bibkey: karpsitnik2009luke
authors: D. Karp and S. M. Sitnik
year: 2009
title: Inequalities and monotonicity of ratios for generalized hypergeometric function
doi: 10.1016/j.jat.2008.10.002
url: https://arxiv.org/abs/math/0703084
claim: "Conjectures 1 and 2 propose Luke lower and upper bounds under a positive sum of denominator-minus-numerator parameters."
strata_touched:
  - D5/S3/Analytic/KarpSitnikLuke/Result
  - D5/S3/Analytic/KarpSitnikLuke/UpperResult
license: citation-only
triage: anchor

The conjectured sum-condition Luke bounds

Verified locator

https://arxiv.org/abs/math/0703084

arXiv: math/0703084. Version: https://arxiv.org/abs/math/0703084v2. DOI: 10.1016/j.jat.2008.10.002. Journal of Approximation Theory 161 (2009), 337–352. Page numbers below are printed pages of the arXiv version.

Source statements

Theorem 3, printed p. 6, equation (23):

For bₖ > aₖ > 0, k = 1, . . . , q, x > −1 and σ ≥ 1 the inequality

$\frac{1}{(1+x{\prod }_{i=1}^q(a_i/b_i){)}^{\sigma }}<{}_{q+1}{F}_q(\sigma ,(a_q);(b_q);-x)$

holds true.

Conjecture 1, printed p. 14:

Theorem 3 is true for all σ > 0 and ∑ᵢ₌₁ᵠ (bᵢ − aᵢ) > 0.

Equation (1), printed p. 1, defines the function as

$${}_{q+1}{F}_q((a_{q+1});(b_q);z)=\sum _{n=0}^{\infty }\frac{(a_1{)}_n\cdots (a_{q+1}{)}_n}{(b_1{)}_n\cdots (b_q{)}_{n}n!}{z}^n.$$

The following sentence reads:

where (a_q) = a₁, a₂, . . . , a_q is the parameter list and (a)ₙ = a(a + 1) · · · (a + n − 1) is the shifted factorial. The series converges for |z| < 1 and can be analytically continued onto the entire complex plane cut along [1, ∞].

Mathematical correspondence

hyperTerm uses the existing TwoBlockResult.pochRatio for the products of rising factorials, adds the upper parameter σ and the n! denominator, and multiplies by zⁿ. hyper is its real tsum. claim restricts the conjecture to −1<x<1, summable defining series and a positive base for Real.rpow. This weaker assertion is already false. The witness has q=2, σ=1, a=(1,1), b=(1/10,2), x=1/10, with sum(b−a)=1/10 and product(a/b)=5. The series is at most 37/66, less than the proposed lower bound 2/3. It fails the componentwise hypothesis of Theorem 3.

The remaining componentwise case 0<σ<1 and Conjecture 2 are not settled by this fixed-parameter result. The bounded literature scope is given in Problems/karp-sitnik-2009-luke-lower-bound.md.

Conjectured upper bound

Conjecture 2, printed p. 14, equation (38):

For 0 < σ ≤ min(a₁, a₂, . . . , a_q), ∑ᵢ₌₁ᵠ (bᵢ − aᵢ) > 0 and x > 0:

${}_{q+1}{F}_q(\sigma ,(a_q);(b_q);-x)<\frac{1}{{\left[1+x{\prod }_{i=1}^q\frac{\Gamma (a_i)\Gamma (b_i-\sigma )}{\Gamma (b_i)\Gamma (a_i-\sigma )}\right]}^{\sigma }}.(38)$

The preceding sentence reads:

The asymptotic formula (16) and numerical experiments suggest the following

Formula (16), printed p. 5, is preceded by:

It follows that for σ < min(a₁, a₂, . . . , a_q) we will have

$${}_{q+1}{F}_q(\sigma ,(a_q);(b_q);-z)=\frac{{\prod }_i\Gamma (a_i-\sigma )\Gamma (b_i)}{{\prod }_i\Gamma (a_i)\Gamma (b_i-\sigma )}{z}^{-\sigma }+o(z^{-\sigma }),z\to \infty .$$

The quotation transcribes the extracted display, which places a $\sigma$ next to the Gamma product. Write $K={\prod }_{i=1}^q\Gamma (a_i)\Gamma (b_i-\sigma )/(\Gamma (b_i)\Gamma (a_i-\sigma ))$. Separately, the asymptotically precise reading via (16) is $1/(1+x{K}^{1/\sigma }{)}^{\sigma }$: the inner exponent $1/\sigma$ makes its leading asymptotic coefficient agree with (16). The counterexample has $\sigma =1$, where both readings coincide.

UpperResult.claim uses the existing Result.hyperTerm and Result.hyper, restricts to $0<x<1$, and assumes summability and $K>0$. Each restriction weakens the universal assertion. At $q=2$, $\sigma =1$, $a=(2,2)$, $b=(3/2,3)$, $x=1/2$, the Gamma recurrence gives $K=1$. The six-term alternating lower bracket is $249826/363825>2/3$, refuting the proposed upper bound. The componentwise upper-bound case $b_k>a_k$ is open here; Problems/karp-sitnik-2009-luke-upper-bound.md specifies the scope.