bibkey: maowang2026narayana
authors: Jianxi Mao and Lijie Wang
year: 2026
title: The Narayana transformation
doi: null
url: https://arxiv.org/abs/2607.01572v1
claim: Conjecture 4.1 asks for real nonpositive roots of the row polynomials of the ordinary squares of the Eulerian and Delannoy triangles.
strata_touched:
  - D5/S1/Recurrence/Algebraic/DelannoySquareRoots
license: citation-only
triage: anchor

The original Delannoy matrix-square question

Verified locator

Source: https://arxiv.org/abs/2607.01572v1

Section 4, Concluding remarks and open problems, Conjecture 4.1, PDF page 11. The primary source TeX MW-Narayana.tex defines the path-count Delannoy triangle, sets G=D² by ordinary matrix multiplication, and asks whether the row generating functions of A² and D² have only real nonpositive roots. Only the D² clause is selected here.

Source and scope

Jianxi Mao and Lijie Wang, The Narayana transformation, arXiv:2607.01572v1, Concluding remarks and open problems, PDF page 11. The source TeX has SHA-256 933bc8b944c095032f70fc562ebd3e7beee68423797b495f28a3fe375e3df871.

The source defines D(n,k) as the number of lattice paths from (0,0) to (n-k,k) using steps (1,0), (0,1) and (1,1). The triangle has row zero [1] and is zero outside 0 <= k <= n. Its square is ordinary matrix multiplication:

[ G(n,k)=\sum_{j=k}^{n}D(n,j)D(j,k),\qquad G_n(x)=\sum_{k=0}^{n}G(n,k)x^k. ]

The source derives

[ \sum_{n\ge0}G_n(x)t^n= \frac{1-t}{(1-t)(1-2t-t^2)-xt(1+t)(1+t^2)}. ]

Conjecture 4.1 states:

The RGFs of A² and D² have only real nonpositive roots.

The selected question is the D² clause. The Eulerian clause, higher powers, simplicity and interlacing are outside its scope. The strict negative-root version is equivalent for this triangle because the constant coefficient of each squared row is positive.

The formal source counts actual ordered step words and derives the displayed generating function from their endpoint enumeration and the finite matrix product. Its endpoint exclusion says that the ordinary Delannoy row polynomial is nonzero at 1-sqrt(2) in every degree. This excludes a boundary zero in the proposed complementary root-count argument; it does not prove the conjecture by itself.

For complex u,v,y satisfying u != 0, v != 0, u+v+uv=1 and yuv=1, the formal bridge also proves, for every natural n,

[ y(v-u)(-1)^nG_n(-y)= \frac{T_n(-u)}{u^{n+1}}-\frac{T_n(-v)}{v^{n+1}}. ]

The original squared-row series is used on the left. The identity is valid in multiplied form even when u=v; its quotient use requires distinct parameters. The upper-circle count is formalized by upper_circle_roots. It produces one distinct actual squared-row root in the central positive transformed interval for each ordinary-row zero above sqrt(2)-1, with all endpoint, continuity, cancellation and injectivity conditions proved. The complementary real-interval count and final degree/factor closure remain unformalized.

The bounded source comparison distinguishes ordinary Delannoy rows from the squared rows. Chebyshev-root formulas and generic continuity or factorization theorems supply reusable ingredients, not a resolution of this original matrix-square question. No exhaustive publication-absence or priority claim is made.