Israel’s Laguerre Polynomial Parity Conjectures

Abstract

Laguerre(n,4) has odd reduced numerator and denominator for every natural n.

Definition 1.1 (The Laguerre polynomial at four).

$$\forall n\in \mathbb{N},L\left(n\right)=\sum _{k=0}^{n}binomial\left(n,k\right)\cdot (-4{)}^k/k!$$

Formalization. D5/S3/ArithSums/IsraelLaguerreFourParity.L (✓ std3).

Citation. N. J. A. Sloane; Robert Israel (2018). OEIS A160627, Numerator of Laguerre(n, 4). URL: https://oeis.org/A160627.

Commentary.

The finite binomial sum is the classical Laguerre polynomial evaluated at x=4. The operator binomial denotes Nat.choose, and the slash denotes rational division after the natural factorial is cast to Q.

Theorem 1.2 (Odd numerator and denominator).

$$\forall n\in \mathbb{N},(Odd\left(num\left(L\left(n\right)\right)\right))\wedge (Odd\left(den\left(L\left(n\right)\right)\right))$$

Proof. Machine-checked in Lean as D5/S3/ArithSums/IsraelLaguerreFourParity.result (✓ std3). ∎

Resolves. Problems/oeis-a160627-israel-laguerre-four-parity (proved) by D5/S3/ArithSums/IsraelLaguerreFourParity.result.

Source. Repository-derived.

Acknowledgement. N. J. A. Sloane; Robert Israel (2018). OEIS A160627, Numerator of Laguerre(n, 4). URL: https://oeis.org/A160627.

Commentary.

Every nonconstant summand has positive 2-adic valuation because the valuation of k factorial is less than k. The ultrametric sum law therefore gives valuation zero for L(n). Reducedness then excludes a factor of two from both num(L(n)) and den(L(n)).

References

• Truth anchor: D5/S3/ArithSums/IsraelLaguerreFourParity.L
• Truth anchor: D5/S3/ArithSums/IsraelLaguerreFourParity.result