162 research outputs found
Algorithms for the indefinite and definite summation
The celebrated Zeilberger algorithm which finds holonomic recurrence
equations for definite sums of hypergeometric terms is extended to
certain nonhypergeometric terms. An expression is called a
hypergeometric term if both and are
rational functions. Typical examples are ratios of products of exponentials,
factorials, function terms, bin omial coefficients, and Pochhammer
symbols that are integer-linear with respect to and in their arguments.
We consider the more general case of ratios of products of exponentials,
factorials, function terms, binomial coefficients, and Pochhammer
symbols that are rational-linear with respect to and in their
arguments, and present an extended version of Zeilberger's algorithm for this
case, using an extended version of Gosper's algorithm for indefinite summation.
In a similar way the Wilf-Zeilberger method of rational function
certification of integer-linear hypergeometric identities is extended to
rational-linear hypergeometric identities
REDUCE package for the indefinite and definite summation
This article describes the REDUCE package ZEILBERG implemented by Gregor
St\"olting and the author.
The REDUCE package ZEILBERG is a careful implementation of the Gosper and
Zeilberger algorithms for indefinite, and definite summation of hypergeometric
terms, respectively. An expression is called a {\sl hypergeometric term}
(or {\sl closed form}), if is a rational function with respect
to . Typical hypergeometric terms are ratios of products of powers,
factorials, function terms, binomial coefficients, and shifted
factorials (Pochhammer symbols) that are integer-linear in their arguments
Formal power series
In this article we will describe the \Maple\ implementation of an algorithm
presented in~\cite{Koe92}--\cite{Koeortho} which computes an {\em exact\/}
formal power series (FPS) of a given function. This procedure will enable the
user to reproduce most of the results of the extensive bibliography on
series~\cite{Han}. We will give an overview of the algorithm and then present
some parts of it in more detail
Representations of orthogonal polynomials
Zeilberger's algorithm provides a method to compute recurrence and
differential equations from given hypergeometric series representations, and an
adaption of Almquist and Zeilberger computes recurrence and differential
equations for hyperexponential integrals. Further versions of this algorithm
allow the computation of recurrence and differential equations from Rodrigues
type formulas and from generating functions. In particular, these algorithms
can be used to compute the differential/difference and recurrence equations for
the classical continuous and discrete orthogonal polynomials from their
hypergeometric representations, and from their Rodrigues rperesentations and
generating functions.
In recent work, we used an explicit formula for the recurrence equation of
families of classical continuous and discrete orthogonal polynomials, in terms
of the coefficients of their differential/difference equations, to give an
algorithm to identify the polynomial system from a given recurrence equation.
In this article we extend these results by presenting a collection of
algorithms with which any of the conversions between the
differential/difference equation, the hupergeometric representation, and the
recurrence equation is possible.
The main technique is again to use texplicit formulas for structural
identities of the given polynomial systems
Algorithms for classical orthogonal polynomials
In this article explicit formulas for the recurrence equation
p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x)
and the derivative rules
sigma(x) p'_n(x) = alpha_n p_{n+1}(x) + beta_n p_n(x) + gamma_n p_{n-1}(x)
and
sigma(x) p'_n(x) = (alpha_n-tilde x + beta_n-tilde) p_n(x) + gamma_n-tilde
p_{n-1}(x)
respectively which are valid for the orthogonal polynomial solutions p_n(x)
of the differential equation
sigma(x) y''(x) + r(x) y'(x) + lambda_n y(x) = 0
of hypergeometric type are developed that depend only on the coefficients
sigma(x) and tau(x) which themselves are polynomials w.r.t. x of degree not
larger than 2 and 1, respectively.
Partial solutions of this problem had beed previously published by Tricomi,
and recently by Y\'a\~nez, Dehesa and Nikiforov
On the De Branges theorem
Recently, Todorov and Wilf independently realized that de Branges' original
proof of the Bieberbach and Milin conjectures and the proof that was later
given by Weinstein deal with the same special function system that de Branges
had introduced in his work.
In this article, we present an elementary proof of this statement based on
the defining differential equations system rather than the closed
representation of de Branges' function system. Our proof does neither use
special functions (like Wilf's) nor the residue theorem (like Todorov's) nor
the closed representation (like both), but is purely algebraic.
On the other hand, by a similar algebraic treatment, the closed
representation of de Branges' function system is derived. In a final section,
we give a simple representation of a generating function of the de Branges
functions
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