I nternat. J. Math. & Math. Si.
Vol. 4 No. 3 (1981) 417-434
417
BOUNDED INDEX, ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL
EQUATIONS AND SUMMABILITY METHODS
G.H. FRICKE
Department of Mathematics
Wright State University
Dayton, Ohio 45431
RANJAN ROY
Department of Mathematics
State University of New York
College at Plattsburh
Plattsburgh, New York 12901
and
Department of Mathematics
University of Kentucky
Lexington, Kentucky 40502
(Received December 17, 1980)
ABSTRACT.
A brief survey of recent results on functions of bounded index and
bounded index summability methods is given.
Theorems on entire solutions of ordi-
nary differential equations with polynomial coefficients are included.
KEY WORDS AND PHPASES.
Bounded Index, Summaby, ieretial Equations,
Ee Solutions.
1980 Mathematics Subject
I.
Classification
Codes.
30D15, 40D05. 34A20, 34A40.
INTRODUCTION ANN DEFINITIONS.
DEFINITION i.i.
An entire function f:
/
is said to be of bounded index
(b.i.) if there exists an integer N >- 0 such that
max
O_<j_<N
for all z
and all n
If(J)
(z)I
J!
0,1,2,
dex of f (see Lepson [30], Shah [40]).
>_
if(n)(z)l
n!
(i.i)
The least such integer N is called the in-
418
G.H. FRICKE, R. ROY and S.M. SHAH
An entire function f is said to be of bounded value distribu-
DEFINITION 1.2.
(b.v.d.) if for every r > 0, there exists a fixed integer P(r) > 0 such that
tion
the eouation f(z)
w has never more that P(r) roots in any disc of radius r and
(see Bayman [16,17]).
for any w e
A survey of the properties of functions of b.i., and of b.v.d., and a list of
references published up to 1975 and some up to lO76, are given in [40].
In Section 2 we ive some extensions of these concepts to meromorphic functions
[3].
If an entire function is of b.i. N, then its growth is (I; N
[37], [13]).
+ I) ([17],
In Section 3 we study extensions of (I.i) suitable for entire func-
tions of finite order.
Here we show, following Bennekemper [19], that if E[O,)
be the set of all entire functions of order not exceeding 0, and not of maximal
N, then
tyDe order 0, and if f be of b.i.
side denotes the ideal in
E[1,,)
finitely
(f,fl
generated
f(N)) I E[I,), where the left
by f,fl,...,f(N). In Section
4 we consider entire solutions of linear differential equations with Dolvnomial
coefficients and give theorems relating to the poperty of b.i., and bounds on the
index
Nf
rate of
.
of an entire solution f of bounded index, and also a bound on the
Here and in the precedin section we have included some new results
and shorter proofs of some known results.
panying references are new.)
property are
rowth
(Theorems and Examples without accom-
The summability methods related to bounded index
iven in Section 5.
inally in Section 6 we give some recent results,
on functions defined by Dirichlet series and on functions of several variables.
2.
vUNCTIONS OF B.I. AND B.V.D.
It is known that if f is of b.i., then it is of exponential type ([17], [13])
but Functions of
exponential
type need not be of b.i.
In fact there exist func-
tions of exponential type and having simple zeros and of unbounded index
[39].
The followin theorem gives a necessary and sufficient condition for an entire
function of exponential type to be
THEOREM 2.1.
(ricke [i0]).
o.
b.i.
Let f be an entire function o
Then f is of b.i. if and only if for each d > 0, there exists M
that
If’(z)
<
Mlf(z)
for all z with
z
a
n
> d for all n.
exponential
type.
M(d) > 0 such
Here
a’s
denote
n
419
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
the zeros of f.
The bound M 4epends on the closeness to the zeros of f.
If we now examine the
behavior of the loarithmlc derivative near a zero a of order m, we see that
t
f’(z)
(z)
is bounded in a neighborhood of
at;
z
<
m+l
Iz
This simple observation is used to improve
For an entire function f, let
Theorem 2.1.
R(z)
where the
t
t
that is
f’ (z)
f(z)
for z sufficiently close to a
a
a’s
n
Rf(z)
max
{i} U {
[z
n
a
1,2
}]
n
are the zeros of f.
THEOREM 2.2. (Fricke [i0]).
