Elsevier A New Symbolic Computation for Formal Integration with Exact Power Series

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1ANew Symbolic Computation for Formal Integration

with Exact Power Series

Onur K ymaza

,#eref Mirasyedio&lub*

a

Department of Mathematics, Gazi University, K󰃕r󰃺ehir Arts and Science Faculty, K󰃕r󰃺ehir, Turkey

b

Department of Mathematics Education, Gazi University, Education Faculty, 06500 Teknikokullar-Ankara, Turkey

Abstract

This paper describes a new symbolic algorithm for formal integration of a class of

functions in the context of exact power series by using generalized hypergeometric series

and computer algebraic technique.

Keywords : Power Series , Generalized Hypergeometric Series, Symbolic Integration

1.Introduction

Computers are very useful for numerical integration, that is the finding of definite

integrals. But Computer Algebra also lets us perform formal integration, that is the

discovery of integrals as formulae. Formal differentiation was undertaken quite early in

the history of computers by Kahrimanian and Nolan (1953), but it was Slagle (1961) who

took the first steps towards integration. So the first moves made by Slagle were based on

the same heuristics as those used by humans. This way was quite quickly outdated by

truly algorithmic methods after the work of Moses (1967).

In 1969, Risch published a complete algorithm for integrating transcendental

elementary functions, which Rothstein (1971) improved by giving a rational algorithm.

Moreover, Risch (1968, 1969, 1970) sketched a procedure for integrating elementary

functions. Then, Davenport (1981) introduceed an algorithm for integrating algebraic

functions. Also Trager (1984) gave a rational algorithm for integrating algebraic

functions [1]. For further backround about “Formal Integration” see [9].

*

Corresponding author. Tel: +90+312-2126470/3931.

E-mail addresses: okiymaz@gazi.edu.tr (O.KCymaz), seref@gazi.edu.tr (D.MirasyedioE

lu).

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2In 1992, Koepf introduced an algorithm for computing the formal power series of

agiven function using generalized hypergeometric series and the definition of a

reccurence equation of hypergeometric type [6-7]. For the Maple code of Koepf’s

algorithm we refer to [2].

In [5], we extend the major techniques of Koepf to obtain the exact power series

solutions of a given second order linear homogeneous differential equation with

polynomial coefficients, if its reccurence equation is hypergeometric type.

The aim of this paper is to develop a new symbolic computation for formal

integration using generalized hypergeometric series [4] and algorithmic techniques [5].

2. Preliminaries

This section introduces some basic Definitions and Lemmas, which will be used

later.

Definition 2.1 A(generalized) hypergeometric function has a series representation

󰂦󰁦

=0nncwith c

n+1/c

nis a rational function of n[4].

Definition 2.2 The notation (a)

ndenotes the Pochammer symbol (or shifted factorial )

defined by

󰂯󰂮󰂭

󰂏󰀐++=

=

INnnaaan

a

n

)1()1(01

)(

….

The ratio c

n+1/c

ncan be factored and it’s usually written as

)1)(())(()())((

2121

1

+++++++

=+

nbnbnbnxananan

cc

qp

nn

……

.(2.1)

Then if c

0=1, equation (2.1) can be solved for c

nas

!)()()()()()(

2121

nbbbxaaa

c

npnnn

npnn

n

……

=.

where (a)

ndenotes the Pochammer symbol and

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3󰂦󰁦

==

󰂸󰂸

󰂹󰂷

󰂨󰂨

󰂩󰂧

02121

2121

!)()()()()()(

,

nn

nqnnnpnn

qp

x

nbbbaaa

x

bbbaaa

pFq

……

󰀢󰀢

is the usual notation [4].

Definition 2.3 Let

F=󰂦󰁦

=

0nnn

nxa(0

0󰁺

na)

be a formal Laurent series for some n

0󰂏Z.Then Fis called to be hypergeometric type if

it has a positive radius of convergence, and if its coefficients satisfy a reccurence

equation of the form

a

n+m =R(n)a

nn󰁴n

0

a

n=A

nn=n

0,n

0+1, ..., n

0+m–1(2.2)

for some m󰂏N,A

n󰂏C(n=n

0+1, ..., n

0+m–1),

0nA󰂏C\{0}, and some rational function

R.The number mis then called the symmetry number of F.Areccurence equation of type

(2.2) is also called to be of hypergeometric type [6].

The following two lemmas on reduction of order are quite useful for our method.

Lemma 2.1 Let fbe a nontrivial solution of the second order linear homogeneous

differential equation 0)()()(

21

22

0=++yxa

dxdy

xa

dxyd

xa.(2.3)

Then the transformation y=fv reduces equation (2.3) to the first order linear

homogeneous differential equation 02

100=

󰂻

󰂼󰂺

󰂫

󰂬󰂪

++wfa

dxdf

a

dxdw

fa(2.4)

in the dependent variable w,where w=dv/dx [8].

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