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Ismail–Miller conjecture on modified Bessel functions

For ν>μ≥0\nu>\mu\geq 0 and b>a>0b>a>0, define G:[0,∞)→RG:[0,\infty)\to\mathbb{R} by G(x;μ,ν)=(ba)μ−νIμ(ax)Iν(bx)Iμ(bx)Iν(ax)G(x;\mu,\nu)=\left(\frac{b}{a}\right)^{\mu-\nu}\frac{I_\mu(a\sqrt{x})I_\nu(b\sqrt{x})}{I_\mu(b\sqrt{x})I_\nu(a\sqrt{x})}, where IαI_\alpha is the modified Bessel function of the first kind. The Ismail–Miller conjecture asserts that there exists an infinitely divisible probability measure P\mathsf{P} on [0,∞)[0,\infty) such that G(x;μ,ν)=∫[0,∞)e−xt dP(t)G(x;\mu,\nu)=\int_{[0,\infty)}e^{-xt}\,d\mathsf{P}(t) for every x≥0x\geq 0; equivalently, G(⋅;μ,ν)G(\mathord{\cdot};\mu,\nu) is the Laplace transform of an infinitely divisible probability distribution.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Bessel-zero-sum monotonicity formulation

    The conjecture is equivalent to the monotonicity, with respect to the order ν\nu, of the function Qν(t)=∑n≥1jν,n2e−tjν,n2Q_\nu(t)=\sum_{n\geq 1}j_{\nu,n}^2e^{-t j_{\nu,n}^2} for t>0t>0, where jν,nj_{\nu,n} denotes the nnth positive zero of the Bessel function of the first kind of order ν\nu.

    source: Baricz, K. Dhivya Prabhu, “On an infinitely divisible distribution involving modified Bessel functions”

References

Primary source

arXiv

Additional references

analysisprobabilityspecial-functions
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed preprint claims to prove the Ismail–Miller conjecture, but the result has not been independently verified.

The conjecture connects inequalities for modified Bessel functions with the distribution of their zeros and infinite divisibility. Its claimed resolution would establish all of these connections.

October 2026 claimed proof

On October 7, 2026, Árpád Baricz and Dhivya Prabhu K reported a monotonicity result for a Bessel-zero sum, derived from an integral representation and Bernstein’s theorem. They claim this proves the Ismail–Miller infinite-divisibility conjecture, but the result appears only in an unrefereed preprint.

Current status (as of October 2026): A preprint claims the conjecture is proved, but independent verification is not recorded, so the resolution remains unestablished.

Sources

Solutions 0

No solutions have been posted yet.