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Formal Reciprocal Branches and Polynomial Exponents

Abstract

Distinct reciprocal roots give two formal branches, and coefficients of integral unit powers are polynomial in the exponent.

Theorem 1.1 (The two reciprocal formal branches).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelBranches.formal_branches

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelBranches.formal_branches (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let K be a field, let phi be a ring homomorphism from the integer polynomial ring in t and s to K, and let alpha be nonzero with alpha - alpha inverse nonzero and phi(t) = alpha + alpha inverse. There is a formal power series z(y) with constant coefficient alpha and z + z inverse = phi(t) - y. The constant coefficient of z - z inverse is nonzero. Among series with constant coefficient alpha, z is the unique solution of w squared - (phi(t) - y)w + 1 = 0. Put a = (phi(s) - y - z inverse)/(z - z inverse) and b = (z - phi(s) + y)/(z - z inverse). For every nonnegative integer r, (-1)^r p_r(y) after applying phi to its coefficients equals a z^r + b(z inverse)^r, while (-1)^(r+1) b_r(y) after the same coefficient map equals a(z inverse)^(r+1) + b z^(r+1). All these identities are formal power series identities over K.

Theorem 1.2 (Polynomial dependence on an integral exponent).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelBranches.unit_power_coefficients

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelBranches.unit_power_coefficients (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let K be a field of characteristic zero, let u be an invertible formal power series with constant coefficient one, let H be any formal power series over K, and let h be a nonnegative integer. There is a polynomial P over K of natural degree at most h such that, for every integer n, P evaluated at n equals the coefficient of y^h in H(y)u(y)^n. The statement includes negative exponents, interpreted using the inverse of u.

References