Open research questions in Advanced Mathematical Identities
113 gap statements mined from Advanced Mathematical Identities papers in our 4.5M-paper local library, which holds 845 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 need for new techniques to prove q-supercongruences. The lack of a parametric version of Theorem 1.5. The need for further generalization of q-supercongruences.
The nonnegativity conjecture of Merca was previously proven for d ≥ 3, but the case d = 2 remained open. The paper aims to fill this gap by proving the conjecture for d = 2.
The existing literature has not fully explored the case where n is a positive integer congruent to 3 modulo 4. There is a need for generalizations of existing supercongruences. The paper identifies a gap in the understanding of q-supercongruences and their relation to the p-adic Gamma function.
The study of q-multiple t-functions of level 3 at roots of unity is not well understood. Prior work has not derived explicit expressions for the values of these functions. The paper identifies a need for new techniques and methods to study these functions.
The existing proofs for the congruences have limitations. The case 11 was viewed as substantially different from 5 and 7.
The relation J₃ = 11³F₃z₁₁ (equation 81) is proved using pole and zero multiplicity arguments, and the paper states this relation 'cannot only be proved but also derived algorithmically' via Lemma 9.6, yet the explicit algorithmic procedure for deriving such congruence relations between module generators and basis elements for general level remains unspecified.
The need for new identities of theta functions at level 16. The lack of a unified analytic combinatorial framework at level 16. The potential for applying q-series techniques to partition theory.
Further research could explore properties of q-Leonardo polynomials. The study of relationships between q-Leonardo polynomials and other polynomials could be continued. The paper may inspire research on q-analogues of other polynomials.
The lack of a proof for Mathar's 2020 conjecture. The need for a self-contained derivation of the harmonic-number closed form.
A short proof of Mathar's 2020 recurrence conjecture for the generalized-Stirling sequence A001711 · 2026Future research can focus on applying the transformation to other areas of mathematics and physics. Future research can also focus on obtaining new identities and evaluations of special series using the transformation.
A Transformation Between r-Stirling and Binomial Coefficient Series Via Alternating Multiple Zeta Values · 2026 · DOIThe lack of understanding of the properties of two-color partitions with even parts in one color. The need for a systematic study of two-color partitions using q-series techniques.
The methods and analysis used are limited to the case k≡2(mod 4). The case k≡0(mod 4) is a consequence of the work of Bennett-Ellenberg-Ng about the generalized Fermat equation of signature (2,4, n).
Sum of consecutive powers as a perfect power · 2026The paper suggests that future research could focus on applying the results to specific fields of research, such as genetics or finance. The authors note that the multivariate formulation could be used to handle high-dimensional problems and conditional constraints. Future research could also explore the use of the expressions for U N, r, j in other areas, such as machine learning or data analysis.
Future research can focus on applying the results to non-commutative algebra, quantum mathematics, and discrete dynamical systems. Further study of the properties of q-Leonardo tessarines can be conducted.
The classical theory of Leonardo numbers has not been extended to the tessarine framework through the incorporation of q-calculus. There is a lack of understanding of the algebraic properties of q-Leonardo tessarines.
The paper suggests that higher (1 + q)-adic refinements of E_{2n}(q) should require a systematic description of canonical representatives with a prescribed small number of free natural pairs. It also mentions the need for higher-rank analogues of the sliding involution.
To explore the potential applications of the results to other areas of mathematics and computer science.
Investigate the conjecture that the supercongruence holds modulo p^3r-1 or p^3r-2 - Generalize the q-supercongruence (1.4) to positive odd integers m…
The case d = 2 was previously left open - The proof relies on Lemma 2.4 - Certain cases were checked via Mathematica - The result is limited to integer…
Further study of the q-multiple t-values - Investigation of the properties of the (3, 1)-step q-Stirling numbers - Exploration of the relationship between the q-multiple t-values and the q-Stirling…
Investigate the q-analogs of supercongruences - Explore the applications of the q-supercongruences in other…
Further exploration of the properties of complete homogeneous symmetric polynomials. Application of the generalized Vieta formulas in various mathematical areas.
The paper identifies a gap in the understanding of q-analogues of Leonardo polynomials. The study aims to fill this gap by introducing q-Leonardo Pisano and q-Leonardo Lucas polynomials of the first and second kind.
The paper identifies a gap in the literature on the transformation between r-Stirling and binomial coefficient series. The paper aims to fill this gap by establishing a general transformation and obtaining new identities and evaluations.
A Transformation Between r-Stirling and Binomial Coefficient Series Via Alternating Multiple Zeta Values · 2026 · DOITo study the equation x^k + (x+1)^k = y^n for k>100. To apply the methods and analysis used to other problems in number theory.
Sum of consecutive powers as a perfect power · 2026
Most-cited papers in Advanced Mathematical Identities
- A chronometric analysis of simple addition. · Psychological Review · 1972 · 459 citations
- Fibonacci Series Modulo m · American Mathematical Monthly · 1960 · 414 citations
- An Algorithm for the Evaluation of Finite Trigonometric Series · American Mathematical Monthly · 1958 · 411 citations
- A Remark on Stirling's Formula · American Mathematical Monthly · 1955 · 328 citations
- Tables of Integrals, Series, and Products · American Journal of Physics · 1988 · 284 citations
- The Formula of Faa Di Bruno · American Mathematical Monthly · 1980 · 284 citations
- A Generalized Fibonacci Sequence · American Mathematical Monthly · 1961 · 253 citations
- Ramanujan's Extensions of the Gamma and Beta Functions · American Mathematical Monthly · 1980 · 230 citations
- Binomial Coefficients Modulo a Prime · American Mathematical Monthly · 1947 · 202 citations
- An Elementary View of Euler's Summation Formula · American Mathematical Monthly · 1999 · 176 citations
Most recent work
- Modular forms and certain ₂𝐹₁(1) hypergeometric series · Proceedings of the American Mathematical Society · 2026
- Linear independence of odd zeta values using Siegel's lemma · Journal of the London Mathematical Society · 2026
- Congruences for hook lengths of partitions · Proceedings of the American Mathematical Society · 2026
- Three Dwork-type 𝑞-supercongruences from the 𝑞-Dixon sum · Proceedings of the American Mathematical Society · 2026
- Asymptotics for t-core partitions and Stanton's conjecture · Advances in Mathematics · 2026
- Finite and infinite sums involving reciprocals of products of Gibonacci numbers · The Fibonacci Quarterly · 2026
- Values of generalized Liouville power series at algebraic numbers · Functiones et Approximatio Commentarii Mathematici · 2026
- On Schultz's generalization of Borweins' cubic identity · Advances in Mathematics · 2026
- On a conjecture of Pappas and Rapoport · Mathematische Annalen · 2026
- Congruences modulo powers of 3 for 6-colored generalized Frobenius partitions · Journal of Number Theory · 2026
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