Open research questions in Advanced Mathematical Identities
73 unresolved questions extracted from the limitations and future-work sections of 511 Advanced Mathematical Identities papers in our library. Each links back to the study that raised it.
What the literature leaves open
Further exploration of the properties of complete homogeneous symmetric polynomials. Application of the generalized Vieta formulas in various mathematical areas.
The existing proofs for the congruences have limitations. The case 11 was viewed as substantially different from 5 and 7.
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 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.
Further research can be done to apply the derived formulas to various fields. The authors' work can be extended to investigate other related expansions of Ramanujan's Master Theorem.
Polynomial augmentation for Ramanujan’s q-beta integral and related expansions of Ramanujan’s Master Theorem · 2026 · DOIThe classical Vieta formulas do not account for partial knowledge of roots. There is a need for generalizing these formulas.
To apply the new algorithmic approach to other congruences. To use the compact representations of Atkin's basis functions in other applications.
The q-beta integral and its expansions are not well understood. There is a need for new formulas and transformations for the q-beta integral.
Polynomial augmentation for Ramanujan’s q-beta integral and related expansions of Ramanujan’s Master Theorem · 2026 · DOIThe connection between Problem 96 and sums of powers is not well understood. The generating functions for these sums are not well explored.
Further exploration of the connections between sums of powers and special polynomials. Potential applications in number theory and algebra.
Further study of the generalized Tribonacci sequence is needed. The results can be applied to the study of other recursive sequences. The derivation of new sequences using the results is a potential area of research.
The paper identifies a gap in the study of the generalized Tribonacci sequence. The paper notes that the quadratic approximation of the generalized Tribonacci sequence has not been derived before.
There is a lack of understanding of series involving the central binomial coefficient. There is a need for new methods to derive series expansions involving the central binomial coefficient. There is a gap in the literature on the application of these series to physical problems.
The paper identifies a gap in the understanding of conditionally convergent series. The paper addresses the gap by studying the rearrangements of the Alternating Harmonic Series.
There is a gap in existing techniques for counting integer solutions of a linear equation with unit coefficients. The paper aims to address this gap with a simpler counting method.
The paper identifies a gap in the understanding and interpretation of Ramanujan's lost notebook. The author notes that prior studies have not fully explored the significance and influence of Ramanujan's work.
The recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. The study of properties of Bn has occupied the attention of many scholars.
There is a gap in the reference works regarding explicit formulas for Bernoulli numbers. There is a lack of emphasis on technical skills needed for series manipulations.
The need for an elementary proof of the properties of the roots of the characteristic equation. The reliance on advanced theorems like Rouche's theorem in prior work.
The need for a more detailed examination of E(n, r) for the special case in which r is two. The lack of an explicit expression for binary digital sums.
Further development of a comprehensive theory that includes all cases. Exploration of applications of the discrete Fourier transform in other areas of research.
The lack of a comprehensive theory that includes all cases. The need for a deeper understanding of the relationships between different mathematical concepts.
The paper does not provide a comprehensive survey of prior work on generating series. The results are limited to the specific problem of finding a sequence of non-negative integers.
Further research can be done to find more examples of the application of generating series. The generating series approach can be applied to other problems in number theory and combinatorial analysis.
The paper suggests that future research should focus on analyzing the computational complexity of the algorithms used to compute the length of the period of the Fibonacci series modulo m. The paper suggests that future research should focus on analyzing the properties of the Fibonacci series modulo m for all possible values of m.
Most-cited papers in Advanced Mathematical Identities
- The Hybrid Numbers of Padovan and Some Identities · Annales Mathematicae Silesianae · 2020 · 9 citations
- Inclusion properties of hypergeometric type functions and related integral transforms · Studia Universitatis Babes-Bolyai Matematica · 2020 · 7 citations
- A degenerate version of hypergeometric Bernoulli polynomials: announcement of results · Communications in Applied and Industrial Mathematics · 2024 · 7 citations
- A note on generalized hybrid Tribonacci numbers · Discussiones Mathematicae - General Algebra and Applications · 2020 · 6 citations
- Degenerate Hermite poly-Bernoulli numbers and polynomials with q-parameter · Studia Universitatis Babes-Bolyai Matematica · 2020 · 5 citations
- Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions · Publications de l Institut Mathematique · 2020 · 5 citations
- On Generalized Jacobsthal and Jacobsthal–Lucas Numbers · Annales Mathematicae Silesianae · 2022 · 5 citations
- Generalized Tetranacci Hybrid Numbers · Annales Mathematicae Silesianae · 2020 · 4 citations
- Summation Formulas, Generating Functions, and Polynomial Division · Mathematics Magazine · 2022 · 4 citations
- On some Diophantine equations involving balancing numbers · Archivum Mathematicum · 2021 · 3 citations
Most recent work
- Complete Homogeneous Symmetric Polynomials and Generalized Vieta’s Formulas · Results in Mathematics · 2026
- Congruences for an Analogue of Lin’s Partition Function · Mediterranean Journal of Mathematics · 2026
- Theta Function Identities of Level 16 and Their Applications to Colored Partitions · Mathematics · 2026
- q-Leonardo polynomials · Notes on Number Theory and Discrete Mathematics · 2026
- A Closed-Form Evaluation of a Continued Fraction from the Ramanujan Machine · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Polynomial augmentation for Ramanujan’s q-beta integral and related expansions of Ramanujan’s Master Theorem · The Ramanujan Journal · 2026
- A Transformation Between r-Stirling and Binomial Coefficient Series Via Alternating Multiple Zeta Values · Bulletin of the Malaysian Mathematical Sciences Society · 2026
- Two-Color Partitions With Evens in one Color · International Journal of Number Theory · 2026
- Explicit Evaluations of Quintic theta function · INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH · 2026
- √π as a Dimensional-Reduction and Transition-Normalization Constant: Classical Beta Integrals, Angular Budget, and Global Completion · Zenodo (CERN European Organization for Nuclear Research) · 2026
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