Perrier's period-one generating functions

Rachel Perrier, Multidimensional Continued Fractions and Riordan Arrays, Enumerative Combinatorics and Applications 6:4 (2026), Section 5, printed pp. 14–15, gives the period-one recurrence with initial values $r_1^{(-1)}=1$ and $r_{j+1}^{(-1)}=a_j$ for $1\le j\le k-1$:

$$ r_1^{(n+1)}=r_k^{(n)},\qquad r_{j+1}^{(n+1)}=r_j^{(n)}+m_jr_k^{(n)}. $$

The source states on printed p. 15:

We conjecture that solving these recurrences yields

$$ R_1(t)=\frac{1+\sum_{j=1}^{k-1}(a_j-m_j)t^{k-j}} {1-\sum_{j=1}^{k-1}m_jt^{k-j}-t^k},\qquad R_k(t)=\frac{t^{k-1}+\sum_{j=1}^{k-1}a_jt^{k-1-j}} {1-\sum_{j=1}^{k-1}m_jt^{k-j}-t^k}. $$

The finite sums reproduce the two printed numerators and their common denominator. The shifted convention is $R_i(t)=\sum_{n\ge0}r_i^{(n-1)}t^n$, explicitly defined on printed pp. 8 and 12. Section 5 introduces higher dimensions with the sentence ‘The same method extends to higher dimensions.’ It does not repeat the definition of the generating functions; retaining that shift is the contextual reading used here.

Set $k=d+1$. Lean time $n$ represents source time $n-1$, coordinate $j:\operatorname{Fin}(d+1)$ represents source coordinate $j+1$, and $m,a:\operatorname{Fin}(d)\to K$ likewise use zero-based indices. For every commutative ring $K$, every natural $d$, every parameter pair, and every sequence satisfying the initial values and both recurrence equations, the formal result is:

theorem result {K : Type*} [CommRing K] {d : ℕ}
    (m a : Fin d → K) (r : ℕ → Fin (d + 1) → K)
    (h : Recurrence m a r) :
    R r 0 * D m = P m a ∧ R r (Fin.last d) * D m = Q a

Here R is the power series with coefficients r; D, P, and Q are exactly the denominator and two numerators above, with constants embedded in $K[[X]]$. All finite sums are empty when $d=0$; the result includes this case. The initial vector and next-vector rule recursively determine a unique sequence for every parameter pair.

This settles the two conjectured generating-function formulas for every dimension in the period-one case. The arbitrary commutative-ring domain includes the paper's integral setting. The denominator has constant coefficient one and hence is a unit in the formal power-series ring; the multiplicative identities therefore express the printed quotients without requiring a field or division syntax.

The article proves the two- and three-dimensional cases and states the general formulas as a conjecture in Section 5. The target and the commutative-ring domain correction are recorded in preregistration #8627. This is a tier-one external conjecture, admitted under admission_basis: open-problem-resolution.

The proof shape of the sole theorem result is bind-only: coefficient identities, finite-sum reindexing, and ring normalization from pinned Mathlib discharge the statement. There is no escape witness and no separate helper theorem declaration. Its direct import is Mathlib.RingTheory.PowerSeries.Inverse; it imports no D5 module. The five definitions supply the recurrence and its series notation. Utility is none: the delivery is a symbolic identity with unbounded dimension, not a finite enumeration, certified instance, checker, or numerical reduction.

1. Translate the coefficient recurrence into power-series equations, including $R_1=1+XR_k$.
2. Multiply the successor-coordinate equations by $X^{k-1-j}$ and sum. The two decompositions of the same finite sum cancel the intermediate coordinates, leaving an equation for $R_k$.
3. Substitute the first-coordinate equation and collect terms to obtain $R_kD=Q$.
4. Normalize $D+XQ=P$ and multiply $R_1=1+XR_k$ by $D$ to obtain $R_1D=P$.

A commutative ring, dimension, parameter pair, and recurrence solution for which a coefficient of either product identity differs from its numerator would refute the formal statement. Comparing an unshifted generating function instead would test a different statement.

The complete proof is result in D5/S1/Recurrence/Algebraic/PerrierPeriodicGeneratingFunctions.lean. Its public surface consists of Recurrence, R, D, P, Q, and result. The corresponding Scribe theorem node binds this dossier to that result with OpenProblemResolutionClaim and ResolutionKind.Proved. The source quotations, printed-page locators, coefficient-domain discussion, and expanded formulas are retained in Library/Recurrence/perrier2026mcf.md.

theorem; resolution proved for the two period-one shifted generating functions in Section 5. The subsequent conjecture concerning the generalized recurrence in equation (7) is outside this result, as are periods greater than one and analytic convergence of the series.

The author's 2023 Washington State University thesis, reference [22] in the article, was not retrieved in the inherited source assessment. Whether it already proves the general case remains unverified. The article's explicit conjecture is the source of the target; historical openness beyond the documented source assessment and exhaustive novelty or priority are not established.