Determine all triples (a,b,c)∈R_>0^3 such that _1F_2!( a b,c ;-x)>0 for every x>0, where _1F_2!( a b,c ;z)=displaystyle∑_n=0^∞frac(a)_n(b)_n(c)_nfracz^nn! and (q)_n denotes the rising factorial.

Askey–Szegő problem

Determine all triples $(a,b,c)\in\mathbb{R}_{>0}^{3}$ such that ${}_1F_2\!\left(\begin{matrix}a\\ b,c\end{matrix};-x\right)>0$ for every $x>0$ , where ${}_1F_2\!\left(\begin{matrix}a\\ b,c\end{matrix};z\right)=\displaystyle\sum_{n=0}^{\infty}\frac{(a)_n}{(b)_n(c)_n}\frac{z^n}{n!}$ and $(q)_n$ denotes the rising factorial.

References

Primary source

Constructive Approximation

Additional references

Correction to: Rational Extension of the Newton Diagram for the Positivity of 1F2 Hypergeometric Functions and Askey–Szegö Problem — Constructive Approximation — Yong-Kum Cho, Seok-Young Chung, Hera Yun

constructive-approximation hypergeometric-functions special-functions

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Open

A formal correction has been issued, but no new proof, counterexample, or change in the problem’s status was reported.

The Askey–Szegő problem is a named problem in approximation theory. The available scan found no verified resolution or substantive advance beyond a correction to a related published treatment.

August 2026 correction

A formal correction to Rational Extension of the Newton Diagram for the Positivity of 1F2 Hypergeometric Functions and Askey–Szegö Problem was recorded for Yong-Kum Cho, Seok-Young Chung, and Hera Yun. The supplied record does not state which claims were corrected or whether the correction affects the conjectural status.

Current status (as of August 2026): The problem remains open; a formal correction has been issued, but its mathematical consequences and any resolution are unreported.

Sources

No solutions have been posted yet.