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Mathematics
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Special Functions, Representations and Applications
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Special Functions, Representations and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C: Mathematical Analysis".

Deadline for manuscript submissions: 31 January 2027 | Viewed by 129

Editor

Prof. Dr. Shigeru Kanemitsu

E-Mail Website
Guest Editor
Mathematical Research Center, Shandong University, Jinan 250100, China
Interests: number theory; special functions; coding theory; negative capacitance condenser
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This new Special Issue is devoted to special functions, representation theory and their applications. Most phenomena occurring in nature are modelled by second order non-linear differential equations (DEs) and some of the solutions are expressed as special functions. Thus, special functions are ubiquitous in the research of science and the range of their applications have been ever growing; however, the development of totally new directions is inactive. To revive innovative applications, it may be necessary to scrutinize the underlying structure and magnify it by representations, thus the motivation of this Special Issue. We also welcome papers which are loosely related to the doctrine; for there are more things in the world than our philosophy can dream of and we aim to make proper and effective use of them to fully explain the underlying structure of the associated DEs. Our philosophy is stimulated by the famous book of N. Ya. Vilenkin, "Special functions and the theory of group representations’’, 1968, in which most of the important special functions, including Bessel functions, Hypergeometric functions, etc., are discussed with their associated group of motions. As we are not very strictly limited to this framework, we call for competitive papers on special functions, representations (of algebraic systems), and their applications to science and technology. Naturally, the scope includes polynomials as a special and important family of special functions. We hope this Special Issue will widen and deepen the research spirit in related fields.

Prof. Dr. Shigeru Kanemitsu
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • special functions
  • Vilenkin
  • group representations
  • applications

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Published Papers (1 paper)

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Research

22 pages, 447 KB  
Article
Parity Bifurcation, PIII(D6) Topology, and a Stieltjes Framework to Jensen Polynomial Hyperbolicity
by Michel Planat
Mathematics 2026, 14(13), 2240; https://doi.org/10.3390/math14132240 (registering DOI) - 23 Jun 2026
Abstract
We investigate the onset of hyperbolicity in Jensen polynomials Jd,n associated with the Riemann Ξ-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees 5≤d≤16 reveals two distinct regimes. [...] Read more.
We investigate the onset of hyperbolicity in Jensen polynomials Jd,n associated with the Riemann Ξ-function and identify a robust parity-driven bifurcation with a natural geometric interpretation. Numerical analysis for degrees 5≤d≤16 reveals two distinct regimes. For even d, the roots form a compact complex cluster whose imaginary extent decreases smoothly, and the transition to hyperbolicity is governed by a single complex-conjugate pair, consistent with a low-dimensional (tame) geometric structure. For odd d, a hierarchy of more intricate onset mechanisms emerges, including single-event transitions (d=11) and intermittent regimes (d≥13) with decoupled geometric invariants, suggestive of dynamics on decorated (wild) character varieties. We interpret this dichotomy through a connection with the PIII(D6) tau-function arising in the Painlevé confluence diagram. Defining τ(t)=Ξ(12+−t)/Ξ(12), we construct a generating function B(w)=∑j≥0bjwj from the cumulants of logΞ(12+z) using high-precision Cauchy/DFT methods (280–400-digit arithmetic), without explicit use of the zero expansion. Two independent numerical diagnostics indicate that B exhibits Stieltjes-type behavior: (i) positivity of Hankel determinants up to order N=30 and (ii) Padé approximants whose poles converge to γk2 (squares of Riemann-zero ordinates) with stabilizing residues. These results provide strong evidence that the parity bifurcation observed in Jensen polynomial onset reflects a finite-dimensional manifestation of an underlying moment-based positivity structure. Motivated by this correspondence, we formulate a conjecture relating the Stieltjes nature of B(w) to the eventual hyperbolicity of Jensen polynomials. This conjecture suggests a bridge between finite-dimensional root geometry and an infinite-dimensional kernel-based positivity framework, while leaving open the problem of establishing such positivity independently of the zero expansion. Full article
(This article belongs to the Special Issue Special Functions, Representations and Applications)
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