bibkey: witulaslota2007order11 authors: Roman Wituła; Damian Słota year: 2007 title: Quasi-Fibonacci Numbers of Order 11 doi: null url: https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.pdf claim: “The printed page-19 Problem asks whether all five order-eleven quasi-Fibonacci polynomials have degree n for every n at least 5.” strata_touched:

• D5/S1/Recurrence/WitulaSlotaQuasiFibonacciOrderElevenDegree license: citation-only triage: anchor

Quasi-Fibonacci numbers of order 11

Roman Wituła and Damian Słota define five polynomial sequences in Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences 10 (2007), Article 07.8.5. The paper lists no DOI and no arXiv version.

Verified locator

• Source URL: https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.pdf
• System (3.12) and initial values: printed page 5.
• Corollary 10: printed page 7.
• The substitution Δ := 1/δ: printed page 18.
• Problem: printed page 19, immediately after Corollary 20.

Source statement

System (3.12), printed page 5, is

$$\begin{array}{rl}{A}_{n+1}(\delta )& =A_n(\delta )+2\delta B_n(\delta )-\delta E_n(\delta ),\\ B_{n+1}(\delta )& =\delta A_n(\delta )+B_n(\delta )+\delta C_n(\delta )-\delta E_n(\delta ),\\ C_{n+1}(\delta )& =\delta B_n(\delta )+C_n(\delta )+\delta D_n(\delta )-\delta E_n(\delta ),\\ D_{n+1}(\delta )& =\delta C_n(\delta )+D_n(\delta ),\\ E_{n+1}(\delta )& =\delta D_n(\delta )+(1-\delta )E_n(\delta ),\end{array}$$

where

$$A_0(\delta )=1,B_0(\delta )=C_0(\delta )=D_0(\delta )=E_0(\delta )=0.$$

Corollary 10, printed page 7, states:

Since ${\mathrm{deg}}_{\delta }({\tau }_m^n)=n$ for every $m=1,2,\dots ,5$, by $(3.2m)$ for $m=1,2,\dots ,5$ it can be easily deduced the following formula

$$\max \{\mathrm{deg}{A}_n(\delta ),\mathrm{deg}{B}_n(\delta ),\mathrm{deg}{C}_n(\delta ),\mathrm{deg}{D}_n(\delta ),\mathrm{deg}{E}_n(\delta )\}=n.$$

The Problem on printed page 19 asks:

Problem. Is it true that $$>\mathrm{deg}{A}_n(\Delta )=\mathrm{deg}{B}_n(\Delta )=\mathrm{deg}{C}_n(\Delta )>=\mathrm{deg}{D}_n(\Delta )=\mathrm{deg}{E}_n(\Delta )=n>$$ for every $n=5,6,\dots$?

Reading

The symbols $A_n(\Delta ),\dots ,E_n(\Delta )$ denote the same five polynomial families as $A_n(\delta ),\dots ,E_n(\delta )$; Section 6 changes the variable by the substitution $\Delta :=1/\delta$. The recurrence and initial values define the polynomials over $\mathbb{Z}$. Degree is polynomial degree, with the zero polynomial excluded by an equality to the natural number $n$. The quantifier is over every natural $n\ge 5$.

Literature status

The paper leaves the displayed question as a Problem. The OpenAlex record lists nine citing works. Four accessible full texts cite the paper without mentioning this degree question. Two later papers could not be retrieved and remain ASSUMED-UNVERIFIED. OEIS A062883, A189235, and A189236 contain no settlement; MathDB has no entry for the question. Semantic Scholar returned HTTP 429 and is ASSUMED-UNVERIFIED. These readings do not establish exhaustive publication priority.