Novel Plücker Operators and a Dual Rodrigues Formula Applied to the IKP of General 3R Chains

2018 ◽  
pp. 65-73 ◽  
Author(s):  
Bertold Bongardt
Keyword(s):  
Rodrigues Formula

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

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

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.

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

2021 ◽  
Author(s):  
Joseph Wilson ◽  
Matt Visser
Keyword(s):  
Efficient Method ◽  
Geometric Algebra ◽  
Clifford Algebras ◽  
Matrix Representation ◽  
Spin Representation ◽  
Lorentz Transformations ◽  
Rodrigues Formula ◽  
Dimension Formula ◽  
Spacetime Algebra ◽  
Geometric Algebras

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.

Solutions of the Al-Salam–Chihara and allied moment problems

2019 ◽  
Vol 18 (02) ◽  
pp. 185-210 ◽  
Author(s):  
Mourad E. H. Ismail
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Infinite Family ◽  
Weight Functions ◽  
Moment Problems ◽  
Order Operator ◽  
Absolutely Continuous ◽  
Rodrigues Formula ◽  
Wilson Operator ◽  
The Moment

We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [Formula: see text]-Hermite, [Formula: see text]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials.

The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space

2016 ◽  
Vol 13 (10) ◽  
pp. 1650116 ◽  
Author(s):  
Derya Kahvecí ◽  
Yusuf Yayli ◽  
Ísmaíl Gök
Keyword(s):  
Rotation Angle ◽  
Rotation Axis ◽  
Symmetric Matrix ◽  
Unit Length ◽  
Exponential Map ◽  
Spatial Displacement ◽  
Split Quaternions ◽  
Rodrigues Formula ◽  
Lightlike Axis ◽  
The Given

The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in [Formula: see text] and angle, the spatial displacement is represented by a [Formula: see text] semi-orthogonal rotation matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-skew-symmetric matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

scholarly journals COMBINATORIAL ASPECTS OF ORTHOGONAL GROUP INTEGRALS

2011 ◽  
Vol 22 (11) ◽  
pp. 1611-1646 ◽  
Author(s):  
TEODOR BANICA ◽  
JEAN-MARC SCHLENKER
Keyword(s):  
Explicit Formula ◽  
Open Problem ◽  
Orthogonal Group ◽  
Exact Computation ◽  
Expansion Formula ◽  
Rodrigues Formula ◽  
Type Formula

We study the integrals of type [Formula: see text], depending on a matrix a ∈ Mp × q(ℕ), whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case a ∈ M2 × q(2ℕ) to the general case a ∈ Mp × q(ℕ), (2) the construction of the "best algebraic normalization" of I(a), in the case a ∈ M2 × q(ℕ), (3) an explicit formula for I(a), for diagonal matrices a ∈ M3 × 3(ℕ), (4) a modeling result in the case a ∈ M1 × 2(ℕ), in relation with the Euler–Rodrigues formula. Most proofs use various combinatorial techniques.

scholarly journals Supersymmetry, shape invariance and the solubility of the hypergeometric equation

2015 ◽  
Vol 30 (06) ◽  
pp. 1550023 ◽  
Author(s):  
Ashok K. Das ◽  
Pushpa Kalauni
Keyword(s):  
Orthogonal Polynomials ◽  
Closed Form ◽  
Hypergeometric Function ◽  
Hypergeometric Equation ◽  
Shape Invariance ◽  
Rodrigues Formula

It has been shown earlier [D. Bazeia and A. K. Das, Phys. Lett. B 715, 256 (2012)] that the solubility of the Legendre and the associated Legendre equations can be understood as a consequence of an underlying supersymmetry and shape invariance. We have extended this result to the hypergeometric equation. Since the hypergeometric equation as well as the hypergeometric function reduce to various orthogonal polynomials, this study shows that the solubility of all such systems can also be understood as a consequence of an underlying supersymmetry and shape invariance. Our analysis leads naturally to closed form expressions (Rodrigues' formula) for the orthogonal polynomials.

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