An entire function of finite order is of b.i. it
and only if there exists a constant M > 0 such that
If’(z) <_
MR(z)
If(z)
for all z
The proof of Theorem 2.2 makes use of the followinK lemma [i0]:
If f is an entire
function of finite order such that for some M > 0,
If’(z)
<
M/(z)
If(z)l
for all z
then there exists an integer N such that any closed disk of radius I contains at
most N zeros of f.
Beauchamp [3] extended tha basic ideas in Theorem 2.2 to meromorphic functions.
To present his results we need the followin notations.
u {=}.
A function f meromorphic on
is said to be
A-b.v.d. if there exists an integer P such that for any w e A, f(z)
w has at most
DEFINITION 2.3,
Let A
c
P zeros in any disk of radius i.
poles in any disk of radius i.
DEFINITION 2.4.
If A
e A this implies that f has at most P
u {=}, we simply say that f is b.v.d.
A function f meromorphic on C is said to be D.I. (differen-
tial ineauality) if
6i)
If w
f is {=}-b.v.d.
G.H. FRICKE, R. ROY and S.M. SHAH
420
(ii)
f satisfies an inequality of the form
[f(N+l)
for all z e
\P,
P
{b
(z)[
<
R(z)
f(J)
max
0<j<N
where N is a positive integer,
n
C
f has a pole at b
n
and R(z) is a real valued function with R(z) >
and R(z) decreasing
with respect to the distance of z to the poles, that is,
D[d(z,P)] where D
R(z)
function.
(0,]
/
[I,=) is a decreasing
(If f is entire we may consider R(z) to be constant.)
C, Hayman [17] showed that the above con-
For entire functions and for R(z)
dition is equivalent to bounded index.
DEFINITION 2.5.
A function f meromorphic on
is said to be L.D.I. (logarith-
mic differential inequality) if
(i)
f is
{0,=}-b.v.d.
(ii) the logarithmic derivative satisfies
f’ (Z)
f(z)
where
D
< L(z)
for all z
is the set of zeros and poles of f and
function with respect to the distance of z to
L(z) is a decreasing
and L(z) > i.
Using the above definitions, Beauchamp was able to obtain among other results
the following:
THEOREM 2.6.
(Beauchamp [3]).
A function f meromorphic on
is D.I. if and
only if it is L.D.I.; in fact if f is D.I. then R(z) in Definition 2.4 may be
chosen to be of the form R(z)
M
ax
{I, d(,P)
where M is a constant > i,
and N and K are integers with 1 < K < N.
THEOREM 2.7.
(Beauchamp [3]).
A function f meromorphic on
and only if f’(z) is D.I.
THEOREM 2.8.
(Beauchamp [3]).
(i)
the function
(ii)
The product h
i
is D.I.
fg is D.I.
Let f and g be D.I.
Then
is b.v.d, if
421
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
(Beauchamp [3]).
THEOREM 2.9.
Lt f be D.I.
Then f is of order not exceed-
ing 2 and finite type.
In [3] Beauchamp also examines and obtains similar results for functions meroHere R(z) and L(z) depend not only on the distance to
morphic on the unit disk.
the zeros, respectively zeros and poles, but also on the distance to the boundary
of the unit disk.
3.
,
BOUNDED INDEX CONCEPT FOR FUNCTIONS OF FINITE ORDER.
It is known that if f, entire on
is of order > i or of order i and maximal
type then the growth rate of the derivative may be larger than that of the function
(Shah [36], Vijayaraghavan [46], KSvari [28]) and so inequalities of the type (i.i)
To overcome this difficulty both sides of the inequality (I.I) are
may not hold.
Thus we have:
multiplied by a factor.
(Beauchamp [3]).
DEFINITION 3.1.
Let f be entire on
and y >- 0.
function f is y-b.i. (y-bounded index) if there exist a number r
o
The
> 0 and an inte-
ger N >- 0 such that
If (n)
z
n
for all
z[
(z)
<
]ny
(3.1)
max
O<)-<N
-> r o and n >- N.
This definition is an extension of
(I.I) to entire functions of finite order.
If f is of b.i. N then f is of growth (i, N+I) ([40]).
THEOREM 3.2.
growth
(y
(Beauchamp [3]).
Here we have
If f is y-b.i, satisfying (3.1) then f is of
N+I
+ i, ).
Another extension of (i.i) is as follows:
DEFINITION 3.3.
R and N is the smallest integer such that for all n,
bounded m-index N if
(i)
max
An entire function f is said to be of
(Hennekemper [19]).
