Limiting distributions for the remaining Graham–Knuth–Patashnik parameter regimes

Determine the limiting distributions of suitably normalized integer-valued random variables associated with Graham–Knuth–Patashnik recurrences in parameter regimes other than b1 = 0 with the remaining five parameters non-negative.

Background

The paper studies random variables encoded by the generating polynomials associated with Graham–Knuth–Patashnik recurrences involving six parameters b1, b, b3, b, b, and b. It gives a complete description of the limiting behavior when b = 0 and the other five parameters are non-negative.

The authors explicitly identify the cases outside that parameter regime as unresolved. Thus, the open problem concerns extending the limiting-distribution theory to the remaining parameter choices, including cases with b = 0 or with parameters outside the stated non-negative range.