Yoshida’s Rapid-Decrease Question

Abstract

A positive sequence satisfies Yoshida’s rapid-decrease inequalities but its first Turan transform does not.

Definition 1.1 (Rapid decrease).

$$\forall a\in \mathbb{N}\to \mathbb{R},RapidlyDecreasing\left(a\right)=\left((\forall k\in \mathbb{N},0<a\left(k\right))\wedge (\exists alpha\in \mathbb{R},(max\left(2,\frac{\mathrm{Real}.\mathrm{sqrt}\left(2\right)}{2}\cdot \left(1+\mathrm{Real}.\mathrm{sqrt}\left(1+a\left(1\right)\right)\right)\right)\le alpha)\wedge (\forall k\in \mathbb{N},(1\le k)\Rightarrow (a{\left(k\right)}^2\ge alph{a}^2\cdot ite\left((k:\mathbb{Z})-1<0,0,a\left(\mathrm{Int}.\mathrm{toNat}\left((k:\mathbb{Z})-1\right)\right)\right)\cdot ite\left((k:\mathbb{Z})+1<0,0,a\left(\mathrm{Int}.\mathrm{toNat}\left((k:\mathbb{Z})+1\right)\right)\right))))\right)$$

Formalization. D5/S0/Certificates/YoshidaRapidDecreaseRefutation.RapidlyDecreasing (✓ std3).

Citation. Rintaro Yoshida (2010). On some questions of Fisk and Brändén. DOI: 10.48550/arXiv.1005.4218. URL: https://arxiv.org/abs/1005.4218v1.

Commentary.

Definition 29 (p. 10): “A sequence of positive real numbers $\{s_k{\}}_{k=0}^{\infty }$ is called rapidly decreasing, if the sequence satisfies the following condition ${\left(s_k\right)}^2\ge {\alpha }^2{s}_{k-1}{s}_{k+1}$, for $\alpha \ge \max \{2,\frac{\sqrt{2}}{2}(1+\sqrt{1+s_1})\}$.”

The sequence a has natural indices, all its terms are positive, and one real alpha satisfies the threshold and every inequality at k>=1.

Definition 1.2 (The transformed coefficients).

$$\forall a\in \mathbb{N}\to \mathbb{R},\forall r\in \mathbb{N},\forall k\in \mathbb{N},sCoeff\left(a,r,k\right)=a{\left(k\right)}^2-ite\left((k:\mathbb{Z})-(r:\mathbb{Z})<0,0,a\left(\mathrm{Int}.\mathrm{toNat}\left((k:\mathbb{Z})-(r:\mathbb{Z})\right)\right)\right)\cdot ite\left((k:\mathbb{Z})+(r:\mathbb{Z})<0,0,a\left(\mathrm{Int}.\mathrm{toNat}\left((k:\mathbb{Z})+(r:\mathbb{Z})\right)\right)\right)$$

Formalization. D5/S0/Certificates/YoshidaRapidDecreaseRefutation.sCoeff (✓ std3).

Citation. Rintaro Yoshida (2010). On some questions of Fisk and Brändén. DOI: 10.48550/arXiv.1005.4218. URL: https://arxiv.org/abs/1005.4218v1.

Commentary.

Open Problem 5 (p. 11) writes Sᵣ[f(x)]=∑ₖ₌₀∞ (aₖ²−aₖ₋ᵣaₖ₊ᵣ)xᵏ:=∑ₖ₌₀∞ bₖxᵏ. The function sCoeff(a,r) is the coefficient sequence of S_r[f]. Definition 8 (p. 3) sets “and aⱼ=0 if j∉{0,1,…,n}.” Only negative indices are outside an infinite sequence. The conditional ite returns zero at negative indices and a(Int.toNat(j)) otherwise. Both k and r are natural numbers; the two shifted indices are cast to integers before subtraction and addition.

Definition 1.3 (Open Problem 5).

$$claim=\left((\exists a\in \mathbb{N}\to \mathbb{R},RapidlyDecreasing\left(a\right))\wedge (\forall a\in \mathbb{N}\to \mathbb{R},(RapidlyDecreasing\left(a\right))\Rightarrow (\forall r\in \mathbb{N},(1\le r)\Rightarrow (RapidlyDecreasing\left(sCoeff\left(a,r\right)\right))))\right)$$

Formalization. D5/S0/Certificates/YoshidaRapidDecreaseRefutation.claim (✓ std3).

Citation. Rintaro Yoshida (2010). On some questions of Fisk and Brändén. DOI: 10.48550/arXiv.1005.4218. URL: https://arxiv.org/abs/1005.4218v1.

Commentary.

Open Problem 5 (p. 11): “Let $f\left(x\right)={\sum }_{k=0}^{\infty }{a}_k{x}^k$ where $\{a_k{\}}_{k=0}^{\infty }$ is a rapidly decreasing sequence. If $S_r[f\left(x\right)]={\sum }_{k=0}^{\infty }({\left(a_k\right)}^2-a_{k-r}{a}_{k+r})x^k:={\sum }_{k=0}^{\infty }{b}_k{x}^k$, is it true that $\{b_k{\}}_{k=0}^{\infty }$ is a rapidly decreasing sequence?”

A power series is represented by its coefficients a. The first conjunct asserts existence of such a sequence; the second asserts closure for every such sequence and every positive natural r. The predicate RapidlyDecreasing and the function sCoeff are expanded above.

Theorem 1.4 (A negative answer).

$$\neg ((\exists a\in \mathbb{N}\to \mathbb{R},RapidlyDecreasing\left(a\right))\wedge (\forall a\in \mathbb{N}\to \mathbb{R},(RapidlyDecreasing\left(a\right))\Rightarrow (\forall r\in \mathbb{N},(1\le r)\Rightarrow (RapidlyDecreasing\left(sCoeff\left(a,r\right)\right)))))$$

Proof. Machine-checked in Lean as D5/S0/Certificates/YoshidaRapidDecreaseRefutation.result (✓ std3). ∎

Resolves. Problems/yoshida-rapid-decrease (refuted) by D5/S0/Certificates/YoshidaRapidDecreaseRefutation.result.

Source. Repository-derived.

Acknowledgement. Rintaro Yoshida (2010). On some questions of Fisk and Brändén. DOI: 10.48550/arXiv.1005.4218. URL: https://arxiv.org/abs/1005.4218v1.

Commentary.

Take aₖ=48ᵏ/32^(k choose 2) and r=1. Every a_k is positive, a_0=1 and a_1=48. The adjacent identity aₖ₊₁²=32aₖaₖ₊₂ gives rapid decrease with alpha=4 sqrt(2). The first transformed coefficients are 1, 2232 and 5022. The output inequality at index 1 forces alpha squared to be at most 992. Its threshold forces alpha squared to be at least 1117+sqrt(2233), which is larger. This contradiction concerns the coefficient question; it asserts no failure of Laguerre-Polya membership.

References

• Truth anchor: D5/S0/Certificates/YoshidaRapidDecreaseRefutation.RapidlyDecreasing
• Truth anchor: D5/S0/Certificates/YoshidaRapidDecreaseRefutation.claim
• Truth anchor: D5/S0/Certificates/YoshidaRapidDecreaseRefutation.result
• Truth anchor: D5/S0/Certificates/YoshidaRapidDecreaseRefutation.sCoeff