bibkey: karpzhang2023pochhammer
authors: Dmitrii Karp and Yi Zhang
year: 2023
title: Log-concavity and log-convexity of series containing multiple Pochhammer symbols
doi: 10.1007/s13540-023-00238-0
url: https://arxiv.org/abs/2305.09029
claim: "Conjecture 5.2 asserts coefficient-wise log-concavity of both the reciprocal-Pochhammer series and the related gamma series for a log-concave coefficient sequence."
strata_touched:
  - D5/S3/Analytic/KarpZhangReciprocal/BirthDeath
  - D5/S3/Analytic/KarpZhangReciprocal/IntervalConvolution
  - D5/S3/Analytic/KarpZhangReciprocal/Result
  - D5/S3/Analytic/KarpZhangGamma/Series
  - D5/S3/Analytic/KarpZhangConjecture/Result
  - D5/S3/Analytic/KarpZhangCatalanPower/Result
  - D5/S3/Analytic/KarpZhangNinh/Result
  - D5/S3/Analytic/SeriesInequalities/KarpQuadraticTruncations
license: citation-only
triage: anchor

Karp–Zhang reciprocal-Pochhammer coefficients

Verified locator

DOI: 10.1007/s13540-023-00238-0

Source: https://arxiv.org/abs/2305.09029

Pinned source: arXiv:2305.09029v3. The locators below use the printed page numbers of that version.

Conjecture 5.2, page 12

Suppose that $\{y_k{\}}_{k=0}^{\infty }$ is a log-concave sequence. Then the functions $\mu \to \tilde{y}(\mu ;x)$ and $\mu \to \hat{y}(\mu ;x)$ are coefficient-wise log-concave on $\mu >0$.

Immediately before the conjecture, the source defines

$$\tilde{y}(\mu ;x)=\sum _{k=0}^{\infty }{y}_k\frac{(2\mu {)}_k}{(\mu {)}_{k}k!}{x}^k,\hat{y}(\mu ;x)=\sum _{k=0}^{\infty }{y}_k\frac{\Gamma (\mu +k)}{\Gamma (2\mu +k)k!}{x}^k.$$

Definition 1.1, page 2

Let $\{f_k{\}}_{k=0}^{\infty }$ be a non-negative nontrivial real sequence. We call $\{f_k{\}}_{k=0}^{\infty }$ log-concave (or Pólya frequency two – $P{F}_2$, or doubly positive) if $f_k^2\ge f_{k-1}{f}_{k+1}$ for each $k\in \mathbb{N}$, and $\{f_k{\}}_{k=0}^{\infty }$ is supported on an interval of integers (i.e., $f_N=0$ implies either $f_{N+i}=0$ for all $i\in {\mathbb{N}}_0$ or $f_{N-i}=0$ for $i=0,\dots ,N$).

The sequence convention includes nonnegativity, nontriviality, the adjacent log-concavity inequalities and absence of internal zeros. A formal power series assertion is a separate inequality in every natural degree, without any convergence requirement.

Mathematical correspondence

Coeff.c is exactly the coefficient $(2t{)}_n/((t{)}_{n}n!)$ of the reciprocal series. Coeff.lambda is its adjacent-degree ratio before division by $n+1$. Coeff.eta, Coeff.psi and Coeff.chi give the adjacent score increment, score and negative logarithmic curvature as explicit rational finite sums. Coeff.coefficient_calculus includes the degree-uniform curvature estimate.

BirthDeath.variance_bound is a finite variance estimate used to control the score variance in interval convolutions. IntervalConvolution.shift_tail and finite Abel summation give KarpZhangReciprocal.Result.reciprocal_half. Exponential smoothing, the one-sign-change reciprocal-gamma kernel and the positive diagonal transformation give KarpZhangGamma.Series.gamma_half. KarpZhangConjecture.Result.claim states the conjunction of the two coefficient assertions using those exact finite coefficients, and KarpZhangConjecture.Result.result proves it for the source’s full sequence and parameter range.

Conjecture 4.2, page 8

Assume that $\{h_k{\}}_{k=0}^{\infty }$ is log-concave (log-convex ) and independent of $\mu$. Then the function $\mu \to h(\mu ;x)$ defined in (15) is coefficient-wise log-concave (log-convex ).

Formula (15), page 8:

We consider the series

$$h(\mu ;x)=\sum _{k=0}^{\infty }{h}_k\frac{(\mu {)}_{2k}}{(\mu +1{)}_{k}k!}{x}^k.$$

Lemma 4.1(2), page 8

Lemma 4.1. [17, Lemma 7] The hypergeometric function $\lambda (\mu ;x)$ has the following properties.

2. Index law

$\lambda (\alpha ;x)\lambda (\beta ;x)=\lambda (\alpha +\beta ;x)$.

The source attributes this lemma to its reference [17], Lemma 7; no independent verification of that earlier reference is asserted here. Equating degree-$m$ Cauchy coefficients gives the finite index law used in KarpZhangCatalanPower.Index.index_law. Formula (15) and this index law are literature statements; the polynomial continuation and Lean proofs are authored formalizations.

