Pell matrix as a Riordan array

TheoremGeneralized Pell matrix as a Riordan array, for every k#✓fully proved

Let \(k\) be a natural number and let \(p(n, m)\), for natural numbers \(n\) and \(m\), be natural numbers such that for every \(n\) the power series \(\sum_{m \ge 0} p(n, m)\, x^m\) over \(\mathbb{Q}\) satisfies \(\left(\sum_{m \ge 0} p(n, m)\, x^m\right)(1 - x)^{n+1} = (1 + x + \dots + x^k)^n\). Then for all natural numbers \(n\) and \(m\), the coefficient of \(x^n\) in \(\frac{1}{1-x} \left(\frac{x(1 + x + \dots + x^k)}{1-x}\right)^m\) is \(p(m, n - m)\) if \(m \le n\) and \(0\) otherwise. That is, the lower-triangular matrix whose entry in row \(n\) and column \(m\) is \(p(m, n - m)\) is the Riordan array \(\left(\frac{1}{1-x}, \frac{x(1 + x + \dots + x^k)}{1-x}\right)\).

theorem pell_riordan_general (k : ℕ) (p : ℕ → ℕ → ℕ)
    (hgen : ∀ n : ℕ,
      PowerSeries.mk (fun m => (↑(p n m) : ℚ)) * (1 - PowerSeries.X (R := ℚ)) ^ (n + 1) =
      (Finset.sum (Finset.range (k + 1)) (fun i => PowerSeries.X (R := ℚ) ^ i)) ^ n)
    (n m : ℕ) :
    (if m ≤ n then (↑(p m (n - m)) : ℚ) else 0) =
      PowerSeries.coeff (R := ℚ) n (Ring.inverse (1 - PowerSeries.X (R := ℚ)) *
        (PowerSeries.X (R := ℚ) *
          Finset.sum (Finset.range (k + 1)) (fun i => PowerSeries.X (R := ℚ) ^ i) *
          Ring.inverse (1 - PowerSeries.X (R := ℚ))) ^ m)

Used by 1

Used byGeneralized Pell matrix as a Riordan array

TheoremGeneralized Pell matrix as a Riordan array#✓fully proved

Let \(k \ge 2\) be a natural number and let \(p(n, m)\), for natural numbers \(n\) and \(m\), be natural numbers such that for every \(n\) the power series \(\sum_{m \ge 0} p(n, m)\, x^m\) over \(\mathbb{Q}\) satisfies \(\left(\sum_{m \ge 0} p(n, m)\, x^m\right)(1 - x)^{n+1} = (1 + x + \dots + x^k)^n\). Then for all natural numbers \(n\) and \(m\), the coefficient of \(x^n\) in \(\frac{1}{1-x} \left(\frac{x(1 + x + \dots + x^k)}{1-x}\right)^m\) is \(p(m, n - m)\) if \(m \le n\) and \(0\) otherwise. That is, the lower-triangular matrix whose entry in row \(n\) and column \(m\) is \(p(m, n - m)\) is the Riordan array \(\left(\frac{1}{1-x}, \frac{x(1 + x + \dots + x^k)}{1-x}\right)\).

theorem pell_riordan :
    ∀ (k : ℕ) (hk : 2 ≤ k) (p : ℕ → ℕ → ℕ)
      (hgen : ∀ n : ℕ,
        PowerSeries.mk (fun m => (↑(p n m) : ℚ)) * (1 - PowerSeries.X (R := ℚ)) ^ (n + 1) =
        (Finset.sum (Finset.range (k + 1)) (fun i => PowerSeries.X (R := ℚ) ^ i)) ^ n),
      ∀ (n m : ℕ), (if m ≤ n then (↑(p m (n - m)) : ℚ) else 0) =
        PowerSeries.coeff (R := ℚ) n (Ring.inverse (1 - PowerSeries.X (R := ℚ)) *
          (PowerSeries.X (R := ℚ) *
            Finset.sum (Finset.range (k + 1)) (fun i => PowerSeries.X (R := ℚ) ^ i) *
            Ring.inverse (1 - PowerSeries.X (R := ℚ))) ^ m)

Sources

• Jhon J. Bravo, Jose L. Herrera, and José L. Ramírez, "Combinatorial Interpretation of Generalized Pell Numbers," Journal of Integer Sequences 23 (2020), Article 20.2.1, Theorem, lines 269–272, https://cs.uwaterloo.ca/journals/JIS/VOL23/Bravo/bravo4.tex It follows from pell_riordan_general; the hypothesis hk is unused and keeps the source's shape.

Uses 1

Proof usesGeneralized Pell matrix as a Riordan array, for every k

2 fully proved

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