Missing:
Finding
All
Hypergeometric
Solutions
Differential Marko
Equations Bruno
Petkoviek
Department
of Mathematics
University
INRIA
of Ljubljana
Le Chesnay, France.
Slovenia. Bruno.
.si
Abstract
index.
We give a generalization to find
linear
recurrences,
of M. Petkov6ek’s
all hypergeometric
hypergeometric equation.
sequence
and we describe
functions
that
a program
solve a linear
where
of
to find all differential
of the effort
linear
differential
on finding equations
solutions,
i.e.,
by application
and field operations
closed-form
solutions
has been focused on finding functions built over rational
of exp, log, J, algebraic
(see [9] for a bibliography
this is that special functions reason is that finding order of the equation
is currently
symbol.
ries
fk
F
=
~k>o
+ k – 1) = r(a
+ k)/r(a)
It is easy to see that
is hypergeometric
if and
a seonly
if
In general (see [2, Ch. IV]), this series converges for all finite z when p < q, and for IzI < 1 when p = q+ 1 so that one can talk of the hypergeometric function in
to
these cases. The series diverges
closure,
g + 1. In all cases, (1) satisfies equation
on this).
While Iiouvillian functions correspond more or less to the intuitive notion of ‘(elementary” functions and their integrals, it is natural to try to extend the existing algorithms to special functions. One reason for doing
complexity
(a)k = a(a + 1) . . (a
is Pochhammer’s
fo = 1 and the ~atio fk+l/ fk is a rational function of k. The ai’s (resp. bi ‘s) are the negatives of the zeros (resp. poles) of this function, counted with their multiplicities, and z is the leading coefficient. If —1 is not a pole we append it to the list of zeros in order to cancel out the factor k!.
Most
within
p + q parameters:
algo-
solutions
Introduction
equation.
f r
sequences are such that the quotient of terms is a fixed rational function of the
rithm
equation
Salvy@inria.
series is a power series in z with
Hypergeometric two successive
functions
Salvy
Rocquencourt,
78153
marko-petkovsekthmi-lj
liouvillian
of Linear
do arise in practice;
dramatically
(Factorisation
of linear
hopeless on equations
algorithms with
where
6 = z(d/d.z).
give a meaning introduction
since their
Special
the order of the
differential
-
differential
l)-zfi(O+aj)]y(z) jzl
=0,
(2)
another
any solution permits to reduce the under study. This may bring the
reach of existing
increases
[Ofi(d+bi i=l
for all .z # O when p > the following
exp(z),
operators
This
equation
makes it possible
to
to (1) even when p > q + 1 through
of Meijer’s
G-functions
cases of hypergeometric
(1 — Z)a, —w
the
(see [2, Ch. V]). functions
(see [7, ch VII]
include for an ex-
tensive list of such special cases). While all the examples we have given are also liouvillian, there is no strict
of large order, see [4].)
The special functions we consider in this article are generalized hypergeometric functions. A hypergeometric
inclusion between the set of hypergeometric functions and the set of liouvillian functions. Thus the Bessel function Jo(2i@) is hypergeometric but not liouvillian,
Permission to copy without fee all or part of this material ie granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwisa, or to republish, requires a fee and/or specific permission. ACM-ISSAC ‘93-7/93/Kiev, Ukraine ~ 1993 ACM 0-89791 -604-21931000710027 . ..$1 .50
and exp(eZ ) is liouvillian
but
not hypergeometric
(be-
cause of the type of its singularity at infinity). The purpose of this paper is to describe an algorithm and a Maple version of it that finds all hypergeometric solutions
27
of linear
differential
equations
with
rational
Computing directly the division tain L = LILm,, + S, where
coefficients, In other words, given a linear differential operator L c Q [z] [d/rt.z], we find whether there exists a factorisation L = L1 LZ with L2 as in (2), and the values of the parameters involved. In Section 1, we describe
a generalization
rn-hypergeometric
of algorithm
solutions
HYPER
to linear
m-l
[6] to find
recurrences.
s=
S=o
Section 2, we use this algorithm to find hypergeometric functions satisfying a linear differential equation. Sec-
Section
4, we conclude
with
a detailed
m-hypergeometric
Definition
sequences
1 Let F be a field
of characteristic
sequence Un is m-hypergeometric a rational
function
r(n)
mial
no linear
coeficienis
homogeneous
recurrence
with
equation.
