Eecient Computation of Orthogonal Polynomials in Computer Algebra Eecient Computation of Orthogonal Polynomials in Computer Algebra

Eecient Computation of Chebyshev Polynomials in Computer Algebra

Efficient Computation of Chebyshev Polynomials in Computer Algebra

Orthogonal polynomials can be calculated by computation of determinants, by the use of generating functions, in terms of Rodrigues formulas, by iterating recurrence equations, calculating the polynomial solutions of differential equations, through closed form representations and by other means. In computer algebra systems all these methods can be implemented. Depending on the application one mi...

Computer Algebra Algorithms for Orthogonal Polynomials and Special Functions

In this minicourse I would like to present computer algebra algorithms for the work with orthogonal polynomials and special functions. This includes • the computation of power series representations of hypergeometric type functions, given by “expressions”, like arcsin(x)/x , • the computation of holonomic differential equations for functions, given by expressions, • the computation of holonomic...

Software for the Algorithmic Work with Orthogonal Polynomials and Special Functions

Modern computer algebra systems use heuristics and algorithms for the fast computation with mathematical formulas. General purpose computer algebra systems like Axiom [12], Derive [20], Macsyma [17], Maple [5], Mathematica [23] or Reduce [11] are for example great with integrals. Even a small system like Derive computes all explicitly given integrals of Bronshtein and Semedyayev’s integral tabl...

Averages of Characteristic Polynomials in Random Matrix Theory

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and...