Computer Algebra Algorithms for Orthogonal Polynomials and Special Functions

Abstract

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 recurrence equations for sequences, given by expressions, like ( ^sun_ink ) x ^k/k!,

• • the identification of hypergeometric functions,

• • the computation of antidifferences of hypergeometric terms (Gosper’s algorithm), the computation of holonomic differential and recurrence equations for hypergeometric series, given the series summand, like

  $$ P_n \left( x \right) = \sum\limits_{k = 0}^n {\left( \begin{gathered} n \hfill \\ k \hfill \\ \end{gathered} \right)\left( \begin{gathered} - n - 1 \\ k \\ \end{gathered} \right)} \left( {\frac{{1 - x}} {2}} \right)^k $$

  (Zeilberger’s algorithm),

• • the computation of hypergeometric term representations of series (Zeilberger’s and Petkovšek’s algorithm),

• • the verification of identities for (holonomic) special functions,

• • the detection of identities for orthogonal polynomials and special functions,

• • the computation with Rodrigues formulas,

• • the computation with generating functions, corresponding algorithms for q -hypergeometric (basic hypergeometric) functions,

• • the identification of classical orthogonal polynomials, given by recurrence equations.