For m =
algorithm
HYPER
wdh polyno-
and let po(n), n such that
the shift
pl (n),
operator,
HY-
Es(n)
= a(n
. . .. pal(n) be rational
position
1. zEIF,
(5)
for the unknown The algorithm m, as we now
zero and function
C(Z + m)
(6)
C(x)
of
z#o,
2. A(x),
B(x),
C(x)
are rnonac polynomials
3. A(z),
C(z)
are relatively
4. l?(z),
C(Z + m) are relatively
5. A(*),
E(z
over IF,
p., pd ~ O. Then
recurrence
defined
operator
in the usual
(3)
With
com-
way, such operators
of order
d.
form
We omit
Lm,Tu
it is desirable
= O and L = LILm,r solution
of recurrence
to have an algorithm
then
prime, prime
for
every
k.
which
is analogous
to the one
case m = 1 (see also [3]).
A(n) = Z— l?(n)
R(n)
C(n + m)
c(n)
where R is from (5) and Z, A, B, C are as in Lemma 1. Inserting this into (5) and clearing denominators gives
hilated by operators of the form Lm,r := Em – r(n) where r(n) is a rational function of n. Clearly, if a mis an m-hypergeometric
the proof
prime,
are relatzveiy
Integer
given in [6] for the special Let
An operator L is reduczble if it is a product of two operators L1 and L2, both of positive degree. Note that m-hypergeometric sequences are those which are anni-
quence u satisfies
+ km)
non-negative
a ring, and they can be divided as polynomials in the indeterminate E except, that factors do not commute.
Therefore
r(n + s).
where
+ 1),
functions
equation
A(x) = Z— B(x)
R(z)
to read this sec-
L := ~p@E~ k=o is a linear
nonlinear
Lemma 1 Let IF be a jield of characteristic m a positive Integer, Every non-zero rational R(x) over IF has a factorisation of the form
tion focusing on the case m = 1,in order to get a feeling of how this algorithm works. Let E denote
1, this
who is not familiar
is encouraged
=
k=o
rational sequence R is solved in [6]. described there generalises to arbitrary show .
zf zt
It reduces to algorithm
1. The reader
m =
R(n)
as
no > 0
of order less than m.
recurrence
PER [6] when
= p~j+$(n),
mk)=O.
J’=o
When m = 1, this corresponds to the usual definition of hypergeometric sequences. In this section, we describe an algorithm to find m-hypergeometric solutions to a linear
tj(n)
(4)
‘j’jtj(.)~~R(n+
zero. A
such that, for all n > no, un+m/un = r(n). An m-hypergeornetric sequence Un w primitive sattsjies
_J_Jr(n + mk + S) = O. knO
(4) can be rewritten
over IF if there exasts
over F and an integer
j-1
s and write
Then
)
Hence for s = O, 1, . . . . m—1, r satisfies
~Prnj+s(n) J=o
example.
Es
k=o
d.
alIn
(n)~r(n+mk+s)
j=o
is the remainder.
Fix
1
(
we ob-
j–l
~P@+.
~
In
tion 3 describes several natural extensions of our gorithm that widen the class of solutions it finds.
d,
of L by L,,,,,,
u
Ly = O.
~
which will
z’~j(n)c(n
+ mj) = O
j=(l
find all right factors of the form Lm,r of a given recurrence operator L. Fix m ~ 1. Let L be as in (3), normalized in such a way that its coefficients are polynomials, and assume that L = LILm,T. Denote d. = [(d – s)/mJ.
where d,-1
j–l
Pj(n)
:= tj(n)
~ k=O
28
A(n + mk)
~ k =J
B(n + mk) ,
(7)
forj=O, l,..., d,. ties of A, B, C that
From
this it follows
the order of the recurrence (9) is r + lkf. What our algorithm useful is that it is not restricted
by the proper-
makes to the
caser+ll=l. A(n)
I to(n) Step 2. Solving the recurrence. PER [6] finds hypergeometric-sequence
and B(n) This
I td, (n – m(d.
leaves us with
a finite
– 1)).
tions
choice for A(n)
and B(n)
of
- they are monic factors of polynomials to(n) and td, (n – rn(d$ – 1)), respectively. Given the choice of
number
A and B, a glance at the leading
verifying
that
Z satisfies
an algebraic
coefficient
equation
at most d$, so the set of possible
the
like (9) with
form
(9).
coefficients
Z a~d three
Given a linear recurrence in Q [n], it outputs an algebraic
monic
and C’(n) in Q[n] such that
of (7) shows
Algorithm HYsolutions of equa-
polynomials
A(n),
B(n),
there exists a solution
of (9)
over F of degree
values for Z is finite
B(n) C(n)fn+l
as
= ZA(n)C(n
(lo)
+ l) f.,
well. For a fixed choice of A, B, and Z, we can use an alwith
gorithm for finding polynomial solutions of recurrences with polynomial coefficients (see [1] or [6]) to determine if (7) has any non-zero After
all rational
polynomial solutions
solution
satisfy
s ~ m – 1, have to be selected. linear
algebraic
Running us all from
equations
for m = 1,2,. right
factors
(3), and hence all primitive
lutions
of recurrence
requires
for the unknown
this algorithm
m-hypergeometric
For each element
(4) for all s, O ~
This
have to compute
solving
k),,. B(no
... d will give of operator
m-hypergeometric
integer
L
produce
consists
a basis of such
first
in computing
of this basis {(ZP, AP, BP, CP)}, we
the first values of the sequence.
be the smallest
constants.
