Explicit formulas for higher-order (d>2) Borel transforms

Develop explicit, compact representations (for example, convergent series or closed integral formulas) for the higher-order Borel transform [\mathcal{B}_{\chi,d} f](\vec u) = \oint_{|z|=R/2} \frac{dz}{2\pi i z} f(\chi+z) \exp\big(\sum_{j=1}^{d} (j-1)!\,u_j z^{-j}\big) for all integers d>2, analogous to the known d=1 power-series expansion with coefficients \partial_\chi^j f(\chi)/j!^2 and the d=2 Hermite-polynomial series representation.

Background

The paper defines the first-order Borel transform \mathcal{B}_{\chi}f and provides two equivalent representations, including a power series with coefficients involving derivatives of f at \chi. For d=2, an explicit series representation in terms of Hermite polynomials is derived.

Beyond d=2, the authors note that additional variables enter the expansion, and no similarly explicit or compact expression is currently known. Establishing such formulas would parallel the d=1 and d=2 cases and facilitate the application of their determinant/Pfaffian derivative results in broader settings.

References

Finding some representation of the higher order Borel transform beyond d=2 which is similarly explicit and compact as~eq: borel transf or~eq: borel transf.2d is still an open problem.

eq: borel transf.2d:

[Bχ,2f](u⃗)=∑j=0∞∂χjf(χ)j!2(−iu2)jHj(iu12u2).[\mathcal{B}_{\chi,2}f](\vec u) =\sum_{j=0}^\infty \frac{\partial_\chi^jf(\chi)}{j!^2} (-i\sqrt{u_2})^{j} H_j\left(i\frac{u_1}{2\sqrt{u_2}}\right).

— Derivative relations for determinants, Pfaffians and characteristic polynomials in random matrix theory  (2603.29510 - Akemann et al., 31 Mar 2026) in Example following Definition (Borel transform), Subsection 2.2 (First derivatives, Borel transforms and combinatorial formulas)