Degenerate Polynomial Structures in Combinatorial Analysis

Degenerate polynomial structures constitute a rapidly expanding area of combinatorial analysis in which classical families of polynomials and numbers are deformed by an additional parameter, often denoted λ. In the limit λ→0 these structures recover their classical counterparts, while for nonzero λ they reveal richer algebraic and analytic behaviour. Central to this framework are generating functions that encode infinite sequences of degenerate polynomials, together with recurrence relations and explicit summation formulae. These tools yield new identities involving degenerate analogues of Stirling numbers, Bell polynomials, Bernoulli and Euler polynomials, Fubini and Changhee numbers, as well as other special sequences. Beyond intrinsic combinatorial interest, degenerate structures bridge to areas such as probability theory—through connections with degenerate gamma or other distributions—quantum operator algebras via normal-ordering problems, p-adic analysis and special-function theory. Practical applications are emerging in statistical mechanics models with deformation parameters and in coding or enumeration problems where classical identities require systematic generalisation. Recent developments emphasise unified approaches to generating functions, the discovery of novel implicit summation formulae and higher-order extensions, underscoring the global significance of these deformations in uniting disparate threads of modern combinatorics.

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Recent work has advanced our understanding of degenerate Changhee–Genocchi polynomials by introducing a modified family of the second kind. Their generating function yields addition formulas, recurrence rules and implicit summation relations. Close connections with degenerate Stirling numbers of both kinds have been established, and higher-order analogues have been defined with a wealth of new combinatorial identities.

In the context of operator theory, researchers have examined the normal ordering of degenerate integral powers of the boson number operator. By expressing these powers in terms of degenerate Stirling numbers of the second kind, they have derived explicit operator expansions and a novel Dobinski-type formula for degenerate Bell polynomials. This work illustrates how degenerate combinatorial structures illuminate problems in quantum algebra and statistical physics.

A further strand of investigation has focused on degenerate derangement polynomials and numbers. Defined as a λ-deformation of classical derangements, these polynomials admit closed-form expressions, generating functions and recurrence relations. Their identities intertwine with fully degenerate Bell and Fubini polynomials as well as with degenerate Stirling numbers. Moreover, probabilistic interpretations link their moments to variants of the degenerate gamma distribution, highlighting interplay between combinatorial deformation and statistical models.

Degenerate Polynomial Structures in Combinatorial Analysis publication trend

The graph below shows the total number of articles in degenerate polynomial structures in combinatorial analysis across all publications each year (not limited to Nature Index journals).

Technical terms

Degenerate polynomial: A one-parameter deformation of a classical polynomial sequence that reduces to the original sequence when the parameter vanishes.

Generating function: A formal power series whose coefficients encode a sequence of numbers or polynomials, used to derive identities and recurrence relations.

Recurrence relation: An equation expressing each element of a sequence in terms of preceding elements, often derivable from its generating function.

Stirling numbers: Two families of combinatorial numbers—of the first and second kinds—enumerating permutations by cycle structure or set partitions, here extended to degenerate versions.

Dobinski-type formula: An explicit summation expression, originally for Bell numbers, adapted to the degenerate setting to represent polynomial values.

References

1. A Note on Modified Degenerate Changhee–Genocchi Polynomials of the Second Kind. Symmetry (2023).
2. Normal ordering of degenerate integral powers of number operator and its applications. Applied Mathematics in Science and Engineering (2022).
3. Degenerate Derangement Polynomials and Numbers. Fractal and Fractional (2021).