We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

We introduce the 𝐵-Stirling numbers of the first and second kinds, which are the coefficients of the potential polynomials when we express them in terms of the monomials and the falling factorials, respectively. These numbers include, as particular cases, the partial and complete Bell polynomials, the degenerate and probabilistic Stirling numbers, and the 𝑆-restricted Stirling numbers, among others. Special attention is devoted to the computation of such numbers. On the one hand, a recursive formula is provided. On the other hand, we can compute Stirling numbers of one kind in terms of the other, with the help of the classical Stirling numbers.

ABSTRACT This paper focuses on solving the Fredholm–Volterra integro‐differential equations (IDEs) using a numerical method based on function approximation with generalized fractional‐order Bell wavelets (GFOBWs). Previous studies have shown that the Legendre polynomials are highly effective for approximating solutions to fractional differential equations, while the Bell polynomial collocation method provides significantly better results than the Legendre collocation method for IDEs. Building on these findings, we propose a novel collocation technique using fractional‐order Bell wavelets to address the Fredholm–Volterra IDEs. An exact formula is derived for the Riemann–Liouville fractional integral operator (RLFIO) of the GFOBW. This formula is combined with the collocation method to convert the IDEs into a system of algebraic equations, enabling an efficient numerical solution method. Additionally, the absolute error of the numerical solution obtained from the proposed method is estimated. Various examples are provided to demonstrate the accuracy of the method.

Abstract Any margin of the multinomial distribution is multinomially distributed. Retaining this closure property, a family of generalized multinomial distributions is proposed. This family is characterized within multiplicative probability measures, using the Bell polynomial. The retained closure property simplifies marginal properties such as moments. The family can be obtained by conditioning independent infinitely divisible distributions on the total and also by mixing the multinomial distribution with the normalized infinitely divisible distribution. The closure property justifies a stochastic process of the family by Kolmogorov’s extension theorem. Over time, Gibbs partitions of a positive integer appear as the limiting distributions of the family.

This paper introduces a new type of nonlinear shrinkage estimators for the precision matrix in high-dimensional settings, where the dimension of the data-generating process exceeds the sample size. The proposed estimators incorporate the Moore–Penrose inverse and the ridge-type inverse of the sample covariance matrix, and they include linear shrinkage estimators as special cases. Recursive formulae of these higher-order nonlinear shrinkage estimators are derived using partial exponential Bell polynomials. Through simulation studies, the new methods are compared with the oracle nonlinear shrinkage estimator of the precision matrix for which no analytical expression is available.

Recurrence Relations for Degenerate Bell and Dowling Polynomials via Boson Operators

Heterogeneous Stirling Numbers and Heterogeneous Bell Polynomials

In this paper, a [Formula: see text]-dimensional generalized Sawada–Kotera (2D-gSK) equation, with significant applications in quantum gravity field theory, is explored. The study begins by bilinearizing the 2D-gSK equation using the Bell polynomial (BP) method and continues by finding its multiple solitons, after verifying integrability properties, through the simplified Hirota method. Some theorems regarding the existence of multi-lump waves are formally presented as direct results of multiple solitons. To complete the studies, some other specific nonlinear waves of the 2D-gSK equation, such as breather, complexiton, and Jacobi waves, are constructed in a detailed manner. In the end, as a result of the multidimensional and density representations, the dynamic features of such nonlinear waves are assessed. This work provides valuable results regarding nonlinear waves of the 2D-gSK equation and their dynamics.

I give for the first time explicit formulas for the coefficients needed for the fourth-order Edgeworth expansions of a multivariate standard estimate. I call these the Edgeworth coefficients. They are Bell polynomials in the cumulant coefficients. Standard estimates include most estimates of interest, including smooth functions of sample means and other empirical estimates. I also give applications to ellipsoidal and hyperrectangular sets.

This paper introduces a new family of q-special polynomials, termed q-general Bell polynomials, and systematically explores their structural and analytical properties. We establish their generating functions, derive explicit series representations, and develop recurrence relations to characterize their combinatorial behavior. Additionally, we characterize their quasi-monomial properties and construct associated differential equations governing these polynomials. To demonstrate the versatility and applicability of this family, we investigate certain examples, including the q-Gould–Hopper–Bell and q-truncated exponential-Bell polynomials, deriving analogous results for each. Further, we employ computational tools in Mathematica to examine zero distributions and produce visualizations, offering numerical and graphical insights into polynomial behavior.

