Wolfram Koepf, Universität
Kassel, Germany
Computer Algebra Algorithms for Orthogonal Polynomials and Special
Functions
5-hour minicourse at the Euro
Summer School in Orthogonal Polynomials and Special Functions
Leuven, Belgium, August 12-17, 2002
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 binomial(n,k)*x^k/k!)
-
the computation of generating 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,x)=sum(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k=0..n))
(Zeilberger's algorithm)
-
the computation of hypergeometric term representations of series (Zeilberger's
and Petkovsek's algorithm)
-
the verification of identities for (holonomic) special functions
-
the detection of identities for orthogonal polynomials and special functions
-
the identification of classical orthogonal polynomials, given by recurrence
equations
All topics are properly introduced, the algorithms are discussed in some
detail and many examples are demonstrated by Maple implementations. The
participants are invited to submit and compute their own examples.