Paper IVStatements and proofs
Statements and proofs
Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.
Statements by chapter
Cyclic projection of analytic functions
The transfer exponential basis
Signed cyclic sections and the distinguished denominator
- Definition 5.1
- Proposition 5.2: Relation with the transfer exponential Read proof
- Theorem 5.3: Signed derivative cycle Read proof
- Corollary 5.4: Wronskian of the signed basis Read proof
- Proposition 5.5: Twisted addition law Read proof
- Theorem 5.6: Finite exponential representation Read proof
- Definition 5.7
- Corollary 5.8: Symmetries Read proof
- Example 5.9
A finite algebra and matrix model
Reciprocal powers and parameter polynomials
- Definition 7.1
- Theorem 7.2: Bell polynomial formula Read proof
- Theorem 7.3: Parameter recurrence Read proof
- Theorem 7.4: Additive parameter law for reciprocal powers Read proof
- Corollary 7.5: A sequence of binomial type Read proof
- Theorem 7.6: Logarithmic Newton recurrence Read proof
- Corollary 7.7: The first three reciprocal polynomials Read proof
- Corollary 7.8 Read proof
- Example 7.9
Logarithmic derivatives and Riccati equations
Residue-section reciprocals and Appell systems
- Definition 9.1
- Theorem 9.2: Triangular inverse system Read proof
- Definition 9.3
- Theorem 9.4: Parameter-addition law for residue-section reciprocals Read proof
- Definition 9.5
- Theorem 9.6: Appell laws Read proof
- Proposition 9.7: Explicit Appell polynomial Read proof
- Theorem 9.8: Joint Appell convolution in the space and order parameters Read proof
- Proposition 9.9: Formal differential-operator representation on polynomials Read proof
- Corollary 9.10: Appell symmetry Read proof
- Example 9.11
Scaled reciprocal coupling and theta polynomials
Growth and the geometry of the large zeros
- Standard fact 11.1: Order from Taylor coefficients
- Proposition 11.2: Orders and exponential type Read proof
- Definition 11.3
- Theorem 11.4: Indicator polygon and dominant sectors Read proof
- Lemma 11.5: Two term dominance Read proof
- Lemma 11.6: Near cancellation of two exponentials Read proof
- Theorem 11.7: Large zero strings Read proof
- Theorem 11.8: Leading displacement term and derivative sign Read proof
- Corollary 11.9: Leading correction for N=3 Read proof
- Corollary 11.10: The case N=2 Read proof
- Corollary 11.11: Eventual simplicity and exact rays Read proof
- Example 11.12: Exact fourth-order strings
- Corollary 11.13: Reduced zeros Read proof
- Theorem 11.14: Zero counting asymptotics Read proof
- Corollary 11.15: Zero exponents Read proof
Hadamard products and zero power sums
- Theorem 12.1: Genus zero product Read proof
- Theorem 12.2: Diagonal Fredholm determinant representation Read proof
- Corollary 12.3: Complete symmetric function form Read proof
- Proposition 12.4: Products for the normalized sections Read proof
- Definition 12.5
- Theorem 12.6: Newton sums for the reduced zeros Read proof
- Corollary 12.7: Convergent Mittag–Leffler partial-fraction form Read proof
- Proposition 12.8: Theta Bell polynomial formula Read proof
- Corollary 12.9: Tangent coefficients as zero sums Read proof
- Corollary 12.10: Reciprocal coefficients from zero data Read proof
- Example 12.11: The cosine orbit sums
Coefficient asymptotics from the nearest zeros
Zero zeta functions
- Definition 14.1
- Theorem 14.2: Abscissae of absolute convergence Read proof
- Theorem 14.3: Meromorphic continuation of the symmetry-orbit zero zeta Read proof
- Corollary 14.4: Exact zeta models in orders two and four Read proof
- Definition 14.5
- Corollary 14.6: Branch-dependent symmetry-orbit continuation Read proof
- Definition 14.7
- Theorem 14.8: Derivative-weighted convergence Read proof
- Remark 14.9