# On the Total Positivity and Accurate Computations of \(r\)-Bell Polynomial Bases
## 1 Introduction  
The Stirling numbers form combinatorial sequences with remarkable properties and many applications in Combinatorics, Number Theory, or Special Functions, among others. The Stirling numbers were so named by [PERSON] (cf. [1]) in honor of [PERSON], who computed them in 1730 (cf. [2]). Interested readers are referred to [3; 4; 5] to find a nice introduction and many nice properties of Stirling numbers. In the literature, many kinds of generalizations of Stirling numbers have been introduced and studied ([3; 6]). In this paper, we shall focus on the \(r\)-Stirling numbers whose combinatorial and algebraic properties, in most cases, generalize those of the regular Stirling numbers. The \(r\)-Bell polynomials (see [7]) are considered as an efficient mathematical tool for combinatorial analysis (cf. [5]) and can be applied in many different contexts: in the evaluation of some integrals and alternating sums [8; 9], for the analysis of the internal relations for the orthogonal invariants of a positive compact operator [10], in the Blissard problem [5], for the Newton sum rules for the zeros of polynomials [11], in the recurrence relations for a class of Freud-type polynomials [12] and in many other subjects. In this paper, a class of \(r\)-Stirling matrices will be introduced. It will be shown that these matrices determine the linear transformations between monomial and \(r\)-Bell polynomial bases and have the property of being totally positive and, therefore, having all of their minors as non-negative. Several applications of the class of totally positive matrices can be found in [13; 14; 15]. Totally positive matrices can be expressed as a particular product of bidiagonal matrices (see [16; 17]). This bidiagonal decomposition provides a representation of the totally positive matrices that exploits their total positivity property and allows us to derive algorithms to high relative accuracy for the resolution of relevant algebraic problems, such as, the computation of their inverses, eigenvalues, singular values, or the solution of some systems of linear equations. A major issue in Numerical Linear Algebra is to obtain algorithms to high relative accuracy, because the relative errors in their computations will have the order of the machine precision and will not be drastically affected by the dimension or conditioning ofthe considered matrices. Consequently, the design of algorithms adapted to the structure of totally positive matrices and achieving computations to high relative accuracy has attracted the interest of many researchers (see [18; 19; 20; 21; 22; 23; 24]). Let us recall that many interpolation, numerical quadrature, and approximation problems require algebra computations with collocation matrices of the considered bases. When solving Taylor interpolation problems, Wronskian matrices have to be considered. On the other hand, Gramian matrices define linear transformations to obtain orthogonal from nonorthogonal bases. Moreover, the inversion of Gramian matrices is also required when approximating, in the least-squares sense, curves by linear combinations of control points and the basis functions. The large range of applications of \(r\)-Bell polynomials has motivated us to exploit the total positivity property of \(r\)-Stirling matrices to design fast and accurate algorithms for the resolution of algebraic problems with collocation, Wronskian, and Gramian matrices of \(r\)-Bell polynomial bases. The layout of this paper is as follows. Section 2 summarizes some notations and auxiliary results. Section 3 focuses on the class of \(r\)-Stirling matrices. Their bidiagonal decomposition is provided and their total positivity property is deduced. In Section 4, \(r\)-Bell polynomials are considered. Collocation, Wronskian, and Gramian matrices of \(r\)-Bell polynomial bases are shown to be totally positive. Then, using the proposed factorization for \(r\)-Stirling matrices, the resolution of the above mentioned algebraic problems can be achieved to high relative accuracy.