Divided differences over even and odd integer nodes are 2-adic units

Abstract

Let z be distinct integer nodes indexed by a finite type, even on a set A of indices and odd off A, and let alpha = #A and beta = #A^c. If the binomial coefficient C(alpha + beta - 2, alpha - 1) is odd, then the partial divided-difference sum over the even nodes, the sum over i in A of 1 / prod_{j != i} (z i - z j), is a nonzero rational number of 2-adic valuation 0.

Theorem 1.1 (The even-node partial sum is a 2-adic unit).

$$\forall iota\in Type,[\mathrm{Fintype}\left(iota\right)],[\mathrm{DecidableEq}\left(iota\right)],\forall z\in iota\to \mathbb{Z},\mathrm{Injective}\left(z\right)\Rightarrow (\forall A\in \mathrm{Finset}\left(iota\right),\mathrm{Nonempty}\left(A\right)\Rightarrow ((\forall i\in A,\mathrm{Even}\left(z\left(i\right)\right))\Rightarrow ((\forall i\in iota,\left(\neg (i\in A)\right)\Rightarrow (\mathrm{Odd}\left(z\left(i\right)\right)))\Rightarrow (\mathrm{Odd}\left(\mathrm{choose}\left(\mathrm{card}\left(iota\right)-2,\mathrm{card}\left(A\right)-1\right)\right)\Rightarrow (\sum _{i\in A}(\prod _{j\in \mathrm{erase}\left(\mathrm{univ},i\right)}((z\left(i\right)-z\left(j\right):\mathbb{Z}):\mathbb{Q}){)}^{-1}\ne 0\wedge \mathrm{padicValRat}\left(2,\sum _{i\in A}(\prod _{j\in \mathrm{erase}\left(\mathrm{univ},i\right)}((z\left(i\right)-z\left(j\right):\mathbb{Z}):\mathbb{Q}){)}^{-1}\right)=0)))))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/ParityNodeDividedDifference.evenOdd_dividedDifference_twoAdicUnit (✓ std3). ∎

Source. Repository-derived.

Commentary.

Write the even nodes as z i = 2 mu i. The sum equals 2^(1 - alpha) times the divided difference, over the integer nodes mu, of g(2 mu) with g(x) = prod_{j not in A} 1 / (x - z j). In the 2-adic integers every z j with j not in A is a unit u_j^(-1), and 1 / (x - z j) agrees at the even nodes, modulo 2^N, with the truncated geometric series -u_j sum_{t < N} (u_j x)^t. The divided difference of mu^i over alpha distinct integer nodes is the coefficient of degree alpha - 1 of the remainder of X^i modulo the monic nodal polynomial: it is an integer, it vanishes for i < alpha - 1 and it equals 1 for i = alpha - 1. Choosing N larger than the 2-adic valuations of the nodal products, the sum is congruent modulo 2 to the coefficient of x^(alpha - 1) in prod_{j not in A} (-u_j sum_{t < N} (u_j x)^t). Modulo 2 this product is (1 + x + … + x^(N-1))^beta, whose coefficient of degree alpha - 1 is C(alpha + beta - 2, alpha - 1). An odd coefficient makes the sum a 2-adic unit. The binomial index is written card(iota) - 2 with natural-number subtraction, which equals alpha + beta - 2 whenever alpha + beta >= 2; for a single node it is 0, matching the value 1 of the sum.

References

• Truth anchor: D5/S3/Quantum/Dynamics/ParityNodeDividedDifference.evenOdd_dividedDifference_twoAdicUnit