Applying the state-of-the-art Bell polynomials to the open unit disk, a differentialoperator θmξ, P is produced. In this paper, we shall provide a family of analytic functions related to the differential operator indicated above. The upper bound for the nonlinear functional |a2a4−a23|, otherwise known as the Hankel determinant, is our primary finding. Aside from using the Bell polynomial, the differential operator is gained using the Hadamard product. Coefficient equating and other fundamentals of classical calculus will be used in the primary finding of the upper bound.

This study investigates two-dimensional Bell polynomials, emphasizing their fundamental properties and applications in mathematical analysis. Utilizing the framework of generating functions, we derive explicit representations, summation formulae, recurrence relations, and addition formulas for these polynomials. Additionally, we present their matrix form and product formula, further enriching their structural analysis. Furthermore, we introduce the 2D Bell-based Stirling polynomials of the second kind and explore their associated properties. This research aims to enhance the theoretical understanding of Bell polynomials and their broader applications in mathematical analysis.

We propose a Stokes expansion ansatz for finite-depth standing water waves in two dimensions and devise a recursive algorithm to compute the expansion coefficients. We implement the algorithm on a supercomputer using arbitrary-precision arithmetic. The Stokes expansion introduces hyperbolic terms that require exponentiation of power series, which we handle efficiently using Bell polynomials. Although exact resonances occur at a countable dense set of fluid depths, we prove that for almost every depth, the divisors that arise in the recurrence are bounded away from zero by a slowly decaying function of the wavenumber. A direct connection between small divisors and imperfect bifurcations is observed. They are found to activate secondary standing waves that oscillate non-uniformly in space and time on top of the primary wave, with different amplitudes and phases on each bifurcation branch. We compute new families of standing waves using a shooting method and find that Padé approximants of the Stokes expansion continue to converge to the shooting method solutions at large amplitudes as new small divisors enter the recurrence. Closely spaced poles and zeros of the Padé approximants are observed, which suggests that the bifurcation branches are separated by branch cuts.

This study investigates a generalized sixth-order Boussinesq equation derived through the extension of its classical bilinear framework. Leveraging Bell polynomials, we propose a streamlined analytical methodology to systematically construct the equation's bilinear representation, enabling the direct derivation of multi-soliton solutions. Furthermore, by employing an extended homoclinic test method, we rigorously obtain both homoclinic breather wave solutions and rogue wave configurations. Notably, these analytical results provide a theoretical foundation for exploring extreme wave phenomena in higher-order nonlinear fields, with implications that transcend disciplinary boundaries and demonstrate transformative potential across multiple domains, including medicine and engineering.

Abstract In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations (in the sense of having three-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in the 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.

ABSTRACTWe provide a generalization of Faà di Bruno's formula to represent the ‐th total derivative of the multivariate and vector‐valued composite . To this end, we make use of properties of the Kronecker product and the ‐th derivative of the left‐composite , which allow the use of a multivariate and matrix‐valued form of partial Bell polynomials to represent the generalized Faà di Bruno's formula. We further show that standard recurrence relations that hold for the univariate partial Bell polynomial also hold for the multivariate partial Bell polynomial under a simple transformation. We apply this generalization of Faà di Bruno's formula to the computation of multivariate moments of the normal distribution.

Abstract In this paper, we systematically investigate the integrability and exact solutions of the non-isospectral KP equation using the binary Bell polynomials. The bilinear form, bilinear B $$\ddot{a}$$ a ¨ cklund transformation and Lax pair are obtained quickly and naturally. Based on the Lax pair, the infinite conservation laws are also derived, with all conserved densities and fluxes expressed through explicit recursion formulas. Furthermore, we present explicit soliton solutions visualized through corresponding three-dimensional plots and heat maps, providing vivid depictions of solitary wave propagation.

The Genocchi numbers were introduced by the Angelo Genocchi, and have been widely studied across various fields of pure and applied mathematics. In this paper, we define three special polynomials that are generalizations of the Genocchi polynomial and numbers, and find some relationships between the probablistic Stirling numbers of the second kind, probabilistic Bell polynomials, probablistic Bernoulli polynomials, the Stirling numbers of the first and the second kind, Genocchi polynomials and those polynomials and numbers.

This paper introduces fully Dowling polynomials of the first and second kinds, which are degenerate versions of the ordinary Dowling polynomials. Then, several important identities for these degenerate polynomials are derived. The relationship between fully degenerate Dowling polynomials and fully degenerate Bell polynomials, degenerate Bernoulli polynomials, degenerate Euler polynomials, and so on is obtained using umbral calculus.

Asymptotic for the rightmost zeros of Bell and Eulerian polynomials