Polynomial Theory and Differential Equations

Polynomial theory and differential equations intersect at the foundations of modern analysis, combining algebraic structures with continuous change. Polynomial theory studies expressions formed by sums of powers of a variable with constant coefficients, offering tools such as generating functions, recurrence relations and orthogonality properties. Differential equations describe how quantities evolve and interact, often admitting polynomial solutions that reveal spectral properties, asymptotic behaviour and stability. Classical families of orthogonal polynomials—such as Hermite, Legendre and Chebyshev—arise as solutions to second-order linear differential equations and underpin methods in physics, engineering and numerical analysis. Recent advances extend these ideas to multi-variable, q-deformed and degenerate cases, deploying operational calculi, factorisation methods and monomiality principles. Such developments enable new analytical representations, integral formulae and symmetry relations, with applications ranging from fluid dynamics to signal processing and quantum mechanics.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Polynomial Theory and Differential Equations publication trend

The graph below shows the total number of articles in polynomial theory and differential equations across all publications each year (not limited to Nature Index journals).

Technical terms

Polynomial: A finite sum of terms consisting of a variable raised to non-negative integer powers, each multiplied by a coefficient.

Differential equation: An equation relating a function to its derivatives, describing the rates at which quantities change.

Orthogonal polynomials: A sequence of polynomials that are pairwise orthogonal under a specific weight function on a given interval.

Generating function: A formal power series whose coefficients encode a sequence of numbers or polynomials for compact representation and manipulation.

Recurrence relation: An equation defining each term of a sequence as a function of preceding terms, often enabling efficient computation of polynomial families.

Monomiality principle: An operational framework linking polynomial families to differential or difference operators, facilitating unified derivation of their properties.

References

1. A Survey on Orthogonal Polynomials from a Monomiality Principle Point of View. Encyclopedia (2024).
2. Sequences of twice-iterated Δw-Gould–Hopper Appell polynomials. Journal of Taibah University for Science (2023).
3. Differential Equations Associated with Two Variable Degenerate Hermite Polynomials. Mathematics (2020).