Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to $$2\times 2$$ Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $$2\times 2$$ 2 × 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

Explicit Baker–Campbell–Hausdorff–Dynkin formula for spacetime via geometric algebra

We present a compact Baker–Campbell–Hausdorff–Dynkin formula for the composition of Lorentz transformations [Formula: see text] in the spin representation (a.k.a. Lorentz rotors) in terms of their generators [Formula: see text]: [Formula: see text] This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension [Formula: see text], naturally generalizing Rodrigues’ formula for rotations in [Formula: see text]. In particular, it applies to Lorentz rotors within the framework of Hestenes’ spacetime algebra, and provides an efficient method for composing Lorentz generators. Computer implementations are possible with a complex [Formula: see text] matrix representation realized by the Pauli spin matrices. The formula is applied to the composition of relativistic 3-velocities yielding simple expressions for the resulting boost and the concomitant Wigner angle.

From special operators (1-Dx )n+α to Laguerre Polynomials

Laguerre polynomials Ln α (x) are shown to be the transforms of monomials by the special operators (1-Dx )n+α . From this their current properties such as Rodrigues formula, Lucas symbolic formula, orthogonality, generating functions, etc… are systematically obtained. This success opens the way for the study of special functions from special operators by the powerful operator calculus.

Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

On the ω-multiple Charlier polynomials

The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice $\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} $ ω N = { 0 , ω , 2 ω , … } , $\omega \in \mathbb{R}$ ω ∈ R . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an $( r+1 )$ ( r + 1 ) th order difference equation is given. As an example we consider the case $\omega =\frac{3}{2}$ ω = 3 2 and define $\frac{3}{2}$ 3 2 -multiple Charlier polynomials. It is also mentioned that, in the case $\omega =1$ ω = 1 , the obtained results coincide with the existing results of multiple Charlier polynomials.

Aircraft Pipe Geometric Feature Modeling and Error Compensation Based on Assembly Constraints

An error compensation method based on the assembly constraints of aircraft pipe is proposed for solving the problem of frequent failures caused by high assembly stress in the actual assembly process. Firstly, the pipe assembly process’ geometric modeling is carried out using geometric modeling method, and a new modeling method based on axis vector is proposed. On this basis, the Rodrigues formula was used to establish a pipe space pose calculation model based on actual assembly conditions. Then, the assembly constraints are analyzed, and the key constraint features are identified based on the pipe assembly requirements. Subsequently, the pipe assembly scenarios under different constraint forms are analyzed, and the pipe error compensation methods under a single constraint and associated constraint are respectively proposed. Finally, a typical flaring pipe is selected for the test; the pipe assembly error compensation calculation and the pipe installation air tightness test are carried out respectively. The results show that the proposed method could effectively realize the theoretical space pose adjustment and the pipe parameter compensation, and the air tightness of the compensated pipe is better than that of the uncompensated one.

Fractional order of Legendre-type matrix polynomials

Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.