The Motzkin Orthogonal Polynomial Basis

Abstract

Monic orthogonal polynomials convert shifted Motzkin Hankel determinants to coefficient determinants of fixed size.

Definition 1.1 (The orthogonal polynomial recurrence).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal

Formalization. D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal (✓ 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.

Over the integer polynomial ring in t and s, the polynomials p_r(y) are defined by p_0 = 1, p_1 = y - s and p_{r+2} = (y - t)p_{r+1} - p_r for every nonnegative integer r. The variable y is distinct from the parameters t and s.

Theorem 1.2 (The monic basis expansion).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal_basis

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal_basis (✓ 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.

For every nonnegative integer n, p_n is monic of degree n, and y^n is the sum of M_{n,k}(t,s)p_k(y) over k from zero through n. Thus the Motzkin array gives the change of basis from these polynomials to monomials.

Theorem 1.3 (The coefficient determinant formula).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.hankelDet_coefficients

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.hankelDet_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.

For all nonnegative integers m and n, d_m(n,t,s) equals (-1)^(mn) times the determinant of the m by m matrix whose row i and column j entry is the coefficient of y^j in p_{n+i}(y), with both indices ranging from zero to m minus one. The matrix size is the shift m, independently of the Hankel matrix size n.

References

• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.hankelDet_coefficients
• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal
• Truth anchor: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal.orthogonal_basis
• Dependency: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelTransfer