Exponential partial Bell polynomials are a main tool for computing explicit formula for a large family of numbers and polynomials. Using these polynomials, we review most of the polynomials already studied in the book of Djordjevic-Milovanovic (2014). This study allows to unify the writing of these polynomials and obtain other extensions. Consequently the obtained results find their application in Number Theory,notably for the arithmetical properties of power products, Fermat numbers, Mersenne numbers and Fermat Last Theorem.

We present a new numerical approach to solve the optimal control problems (OCPs) with a quadratic performance index. Our method is based on the Bell polynomials basis. The properties of Bell polynomials are explained. We also introduce the operational matrix of derivative for Bell polynomials. The chief feature of this matrix is reducing the OCPs to an optimization problem. Finally, we discuss the convergence of the new technique and present some illustrative examples to show the effectiveness and applicability of the proposed scheme. Comparison of the proposed method with other previous methods shows that this method is accurate.

This paper introduces a Tau-based numerical approach utilizing Bell polynomials to solve a fractional HIV model involving $CD4^+ T$ cells. The model comprises a system of three fractional nonlinear ordinary differential equations. The method begins with deriving the operational matrix for the fractional derivative of Bell polynomials. Next, any function within the space \(L^2[0,1]\) is approximated using Bell polynomials, and these approximations are substituted into the model equations. The resulting expressions are used to compute Chebyshev collocation points within the interval \([0,1]\). By applying the Tau method along with the boundary conditions, the problem is converted into a system of algebraic equations that can be solved using standard numerical techniques such as Newton's method. An example is provided to illustrate the accuracy and effectiveness of the proposed method.

In this paper, an operational matrix method based on the Bell polynomials  has been presented to find approximate solutions of high-order Volterra integro-differential equations. This method  uses a simple computational manner to obtain a quite acceptable approximate solution. The main characteristic behind this method lies in the fact that on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the efficiency and accuracy of the Bell polynomials  for solving these equations are acceptable. The convergence analysis of  this method will be shown by preparing some theorems. Moreover, we will obtain an estimation of the error bound for this algorithm. Finally, some examples are presented to illustrate the applicability, efficiency and accuracy of this  scheme in comparison with some  other well-known methods such as Legendre, Bernoulli, Taylor and Bessel polynomial algorithms

In this paper we demonstrate the existence of a set of polynomials Pi, 1£ i £ n , which are positive semi-definite on an interval [a, b] and satisfy, partially, the conditions of polynomials found in the Lagrange interpolation process. In other words, if a=a1<…<an=b is a given finite sequence of real numbers, then Pi (aj)=dij (dij is the Kronecker delta symbol); moreover, the sum of Pi 's is identically 1.

Zernike circle polynomials use for wavefront analysis because of their orthogonality over a circular. ‏‎ So far, different representations have been presented for these polynomials. In this paper, ‎ ‎‎we introduce Zernike polynomials and their important properties. Also, we present some other orthogonal polynomials such ‎as‎ Bessel ‎functions, ‎ Legendre polynomials and ‎ Jaccobi polynomials ‎and ‎give some important results of them‏. ‎‎ Finally, we express the relationship between Zernike orthogonal polynomials and these well-known orthogonal polynomials. ‏‎ For different cases, the ‎radial‎ polynomials in the Zernike functions are obtained using the well-known orthogonal polynomials‏. These relations frequently used for representing‎ wavefront ‎aberrations.