Lagerra–Kelly functions and related polynomials

Abstract

The origin of the Laguerre–Kelly polynomials is related to the solution of the Cauchy problem for an abstract homogeneous evolutionary equation of a fractional order with an unbounded operator coefficient A. Using the representation of its solution through the Mittag-Leffler operator function with the replacement of the operator A by its Kelly transform A = (I −q)^–1q and the subsequent expansion into a power series of q, the basic formula of the transform method is obtained. The coefficients of this series are the Laguerre–Kelly functions. Since the Kelly transform method is exponentially convergent and in some cases more efficient than existing methods (in terms of the Kelly algorithmic implementation), the study of the Laguerre–Kelly functions is an important and relevant problem. The main properties of the Laguerre–Kelly functions and related polynomials are investigated. An explicit form of these functions and recurrent formulas of two types (with and without an integral term) which they satisfy are found. It is proved that the Laguerre–Kelly polynomials do not satisfy the three-term recurrent relation and therefore do not form an orthogonal system. Moreover, they are not the solutions of finite-order differential equations with variable polynomial coefficients independent of the degree of the polynomial. A number of properties of zeros of the Laguerre–Kelly polynomials are studied. Using the computer algebra system Maple, we explore the asymptotic behaviour of these functions, which is very important for substantiating the exponential convergence rate of the Kelly transform method.