Kurkov’s Reversed q-Bell Row Polynomials

Abstract

Kurkov’s triangular recurrence produces the reversed q-Bell row polynomials.

All indices are natural numbers, including zero, and all polynomials lie in Z[X]. The expression binom(n,k) is the natural binomial coefficient cast to an integer constant polynomial. A sum with upper index k-1 ranges over j=0,…,k-1 and is empty when k=0. The reversal operator reflects coefficients about the natural degree.

Definition 1.1 (The q-Bell row polynomials).

$$\begin{array}{r}\mathrm{qBell}\left(0\right)=1\\ \forall n\in \mathbb{N},\mathrm{qBell}\left((n+1)\right)=\sum _{k=0}^n(\mathrm{binom}\left(n,k\right)\cdot \mathrm{qBell}\left(k\right)\cdot X^k)\end{array}$$

Formalization. D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.qBell (✓ std3).

Citation. Paul D. Hanna; Mikhail Kurkov (2025). OEIS A126347 q-Bell coefficient triangle and Kurkov’s reversed-row conjecture. URL: https://oeis.org/A126347.

Commentary.

The initial row is one. Wagner’s recurrence forms row n+1 from all rows through n, weighted by binom(n,k) X^k. These are the row polynomials whose coefficients form OEIS A126347.

Definition 1.2 (Kurkov’s triangular recurrence).

$$\begin{array}{r}R\left(0,0\right)=1\\ \forall k\in \mathbb{N},R\left(0,(k+1)\right)=0\\ \forall n,k\in \mathbb{N},R\left((n+1),k\right)=R\left(n,n\right)+X^{(n+1)}\cdot (\sum _{j=0}^{k-1}(R\left(n,j\right)))\end{array}$$

Formalization. D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.kurkovR (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul D. Hanna; Mikhail Kurkov (2025). OEIS A126347 q-Bell coefficient triangle and Kurkov’s reversed-row conjecture. URL: https://oeis.org/A126347.

Commentary.

Row zero is one at column zero and zero at later columns. Every later entry adds the preceding diagonal to X^(n+1) times the prefix of row n. On the diagonal of row n+1 that prefix only reads columns j<=n, so the row-zero totalization cannot affect the theorem.

Theorem 1.3 (Kurkov’s reversed-row identity).

$$\forall n\in \mathbb{N},R\left(n,n\right)=\mathrm{reverse}\left(\mathrm{qBell}\left((n+1)\right)\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.result (✓ std3). ∎

Resolves. Problems/oeis-a126347-kurkov-reversed-qbell-row-polynomial (proved) by D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna; Mikhail Kurkov (2025). OEIS A126347 q-Bell coefficient triangle and Kurkov’s reversed-row conjecture. URL: https://oeis.org/A126347.

Commentary.

For every n, the diagonal entry R(n,n) is the coefficient reversal of qBell(n+1). The q-Bell polynomial is monic of degree binom(n+1,2), so natural-degree reversal agrees with reversal of the fixed-length OEIS row. A Pascal expansion of the R-prefix sums supplies the binomial convolution used by Wagner’s recurrence.

References

• Truth anchor: D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.kurkovR
• Truth anchor: D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.qBell
• Truth anchor: D5/S1/Recurrence/Algebraic/KurkovReversedQBellRowPolynomial.result