Abstract
The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work.
Highlights
The classic subject of special functions has emerged from a wide variety of practical problems that interest mathematicians and other researchers in science to study their properties, characteristics, and applications.Traditionally, the special functions of mathematical physics are defined using power series representations
Another standard method to generate a sequence of orthogonal polynomials is to use the Rodrigues’ formula
Formula for the shifted Legendre type matrix polynomials and functions starting by setting up the analog of Rodrigues’ formula in matrix framework. Using this established formula, we investigate several properties of these polynomials including the recurrence relations, differential relations, integral representation, and hypergeometric representation
Summary
The classic subject of special functions has emerged from a wide variety of practical problems that interest mathematicians and other researchers in science to study their properties, characteristics, and applications. The fractional-order analogs of the standard Rodrigues’ formulas were recently used to either define new classes of special functions or to study and classify the special functions of fractional calculus as generalized calculus operators of some basic functions [12,13,14] Such an approach allows for establishing several new integral and differential representations that can be useful in applications and for examining related numerical algorithms. Formula for the shifted Legendre type matrix polynomials and functions starting by setting up the analog of Rodrigues’ formula in matrix framework Using this established formula, we investigate several properties of these polynomials including the recurrence relations, differential relations, integral representation, and hypergeometric representation. We should mention that the present work is mostly analytic and designed to develop new properties with proposed algorithm, which are necessary for the forthcoming applications
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
