Complete Bell Polynomials Research Articles

On Some Relations Involving the Ramanujan’s Tau Function

It is known a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms of complete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where participate the values 𝝉(𝒌).

Laplace transform of analytic composite functions via Bell’s polynomials

Bell polynomials have already been used in many different fields, ranging from number theory to operator theory. Here another application in Laplace Transform (LT) theory is shown, describing a method for computing the LT of nested functions. A table containing the first few values of complete Bell polynomials is shown, and a program for deriving the next ones is given. A program for approximating the Laplace Transform of composed analytic functions is also presented.

ON RAMANUJAN’S TAU-FUNCTION

We exhibit a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, whose solution gives a closed formula for τ(n) in terms of complete Bell polynomials. Besides, we show that it is possible to write τ(n) in terms of the compositions of n.

Ramanujan’s Tau-Function in Terms of Bell Polynomials

We obtain a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, and the solution of this recurrence gives a closed formula for 𝝉(𝒏) in terms of the complete Bell polynomials

SUM OF DIVISORS FUNCTION

We show that the recurrence relation deduced by Robbins and Osler et al for the sum of divisors function can be solved in terms of the complete Bell polynomials. Besides, the connection between and the number of representations of n as the sum of four triangular numbers allows obtain arecurrence relation where only participate the values of with m odd.

Complete Bell Polynomials and Recurrence Relations for Arithmetic Functions

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials.

Laplace Transform of nested analytic functions via Bell’s polynomials

Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.

ON THE CHARACTERISTIC POLYNOMIAL OF CHEBYSHEV MATRICES

We exhibit that the coeficients of the characteristic polynomial of any matrix Anxn can be written in terms of the complete Bell polynomials, and this result is applied to Chebyshev matrices which generates the concept of Associated Polynomials of Chebyshev.

Analytical Formulas Using Affine Transformation for Pricing Generalized Swaps in Commodity Markets with Stochastic Convenience Yields

This paper presents analytical formulas for pricing generalized swaps, including the moment swap, gamma swap, entropy swap and self-quantoed variance swap. The formulas are based on closed-form formulas for the conditional expectations of the product of the price and its logarithm and the product of the price and the convenience yield obtained by solving a partial differential equation corresponding to the infinitesimal generator for the two-dimensional diffusion process. In this respect, the formulas obtained are combinatorial in nature and are solved via an affine transformation involving the complete Bell polynomials. The formulas are quite suitable for practical usage with symmetric and skew-symmetric properties, i.e., they are simpler and more compact compared with those existing in the literature. Moreover, for moments swaps, we show in general that the strike price does not depend on the initial spot price but depends only on the initial convenience yield, which highlights the resulting versatility in this respect.

An Extension of the Abel–Liouville Identity

Abstract In this note, we present an extension of the celebrated Abel– Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of n-dimensional vector-valued functions in the subclass of n-times differentiable functions with a nonvanishing Wronskian.

Note on the Higher-Order Derivatives of the Hyperharmonic Polynomials and the r-Stirling Polynomials of the First Kind

In this paper, we focus on the higher-order derivatives of the hyperharmonic polynomials, which are a generalization of the ordinary harmonic numbers. We determine the hyperharmonic polynomials and their successive derivatives in terms of the r-Stirling polynomials of the first kind and show the relationship between the (exponential) complete Bell polynomials and the r-Stirling numbers of the first kind. Furthermore, we provide a new formula for obtaining the generalized Bernoulli polynomials by exploiting their link with the higher-order derivatives of the hyperharmonic polynomials. In addition, we obtain various identities involving the r-Stirling numbers of the first kind, the Bernoulli numbers and polynomials, the Stirling numbers of the first and second kind, and the harmonic numbers.

Geometric and arithmetic realized comoments

The author investigates realized comoments that overcome the drawback of conventional ones and derive the following findings. First, the author proves that (even generalized) geometric implied lower-order comoments yield neither geometric realized third comoment nor fourth moment. This is in contrast to previous studies that produce geometric realized third moment and arithmetic realized higher-order moments through lower-order implied moments. Second, arithmetic realized joint cumulants are obtained through complete Bell polynomials of lower-order joint cumulants. This study’s realized measures are unbiased estimators and they can, therefore, overcome the drawbacks of conventional realized measures.

Moment Functions on Groups

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

Some new formulas of complete and incomplete degenerate Bell polynomials

The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.

Some properties of degenerate complete and partial Bell polynomials

In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.

Complete and incomplete Bell polynomials associated with Lah\u2013Bell numbers and polynomials

The nth r-extended Lah–Bell number is defined as the number of ways a set with n+r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

Realized cumulants for martingales

Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).

Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration

We considered the time-inhomogeneous linear birth–death processes with immigration. For these processes closed form expressions for the transition probabilities were obtained in terms of the complete Bell polynomials. The conditional mean and the conditional variance were explicitly evaluated. Several time-inhomogeneous processes were studied in detail in view of their potential applications in population growth models and in queuing systems. A time-inhomogeneous linear birth–death processes with finite state-space was also taken into account. Special attention was devoted to the cases of periodic immigration intensity functions that play an important role in the description of the evolution of dynamic systems influenced by seasonal immigration or other regular environmental cycles. Various numerical computations were performed for periodic immigration intensity functions.

On r-Central Incomplete and Complete Bell Polynomials

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.

On Central Complete and Incomplete Bell Polynomials I

In this paper, we introduce central complete and incomplete Bell polynomials which can be viewed as generalizations of central Bell polynomials and central factorial numbers of the second kind, and also as ’central’ analogues for complete and incomplete Bell polynomials. Further, some properties and identities for these polynomials are investigated. In particular, we provide explicit formulas for the central complete and incomplete Bell polynomials related to central factorial numbers of the second kind.