Open research questions in Mathematical functions and polynomials
92 gap statements mined from Mathematical functions and polynomials papers in our 4.5M-paper local library, which holds 486 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The lack of a comprehensive treatment of ladder operators for Laguerre-type and Jacobi-type orthogonal polynomials. The need for a generalization of existing results from λ > 0 to λ > -1.
Future research can focus on further developing the theory of multiple orthogonal polynomials under the Christoffel transform. Future research can also explore the applications of the results in other fields. Future research can investigate the properties of multiple orthogonal polynomials under other types of transforms.
The paper identifies a gap in the current research as the need for a deeper understanding of the properties of multiple orthogonal polynomials under the Christoffel transform. The paper also identifies a gap in the existing literature on the Christoffel transform and its applications.
The authors suggest that further research could explore the applications of symmetric truncated Freud polynomials. They note that the electrostatic interpretation of the zeros of the polynomials could be developed further. The paper's contributions could also be extended to other families of orthogonal polynomials.
The authors note that the properties and equations satisfied by these polynomials are not well understood. The paper aims to fill this gap by deriving new results and providing a deeper understanding of these polynomials.
The problem of approximate polynomial decomposition with regard to a given norm is not well-studied for multivariate polynomials. There is a need for an algorithm that can compute an approximate decomposition of a multivariate polynomial in the Hamming distance.
The paper suggests that future research could focus on understanding the properties of elliptic orthogonal polynomials. The study notes that the results have implications for several areas, including point processes and random matrices, and that further research could explore these connections.
Recurrence relations and the Christoffel-Darboux formula for a special class of elliptic orthogonal polynomials · 2026 · DOIThe lack of a general method to construct the tau functions for the constrained CKP hierarchy. The need to investigate the CKP Darboux transformation by the results of the KP Darboux transformation.
The tau functions of the constrained CKP hierarchy · 2026The gap is the lack of exact representations of the associated complex exponents. The gap is the lack of understanding of the behavior of truncated expansions and the structure of their remainder terms.
The dimension of the space of k-modified harmonic polynomials is not known for exceptional values of k. Prior work has not completed the dimensional studies for these exceptional values.
The lack of a purely analytic proof of the half-shift reflection identity for the digamma function. The need for a deeper understanding of the properties of the digamma function.
The need for a deeper understanding of the relationships between different combinatorial models. The lack of connections to representation theory and lattice paths.
The paper does not provide a detailed analysis of the applications of the results. The methodology used is limited to the study of the pre-critical phase of the underlying two-dimensional Coulomb gas system.
The lack of a Rademacher-type exact formula for cubic overpartitions. The need to derive higher-order Turán inequalities for the cubic overpartition function.
Rademacher-type exact formula and higher order Turán inequalities for cubic overpartitions · 2026 · DOIFuture research can apply the results to various mathematical fields, such as combinatorics and geometry. The paper's contributions can be used to study other refinements of Ehrhart polynomials.
The gap in current research is the lack of understanding of the algebraic structure of ehrλ P (q, t). The paper identifies the need for a new perspective on Chapoton's work.
The lack of a method to determine the approximation of the Appell polynomials in terms of other special polynomials. The need for a powerful tool for approximating transcendental functions.
The lack of a generalization of the known lower and upper Legendre conjugate for weight functions. The need for a deeper understanding of the properties and applications of the generalized Legendre conjugates.
The paper notes that the theory of orthogonal polynomials on the complex plane is well-established, but the study of elliptic orthogonal polynomials is less developed.
Recurrence relations and the Christoffel-Darboux formula for a special class of elliptic orthogonal polynomials · 2026 · DOIThe method requires a modified basis to reduce cancellation.
To explore monomiality conditions and present additional illustrative examples. To develop the formalism further to establish Padé approximants using the umbral notation.
The analysis of the asymptotic properties of the orthogonal polynomials. The application of the derived ladder operators to other types of orthogonal polynomials.
To analyze the difference between the true solution and the output of the algorithm for indecomposable polynomials. To improve the algorithm to work well for polynomials with a large number of terms.
To further study the properties of the tau functions for the constrained CKP hierarchy. To apply the results to 2D conformal field theories, topological field theories, and matrix models.
The tau functions of the constrained CKP hierarchy · 2026Study of the applications of the generalized Legendre conjugates in different contexts. Analysis of the properties and applications of the generalized Legendre conjugates in specific weighted spaces.
Most-cited papers in Mathematical functions and polynomials
- Handbook of Mathematical Functions · American Journal of Physics · 1966 · 1,921 citations
- Line spectrum representation of linear predictor coefficients of speech signals · The Journal of the Acoustical Society of America · 1975 · 431 citations
- Faber Polynomials and the Faber Series · American Mathematical Monthly · 1971 · 241 citations
- A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables · Computational Statistics & Data Analysis · 2008 · 203 citations
- Some Generalizations of Vandermonde's Convolution · American Mathematical Monthly · 1956 · 187 citations
- A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa · The American Statistician · 1995 · 159 citations
- A Generalization of Hermite's Interpolation Formula · American Mathematical Monthly · 1960 · 130 citations
- Gershgorin's Theorem and the Zeros of Polynomials · American Mathematical Monthly · 1965 · 128 citations
- How to Integrate A Polynomial Over A Sphere · American Mathematical Monthly · 2001 · 99 citations
- Derivatives of Noninteger Order · Mathematics Magazine · 1995 · 98 citations
Most recent work
- Christoffel transform and multiple orthogonal polynomials · Journal of Computational and Applied Mathematics · 2026
- Inhomogeneous SSH models and the doubling of orthogonal polynomials · SciPost Physics · 2026
- Generalized upper and lower Legendre conjugates for weight functions · Bulletin of the Belgian Mathematical Society - Simon Stevin · 2026
- Distributions of zeros of a complex polynomial · Journal of Mathematical Analysis and Applications · 2026
- Zeros and Exponential Profiles of Polynomials II: Examples · Constructive Approximation · 2026
- On the accuracy of averaged Gauss quadrature rules · Journal of Computational and Applied Mathematics · 2026
- A mixed interpolation-regression method for numerical integration on the unit circle using zeros of para-orthogonal polynomials · Journal of Computational and Applied Mathematics · 2026
- Interpolatory quadrature rules over general non-rectangular domains · Applied Mathematics and Computation · 2026
- Necessary and Sufficient Conditions for Absolute Monotonicity of Functions Related to Gaussian Hypergeometric Functions · Mediterranean Journal of Mathematics · 2026
- Zeros of Linear Combinations of Orthogonal Polynomials · Mediterranean Journal of Mathematics · 2026
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