If (j)
(z)
>-
j,
0_<j_<N
If (n)
for all z
n!
zl
< i,
and
(ii)
max
0_<j_<N
If (j)
j
(z)
}
Iz] j
>
If (n)
n!
(z)
Izl an
for all z
Iz[
>_ i.
422
G.H. FRICKE, R. ROY and S.M. SHAH
This definition is a slight variation of the one given by G. Frank
[6], and
is
used by Hennekemper to prove Theorem 3.4 below.
Let E[0,) be the set of all entire functions which are of order not exceeding
) and not of maximal type order
0 (0 < 0 <
{f entire
E[0, )
llf(z)
constants C
Let
fl,f2
I
that is,
exp(C21zl 0)
l(f)
and C
for all z
C
2
and
2(f)}.
(fl f2 "’’’fn)0
the ideal in E[0,)
fl’f2"’’’ fn"
(Hennekemper [19]).
THEOREM 3.4.
> 0.
C
I
E[0 ) and denote by
fn
finitely generated by
let
< C
0,
E[0,) be of bounded m-index N and
Let f
Then
f(N))0+e
(f,f,,
E[0+e’)"
We give below a different proof of this theorem when p
i and
0.
For another
proof see [19].
THEOREM 3.5.
Let f be an entire function of b.i. N, not identically zero.
Then
f,N,( "i
(f,f,
PROOF.
Thus, for C
N, we have for any j
(k)
max {If
(z) I}
Since f is of b.i.
If (j)
(N
(z)
<-j
for all z
O<k_<N
.
+ i)
If (j)
for all z, and j
E[I =)
1,2
N+I.
N
< C
(z)
Let m
N
G(r)
l
k=0
l
k=O
if(k)
If (k)
(z)
II
I and
(mr)[
.len G(r) is continuous and piecewise continuously differentiable.
Also because of
the definition of b.i.,
max
O<<N
If () (z)[
> 0
and thus
G(r) > 0
Hence for all r, except possibly for a set of measure zero,
for
r > 0
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
N
IG’ (r)
d
lf(k)
N
(r)
-<
dr
k=0
If (N+I)
<
G(r) +
<
G(r) + CG(r)
l
k=O
]f(k+l)
423
(r)
(er)
Thus
IG’(r)]/G(r)
+
_< C
G’(r)IG(r) >-(C + i)
1
Hence
log
Now if we let C
G’ (x)
"(x) dx
G(r)
G(0)
I/G(0) and C 2
I
G(.r)
C
+
C G(r) >
i
G(O)
>
-(C +l)r
i, then for all r >_ 0
exp(-C2r)
and thus for all r >_ 0
C
I
> i
exp(C2r)G(r)
was arbitrary, we obtain by considering z
Since arg
N
i < C
exp(C21zl) j__Z 0 If (j)
I
The proof can now be concluded by applying the following ([21], [23],
for all z.
[19])
LEMMA
Let f
if there exist
e E[p =).
’fl "’’fn
CI,C 2
The
(fo fl
’fn)p
E[O,=) if and only
> 0 such that
.
for all z e
4.
o
i
-< C I
exp(C2]z]0)
n
{
Z
j=O
Ifj(z)
ENTIRE SOLUTIONS OF DIFFERENTIAL EQUATIONS.
(i)
Consider the differential equation (d.e.)
w,a#
a w 3.n.
o
L (w P)
P (z)w
L
n
+
alw(n-l) +
+
a w
n
0
a
o
0
(4.1)
0
(4.2)
and the d.e.
n
where a e
k
, Pk
o
"n’( +
Pl(Z)w(n-l) +
are all polynomials and
+ P n (z)w
G.H. FRICKE, R. ROY and S.M. SHAH
424
zk(l +
Pk(Z)
o(i))
Izl
(4.3)
/
The following results are known.
THEOREM 4.1.
(Shah [37]).
All solutions of the equation L (w a)
0
n
a
k
e
are entire functions of b.i. and b.v.d.
There it is shown that every solution f(z) is of b.i., and a bound on the
index of f is also given.
By differentiating (4.1), one sees easily that every
solution is of b.v.d.
THEOREM 4.2.
(Shah [38]).
If
deg P o > max deg P k
l<k_<n
(4 4)
then all those solutions of the equation L (w,P)
n
0, which are entire functions,
are of b.i. and b.v.d.
Extensions of this theorem are given by Fricke and Shah ([ii],
for the index
Nf
[13]).