1. It is shown in [6] that
HYPER can actually
Our step 2 thus this basis for (9).
C.
of (4) have been found
for one value of s, those which
(Z, A, B, C) as in Lemma
algorithm solutions.
non-negative
integer
such that
+ k),,. C(no + k) are different k ~ O. Then
for any constant
from
Let no A(no +
O for every
K, the sequence”
so-
f. = ~~zn-fl.c
Ly = O.
‘-1
A(i)
n~no,
(“)1-l~!
i=no
Basic
2
Algorithm
In this section
we apply
satisfies (9) for all n ~ no + M. If no >0, we still have to determine
algorithm
HYPER
— the al-
To get these values,
gorithm of the previous section with m = 1 — to find hypergeometric solutions of linear differential equations. Our algorithm
consists of three steps which
we now de-
If the dimension
of the solution
then isolate solutions dimension 1. Recurrence
tions.
for
formal
power
We start from a homogeneous
equation
series
solu-
X(Z i=o
linear differential
at n = a linear
d, jzO
where we assume that different
of the solution
%3
)
y(’)(z)
=
the leading
o,
coefficients
from O. Suppose the formal
extra
(8)
p$d, are
may also be O, when A(no coefficient
the –
of(9)
vanishes
can be seen from
the way
on the f~’s that has
been
may be O or not.
done
for
each
element
of solutions
of
of two
power series ~(.z) =
{f~,P = a~,f) O S ~
dsolvehyper(eqn, y(x)) ; _Cl hypergeom( [4/3, 5/3,
31
I] , [5/2,
sequence to satisfy.
3] , 2i’/4
X)
The cated:
following
equation
of order
3 is more
As a final note, our algorithms extend without any modification to fields of coefficients containing Q. The
compli-
only difficulty (2’ - 2.22 + 2)y(3)(~) +(102 When
translated
a recurrence
+ (722 – 152 + $)y’’(z)
– 13)y’(z) into
of order
this equation
gives
Acknowledgement by the ESPRIT
2:
+(n2
of polynomials
solution as well
into linear
factors.
This work was supported in part Basic Research Action Programme
III
of the E.C. under
(n2 + 10n + 16)Un+Q – (2922 + 13n + 13)un+l
of integer
over the field has to be effective,
aa the decomposition
– 2y(%) = o.
a recurrence,
is that the computation
of polynomials
contract
ALCOM
II (#7141).
– 3n – 2)un = O. References
This is where HYPER becomes necessary. of hypergeometric vector:
solutions
which
It finds a basis
consists
of only one
[1] ABRAMOV, S. A. that are connected solutions
2=1,
A(n)=
n+2,
l?(n)
=n+7,
C(n)=l.
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) ;
hypergeom(
[1,
2] ,
[7] , z)
algebra.
with
tional
In many cases, a linear all the information
a function.
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However, globally,
differential
equa-
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which is needed to work
Inc., Malabar,
these computations
when one is interested
in us-
then it becomes useful to have
are rather
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1981.
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expensive.
heuristically
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[5] KOEPF,
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W.
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implementing
both
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[7] PRLTDNIKOV, MARICHEV,
and in which order.
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of a subsequent the algorithm
of
series
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Section 1 and the algorithm of Section 2 will soon be part of the Maple share library. Combinatorialists might use it fruitfully in conjunction with the GFUN package which provides tools for manipulating linear re-
3:
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a Maple
package for the manipulation of generating and holonomic functions in one variable. Technical Report 143, Institut National de Recherche en Informatique
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tensions
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Transcendental R. E. Krieger
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interesting
kibernet.
40-42.
The algorithm we describe in this paper is a first step towards a better use of hypergeometric functions in computer
equa-
[3] GO SPER, R. W. Decision hypergeometric summation.
Conclusion
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and difference
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_Cl
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Moscow University Compui. Maih. Cybernei., 3 (1989), 63–68. Transl. from Vestn. Moskov.
gets: y(z)
of linear
tions.
Since in this case M = O and no dsolvehyper(eqn2,
Problems in computer algebra with a search for polynomial
ACM
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33
