Eventual sign pattern for trivariate Cauchy-type coefficients

Prove that for every fixed pair of integers j, ell >= 1 and every composition k, the coefficients C_n^{j;k;ell} satisfy C_n^{0;k;ell}<0 and C_n^{j;k;ell}>0 for all sufficiently large n, and, when k is admissible, satisfy C_n^{j;k;ell}>\binom{n+j-1}{n}/\zeta(k)^\ell for all sufficiently large n.

Background

The paper defines the trivariate coefficients C_n{j;k;ell} through the generating function (1-x){-j}(x{dep(k)}/Li_k(x))ell. These coefficients simultaneously generalize higher-order Gregory coefficients and higher-order Nörlund-type numbers.

The authors prove the conjectured sign behavior in the cases of ordinary polylogarithms and double polylogarithms, but leave the general multiple-polylogarithm case unresolved. The conjecture would establish eventual positivity properties for reciprocal multiple polylogarithms and provide lower bounds connected with reciprocals of multiple zeta values.

References

In fact, from over-whelming evidence, we are confident that the following conjecture should be true.

— Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers  (2609.11072 - Xu et al., 10 Sep 2026) in Conjecture 1, Section 1, subsection “A trivariate extension”

We would like to know if the general coefficients $C_{n}{j;;\ell}$ defined by equ:trivariateDefn have any significant arithmetic properties such as congruences or if they are related to some other interesting objects in number theory such as those discovered in .

equ:trivariateDefn:

$\frac{1}{(1-x)^j}\cdot \left(\frac{x^{\dep()}}{\Li_(x)}\right)^\ell =\sum_{n=0}^\infty C_{n}^{j;;\ell}x^n. $

— Rational Approximations for Reciprocals of Multiple Zeta Values and Trivariate Cauchy Numbers  (2609.11072 - Xu et al., 10 Sep 2026) in Section “Concluding remarks”