 10.1007/s10476-017-0503-z

Source: https://arxiv.org/abs/1611.06724

Pinned version: https://arxiv.org/abs/1611.06724v1, Section 3, equations (2), (3), (9), (10), (13), Theorem 3 and Conjectures 1 and 2. Journal publication: Analysis Mathematica 43 (2017), 567–580.

Mathematical correspondence

Equation (10) defines

$$f(\mu ;x)=\sum _{n=0}^{\infty }\frac{\Gamma (a+\mu +n)}{\Gamma (b+\mu +n)}\frac{x^n}{n!}.$$

In the generalized parameter notation, the coefficient ratio is

$$c(t)=\frac{{\prod }_{i=1}^p\Gamma (a_i+t)}{{\prod }_{j=1}^q\Gamma (b_j+t)},f(\mu ;x)=\sum _{n=0}^{\infty }c(\mu +n)\frac{x^n}{n!}.$$

Equation (9) defines the shifted Turanian

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

and its degree-$m$ coefficient is

$${\delta }_m=\sum _{k=0}^m\frac{c(\mu +\alpha +k)c(\mu +\beta +m-k)-c(\mu +k)c(\mu +\alpha +\beta +m-k)}{k!(m-k)!}.$$

Condition (13) is the descending chain of consecutive ratios of elementary symmetric functions of the denominator and numerator parameters:

$$\frac{e_{q-j}(\mathbf{b})}{e_{p-j}(\mathbf{a})}\le \frac{e_{q-j-1}(\mathbf{b})}{e_{p-j-1}(\mathbf{a})}(0\le j<p).$$

Theorem 3 proves ${\Delta }_f(\alpha ,\beta ;x)\ge 0$ for $x\ge 0$ when $\mu ,\beta \ge 0$ and $\alpha \in \mathbb{N}$, under $p\le q$ and (13); if $\alpha \le \beta +1$, it also proves ${\delta }_m\ge 0$ for every $m\in {\mathbb{N}}_0$. Conjecture 1 states verbatim:

All conclusions of Theorem 3 hold for all α, β ≥ 0 if µ ≥ 1.

The Lean definitions Defs.coeff, Defs.turanCoeff, Defs.esymm and Defs.cond13 encode the coefficient ratio, the finite Cauchy coefficient, the elementary symmetric functions and condition (13). Defs.claim records the coefficient conclusion in the $p\le q$ branch, while Result.result refutes that coefficient clause.

The bounded literature scope and its limitations are stated in Problems/kalmykov-karp-pfq-conjecture-one-coefficients.md.

Two parameter blocks and Conjecture 2

Conjecture 2 on page 6 of arXiv:1611.06724v1 states verbatim:

Conjecture 2. Suppose p1 ≤ q1 , p2 ≤ q2 and conditions (13) hold for a1 , b1 and a2 , b2 . Then the function µ → f (µ; x) defined in (3) is log-concave on [1, ∞) and the corresponding generalized Turánian ∆f (α, β; x) has nonnegative coefficients at all powers of x.

Section 2, page 3, states verbatim:

We will always assume that a2 is not empty (i.e. p2 ≥ 1), while all other subvectors are allowed to be empty.

Equation (3) on page 2 is

$$f(\mu ;x)=\frac{\Gamma ({\mathbf{a}}_2+\mu )}{\Gamma ({\mathbf{b}}_2+\mu )}{}_p{F}_q({\mathbf{a}}_1,{\mathbf{a}}_2+\mu ;{\mathbf{b}}_1,{\mathbf{b}}_2+\mu ;x),p=p_1+p_2,q=q_1+q_2.$$

Vector Gamma and rising factorial expressions mean products over the entries. The unshifted vectors are ${\mathbf{a}}_1,{\mathbf{b}}_1$; the shifted vectors are ${\mathbf{a}}_2,{\mathbf{b}}_2$. Its Taylor coefficients are

$$F_n(\mu )=\frac{{\prod }_i(a_{1,i}{)}_n}{{\prod }_j(b_{1,j}{)}_n}\frac{{\prod }_i\Gamma (a_{2,i}+\mu +n)}{{\prod }_j\Gamma (b_{2,j}+\mu +n)}\frac{1}{n!}.$$

TwoBlockResult.pochRatio is the unshifted rising factorial ratio, TwoBlockResult.seriesCoeff is $F_n(\mu )$, and TwoBlockResult.twoBlockTuranCoeff is the degree-$m$ finite Cauchy coefficient of equation (2). TwoBlockResult.claim imposes positivity, $p_1\le q_1$, $1\le p_2\le q_2$, and Defs.cond13 separately on both blocks, with $\mu \ge 1$ and arbitrary nonnegative shifts. TwoBlockResult.result refutes that coefficient clause with empty unshifted blocks, shifted numerator $(1)$, seven shifted denominator entries equal to $1/2$, and shifts $\alpha =\beta =1/2$ at $\mu =1$. The log-concavity clause at a fixed nonnegative $x$ is not settled here.

The shared half-integer Gamma values in Defs follow from Mathlib’s Real.Gamma_nat_add_half; their reciprocal power products and the bound $640/567<32/(9\pi )$ supply both coefficient calculations. The two-block family, correspondence and bounded literature scope are stated in Problems/kalmykov-karp-pfq-conjecture-two-coefficients.md.

Function clause of Conjecture 1

Theorem 3, page 5 of arXiv:1611.06724v1, states verbatim (line breaks joined):

Theorem 3. If p ≤ q and conditions (13) are satisfied, then ∆f (α, β; x) ≥ 0 for x ≥ 0, µ, β ≥ 0 and α ∈ N, where ∆f (α, β; x) is defined in (9) with f from (10). Moreover, if α ≤ β + 1, then δm ≥ 0 for all m ∈ N0. If p ≥ q and conditions (12) are satisfied, then ∆f (α, β; x) ≤ 0 for x ≥ 0, β ≥ 0 and α ∈ N. Moreover, if α ≤ β + 1, then δm ≤ 0 for all m ∈ N0.

Conjecture 1, page 6, states verbatim:

Conjecture 1. All conclusions of Theorem 3 hold for all α, β ≥ 0 if µ ≥ 1.

FunctionResult.fval is the real infinite sum in equation (10), using the frozen Gamma ratio Defs.coeff. FunctionResult.turanian is equation (9). FunctionResult.claim imposes $1\le p\le q$ and the remaining function hypotheses, without the extra restriction needed only for coefficients. The nonempty-$a_2$ sentence in Section 2 is followed by “In this section we will consider log-convexity of the function f (µ; x) defined in (3) under these assumptions.” Its scope in Section 3 need not be decided: the claim with $1\le p$ is refuted by a witness valid under both the standing and section-local readings. FunctionResult.result uses $p=1$, $a=(1)$, $q=12$, $b_j=1/2$, $\mu =1$, $\alpha =\beta =1/2$, $x=2000$. The proof establishes convergence and estimates the infinite sums; it does not infer a function sign from a negative Taylor coefficient. Its exact scope is recorded in Problems/kalmykov-karp-pfq-conjecture-one-function.md.