Generalized 2D Laguerre polynomials and their quaternionic extensions

Abstract

A class of orthogonal polynomials in two quaternionic variables is introduced. This class serves as an analogous to the classical Zernike polynomials Zm,n(β)(z,z¯) (arXiv:1502.07256, 2014). A number of interesting properties such as the orthogonality condition, recurrence relations, raising and lowering operators are discussed in detail. Particularly, the ladder operators, realized as differential operators in terms of the so-called Cullen derivatives, for these quaternionic polynomials are also studied. Some physically interesting summation and integral formulas are also proved, and their physical relevance briefly discussed.