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Open research questions in Mathematical functions and polynomials

49 unresolved questions extracted from the limitations and future-work sections of 267 Mathematical functions and polynomials papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • 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 · DOI

  • The study identifies a gap in the understanding of elliptic orthogonal polynomials. 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 · DOI

  • Existing approaches for evaluating integrals of products of Hermite polynomials have limitations. These limitations include numerical instabilities stemming from large-integer factorials and quadrature methods.

    A general recursion for integrals involving products of Hermite polynomials and its applications · 2026 · DOI

  • The finite-order gap left by recent investigations. The lack of understanding of the relation between the exact finite threshold and the zero geometry of the Riemann Xi function.

    Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel · 2026 · DOI

  • The difficulty of proving the identity using topographs. The need for a purely analytic proof of the identity.

    A half-shift reflection identity for the digamma function · 2026 · DOI

  • To extend the results to handle multiple-root boundary events. To understand the implications of the results for number theory and mathematics.

    Paper 134H–M: Recursive Jensen Corridors and Intrinsic Moment Geometry for the Riemann Xi Function Exact normalized derivative nesting, critical-area hyperbolicity criteria, and Riemann-kernel interpretations · 2026 · DOI

  • Finite hyperbolicity is not well understood. The Riemann Hypothesis is a long-standing problem in mathematics.

    Paper 134H–M: Recursive Jensen Corridors and Intrinsic Moment Geometry for the Riemann Xi Function Exact normalized derivative nesting, critical-area hyperbolicity criteria, and Riemann-kernel interpretations · 2026 · DOI

  • The paper establishes that perfectness of systems ((Φ₁...Φₛ)ν, ωₛ) is necessary for the CC algorithm to not break down in the repeated Christoffel transform setting. However, no algorithm or criterion is provided to verify perfectness for arbitrary measure systems beyond the sufficient conditions given (Angelesco systems or AT systems with spectral support constraints).

    Christoffel transform and multiple orthogonal polynomials · 2026 · DOI

  • The relationship between equation (83) for Type I polynomials (Kₖ(z₀, x) = −Pₖ(z₀)Âₖ(x)) and equation (84) for Type II polynomials is noted in Remark 3.18, but the algebraic structure underlying these dual formulas and potential generalizations to higher-order Christoffel transforms are not fully developed.

    Christoffel transform and multiple orthogonal polynomials · 2026 · DOI

  • The differential operator Ln in Corollary 4 satisfies LnPn(x; z) with explicit coefficients depending on γn and γn+1, but the spectral properties of this operator (eigenvalue distribution, self-adjointness conditions) and their implications for the polynomial zeros have not been investigated.

    Symmetric truncated Freud polynomials · 2026 · DOI

  • The relationship between the linear functional v defined in Section 7.3 and the original linear functional u through the scaling transformation Qn(x; z) = z^{-n/4}Pn(z^{1/4}x; z) is established algebraically in equation (80), but the geometric or physical interpretation of this transformation in the context of symmetric truncated Freud polynomials remains unexplored.

    Symmetric truncated Freud polynomials · 2026 · DOI

  • The greedy strategy in Algorithm 2 (Steps 3-6) updates the remainder δ by maximizing zero coefficients per iteration, but there is no analysis of whether this greedy approach yields near-optimal approximate decompositions or if alternative coefficient selection strategies might produce better approximations in Hamming distance.

    An Approximate Decomposition of a Multivariate Polynomial and Its Application · 2026 · DOI

  • Algorithm 1 requires computing coefficients c_{−j_d}^{s_d} with a specific hierarchical dependency structure amenable to parallelization (Remark 9), but no parallel implementation complexity analysis, communication costs, or actual parallel performance metrics are provided for multicore or distributed computing environments.

    An Approximate Decomposition of a Multivariate Polynomial and Its Application · 2026 · DOI

  • The 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.

    Joseph Heyward, Meaza Bogale, Daniel Ntiamoah and Alemayehu Negash · 2026 · DOI

  • The singularity swap quadrature method suffers from catastrophic cancellation for near-singular line integrals. The method requires a modified basis to reduce cancellation.

    Stabilizing the singularity swap quadrature for near-singular line integrals · 2026 · DOI

  • 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.

    On k-modified harmonic polynomials and their dimensions · 2026 · DOI

  • 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.

    A half-shift reflection identity for the digamma function · 2026 · DOI

  • 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.

    Orthogonal Polynomials in the Spherical Ensemble with Two Insertions · 2026 · DOI

  • The paper suggests that future research should focus on the study of the applications of the results. The paper suggests that future research should focus on the study of other problems in the field using the methodology introduced in the paper.

    Orthogonal Polynomials in the Spherical Ensemble with Two Insertions · 2026 · DOI

  • 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 · DOI

  • Future 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.

    A Closer Look at Chapoton’s q-Ehrhart Polynomials · 2026 · DOI

  • 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.

    A Closer Look at Chapoton’s q-Ehrhart Polynomials · 2026 · DOI

  • To explore monomiality conditions and present additional illustrative examples. To develop the formalism further to establish Padé approximants using the umbral notation.

    A novel advancement in the study of Appell polynomials via Padé rational approximants · 2026 · DOI

  • 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.

    A novel advancement in the study of Appell polynomials via Padé rational approximants · 2026 · DOI

  • Study 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.

    Generalized upper and lower Legendre conjugates for weight functions · 2026 · DOI

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49 open questions have been extracted from the limitations and future-work passages of 267 Mathematical functions and polynomials papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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