Special Functions

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1 \FrontChapter{References}\label{references}
2 The references below identify classical definitions, proof inputs and adjacent
3 constructions. They do not constitute an exhaustive priority review. The
4 arithmetic seed and the Gamma--Laguerre comparisons inherited from the
5 project are documented in the preserved research archive.
6 
7 \begingroup
8 \makeatletter
9 \patchcmd{\thebibliography}{\chapter*{\bibname}}{}{}{\PackageError{RC}{Bibliography heading patch failed}{}}
10 \makeatother
11 \begin{thebibliography}{15}
12 \setlength{\itemsep}{11pt}
13 \setlength{\parskip}{3pt}
14 \bibitem{Mendez2024} Miguel A. Méndez.
15 \emph{Combinatorial differential operators in: Faà di Bruno formula,
16 enumeration of ballot paths, enriched rooted trees and increasing rooted trees}.
17 Australasian Journal of Combinatorics 89(3) (2024), 350--384.
18 Example 2.4 contains the diagonal feedback equation.
19 \url{https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p350.pdf}
20 
21 \bibitem{Dascaliuc2025} Radu Dascaliuc, Tuan N. Pham, Enrique Thomann and Edward C. Waymire.
22 \emph{Transformation of Stochastic Recursions and Critical Phenomena in the
23 Analysis of the Aldous-Shields-Athreya Cascade and Related Mean Flow Equations}.
24 arXiv:2411.00629v2 (2025).
25 The nonlinear equation $u'=-u+u(\alpha t)^2$ is an adjacent model, not the
26 equation studied here.
27 \url{https://arxiv.org/abs/2411.00629v2}
28 
29 \bibitem{Coffey2006} Mark W. Coffey.
30 \emph{On the coefficients of the Baez-Duarte criterion for the Riemann
31 hypothesis and their extensions}. arXiv:math-ph/0608050 (2006).
32 Its signed Möbius weights differ from the positive squarefree weights used here.
33 \url{https://arxiv.org/abs/math-ph/0608050}
34 
35 \bibitem{DLMF} NIST Digital Library of Mathematical Functions.
36 Sections 5.5 and 5.12 give the gamma reflection and beta identities.
37 Section 10.46 defines the Mittag--Leffler function. Sections 16.8 and 16.21
38 give the generalized hypergeometric and Meijer $G$ differential equations.
39 Section 18.2 develops general orthogonal-polynomial theory.
40 \url{https://dlmf.nist.gov/}
41 
42 \bibitem{Uvarov1969} V. B. Uvarov.
43 \emph{The connection between systems of polynomials that are orthogonal
44 with respect to different distribution functions}.
45 USSR Computational Mathematics and Mathematical Physics 9(6) (1969), 25--36.
46 \url{https://doi.org/10.1016/0041-5553(69)90124-4}
47 
48 \bibitem{Elkies} Noam D. Elkies.
49 \emph{Functions of finite order: product formula and logarithmic derivative}.
50 Harvard Math 259 course notes. The finite-order Hadamard theorem supplies the
51 classical factorization input.
52 \url{https://people.math.harvard.edu/~elkies/M259.02/prod.pdf}
53 
54 \bibitem{Byrnes2013} Alyssa Byrnes, Lin Jiu, Victor H. Moll and Christophe Vignat.
55 \emph{Recursion Rules for the Hypergeometric Zeta Functions}.
56 arXiv:1305.1892 (2013). Hypergeometric zero spectra and coefficient recursions
57 provide related constructions.
58 \url{https://arxiv.org/abs/1305.1892}
59 
60 \bibitem{Voros2003} André Voros.
61 \emph{Zeta functions for the Riemann zeros}.
62 Annales de l'Institut Fourier 53(3) (2003), 665--699.
63 Sections 2.1--2.2 develop related canonical-product and Mellin methods.
64 \url{https://doi.org/10.5802/aif.1955}
65 
66 \bibitem{Voros2004} André Voros.
67 \emph{Erratum: Zeta functions for the Riemann zeros}.
68 Annales de l'Institut Fourier 54(4) (2004), 1139.
69 \url{https://doi.org/10.5802/aif.2046}
70 
71 \bibitem{OEIS} The OEIS Foundation Inc.
72 \emph{The On-Line Encyclopedia of Integer Sequences}, sequence A000182.
73 This is the tangent-number anchor. No unmatched-prefix novelty claim or
74 submission to OEIS is asserted.
75 \url{https://oeis.org/A000182}
76 
77 \bibitem{Alrabaiah2020} Hussam Alrabaiah, Israr Ahmad, Kamal Shah and Ghaus Ur Rahman.
78 \emph{Qualitative analysis of nonlinear coupled pantograph differential
79 equations of fractional order with integral boundary conditions}.
80 Boundary Value Problems 2020, article 138.
81 This is a coupled fractional boundary-value system with different hypotheses.
82 \url{https://doi.org/10.1186/s13661-020-01432-2}
83 
84 \bibitem{Blasiak2011} Pawel Blasiak and Philippe Flajolet.
85 \emph{Combinatorial Models of Creation--Annihilation}.
86 Séminaire Lotharingien de Combinatoire 65 (2011), B65c.
87 Proposition 24 and Note 25 discuss multitype increasing trees and differential systems.
88 \url{https://arxiv.org/abs/1010.0354v3}
89 
90 \bibitem{Mauldin1988} R. Daniel Mauldin and S. C. Williams.
91 \emph{Hausdorff Dimension in Graph Directed Constructions}.
92 Transactions of the American Mathematical Society 309(2) (1988), 811--829.
93 Its spectral-radius balance describes a geometric dimension. The analytic
94 order law here requires its own proof.
95 \url{https://sites.math.unt.edu/~mauldin/papers/no67.pdf}
96 
97 \bibitem{Knessl2006} Charles Knessl and Wojciech Szpankowski.
98 \emph{Binary Trees, Left and Right Paths, WKB Expansions, and Painlevé Transcendents}.
99 Proceedings of ANALCO (2006), 198--204.
100 The ordinary generating-function identity has different normalization and
101 coefficient rules from the differential system here.
102 \url{https://www.cs.purdue.edu/homes/spa/papers/analco06.pdf}
103 
104 \bibitem{Schilling2010} René L. Schilling, Renming Song and Zoran Vondraček.
105 \emph{Bernstein Functions: Theory and Applications}.
106 First edition. De Gruyter, 2010. ISBN 978-3-11-021530-4.
107 Proposition 1.2 and Theorem 1.4 give the classical Laplace uniqueness and
108 Bernstein representation results.
109 \url{https://doi.org/10.1515/9783110215311}
110 \end{thebibliography}
111 \endgroup

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