The log-convex clause of Definition 1.1, page 2, reads:

If $f_k^2\le f_{k-1}{f}_{k+1}$ for all $k\in \mathbb{N}$, the sequence is called log-convex (this inequality implies that a nontrivial sequence is strictly positive).

The initial nonnegative, nontrivial sequence assumption applies to both classes. Here $\mathbb{N}=\{1,2,\dots \}$. Taken literally, the log-convex inequalities also allow a nontrivial sequence supported on $\{0\}$; the coefficient assertion includes that case and imposes no extra positivity hypothesis.

KarpZhangCatalanPower.Coeff.a is the quotient in (15). KarpZhangCatalanPower.Index.index_law is its convolution index law. KarpZhangCatalanPower.Index.b_eq_centralBinom identifies the positive derivative generator. KarpZhangCatalanPower.Interval.tail_nonneg proves nonnegative central tails by reflected-interval cancellation and balancing. KarpZhangCatalanPower.Result.result proves both coefficient signs for all nonnegative base parameters and nonnegative real shifts, reusing the sequence class and finite Abel identity from KarpZhangReciprocal.Result.

Coefficient-wise convention, page 1

if the coefficients ${\delta }_k$ at all powers of $x$ in (2) are non-negative (non-positive) for all $\alpha ,\beta \ge 0$ we will say that $\mu \to f(\mu ;x)$ is coefficient-wise log-concave (log-convex ).

Lemma 3.1, page 4

Let $k,m\in {\mathbb{N}}_0$ with $k\le m/2$, $\mu \ge 0$ and $\alpha ,\beta >0$. Set

$$>T_{k,m}:=(\mu +\alpha {)}_k(\mu +\beta {)}_{m-k}+(\mu +\alpha {)}_{m-k}(\mu +\beta {)}_k>-(\mu {)}_k(\mu +\alpha +\beta {)}_{m-k}-(\mu {)}_{m-k}(\mu +\alpha +\beta {)}_k.>$$

If $T_{k,m}\le 0$ for some $1\le k\le m/2$, then $T_{k-1,m}<0$.

This is formula (8). KarpZhangNinh.Result.Kernel.pairPoch uses $a,b,c,N$ for $\mu ,\alpha ,\beta ,m$ respectively. KarpZhangNinh.Result.Kernel.paired_sign_step proves the positive-$\mu$ specialization of the published backward-sign implication. The paired length argument is part of D5/S3/Analytic/KarpZhangNinh/Result.lean.

Conjecture 3.8, page 8

Assume that $\{f_n{\}}_{n=0}^{\infty }$ is log-concave and independent of $\mu$. Then the function

$$>\mu \to f(\mu ;x)=\sum _{n=1}^{\infty }{f}_n\frac{(\mu {)}_{3n}}{(3n-1)!}{x}^n>$$

is coefficient-wise log-concave.

The sum starts at $n=1$, so its degree-zero coefficient is zero. The sequence class is Definition 1.1, including interval support. Coefficient-wise log-concavity means nonnegativity of every finite Cauchy coefficient of $f(\mu +\alpha ;x)f(\mu +\beta ;x)-f(\mu ;x)f(\mu +\alpha +\beta ;x)$, for nonnegative real $\mu ,\alpha ,\beta$.

Remark 3.7, on the same page, states that the analogous unweighted family with $(\mu {)}_{rn}/(rn-1)!$ is neither coefficient-wise log-convex nor coefficient-wise log-concave for $r\ge 4$ by numerical computations. This is a source-reported numerical boundary, not an all-parameter formal refutation supplied by the $r=3$ assertion.

Rising-factorial log-concavity, equation (6), page 4

Note that

$$>[\log ((\mu {)}_{nr}){]}^{\prime \prime }={\psi }^{\prime }(\mu +rn)-{\psi }^{\prime }(\mu )\le 0,>\text{\hspace{1em}(6)}$$

Here $\psi$ is the digamma function. For any degree $d\ge 2$, take the source index $n=1$ and multiplier $r=d$. The log-concavity assertion gives the four-point inequality below: for $F(t)=\log ((t{)}_d)$, concavity makes $F(t+\alpha )-F(t)$ nonincreasing in $t$. Comparing $t=\mu$ and $t=\mu +\beta$ and exponentiating gives

$$(\mu +\alpha {)}_d(\mu +\beta {)}_d\ge (\mu {)}_d(\mu +\alpha +\beta {)}_d(\mu >0,\alpha ,\beta \ge 0).$$

Degree zero is equality and the degree-one difference is $\alpha \beta$. This is KarpQuadraticTruncations.pochhammer_turan_nonneg; its owner is D5/S3/Analytic/SeriesInequalities/KarpQuadraticTruncations.lean. The homogeneous trisection identities for Conjecture 3.8 are in the Homogeneous namespace of D5/S3/Analytic/KarpZhangNinh/Beta.lean; the same module supplies symmetric beta lifting.