N(f) of an entire solution f of (4.2) are known in some particu-
lar cases ([31], [22], [44]).
tion w is of exponential type
Note that Theorem 4.2 implies that any entire solu-
+
N(fw)
A bounded index on the growth rate of a
i.
solution, without the hypothesis (4.4), is given in the next theorem.
max
ek
if
k
L (w,P)
n
0
if
(Beauchamp [3]).
O, and if y
>
if
ak
k
l_<k<n
THEOREM 4.3.
Bounds
Write
0
Pk
k
k
ek
o
<
k
If f is an entire solution of the equation
0, then f has growth
n
{y + i,
Z
k=l
lkl
/ (7 + i)
IAo
}.
(4.5)
The following examples show that the growth bound (4.5) cannot, in general,
be improved.
EXAMPLE 4.4.
(Beauchamp [3]).
Let f(z)
exp(z
kz(k-l)w
0
Then f satisfies the equation
w’
k)
where k >_ i is an integer.
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
k- I, A
Here y
-k and f has growth (k,l).
i,
EXAMPLE 4.5.
425
The Bessel function J (z) of order n, where n is a positive or
n
negative integer or zero, satisfies the equation
z
Here y
A
0
(ii)
o
1
i
2w,
0
+
2
+ (z 2
zw
n )w
1 and J
2
0
has growth (i I).
n
We now consider one type of converse of Theorem 4.2.
entire functions
We seek a set of
such that every entire function f of b.i. satisfies a linear
gk
d.e. of the form
f(n)
where
go,gl,...,gn_ I
THEOREM 4.6.
+ gn-i
f(n-l)
+
+
gof
0
are entire functions.
(Hennekemper [19],[20]).
Let f be of b.i.N. Then f satisfies
a linear differential equation of the form
f(N+l)
with
gk
E
+
gNf
(N) +
+
go f
(4 6)
0
E[I,).
COROLLARY 4.7.
(Hennekemper [20]).
Any entire function of exponential type
can be written as the difference of two functions each satisfying a linear d.e. of
order N with coefficients from
E[I,).
For the proof of Theorem 4.6 we only need to note that (Theorem 3.5)
f(N+l)
e
E[I,=)
f(N)) I
(f,fl
The Corollary relies on the fact that any function of exponential type can be
written as the sum of two functions of b.i.
REMARK.
([42]).
Simple examples such as Sin z, Cos z (N
i), e
Z
(N
O) show that
the order of the equation (4.6) is best possible.
(iii)
Another type of converse to Theorem 4.2 is as follows:
THEOREM 4.8.
If all n solutions of d.e. L (w,P)
n
exponential type tendeg P
>
O
deg
Pk’
for k
0 are entire functions of
1,2 ...,n; and thus the solutions
are of b.i. and b.v.d.
We shall deduce this from
THEOREM 4.9.
If all n solutions are entire functions and
deg
Po
< max deg
k>O
Pk
(4.7)
426
G.H. FRICKE, R. ROY and S.M. SHAH
then at least one solution w is of order %
Wittich
where
k- So
max
k>0
PROOF.
i
> i
(4.8)
k
This result follows easily from the results and methods of Knab
[47], Poschl [34], Boehmer [4] and Frank [5].
argument.
[24-27],
We sketch briefly the main
By our hypothesis
n
o<
max {a
i
i=l
g
+
i}
(4.9)
Hence there is a region S, the plane cut along a half ray, in which a single-valued
branch W of the solution w can be defined with the property that if
M(r,W)
{IW(z) l,
max
z
S
(
then
%(W)
10glg M(r,W)
lira sup
r-o
log r
> 0
Since all the solutions, by our hypothesis, are entire, we can choose the branch
to be the solution itself and this implies that there is at least one transcendental
solution with positive order (see also Ince
[22, 424-427]).
Now we use Wiman-Valiron central index method (cf:
Valiron
[45, 105-109; 177-181]).
()
w()
1,2
,k
II
except for a set of values of r
[47, 4-11; 65-73],
For the transcendental solutions
()k
w(k)
Wittich
is the central index and the points
+
as
Here N
of finite logarithmic measure.
,
zl
on
r, are the points
N(r)
at which the
maximum modulus is attained:
lw(z)
()
l/Y,
We substitute this asymptotic relation in (4.2) and put N
I/X and then
the equation (4.2) becomes
n
Z
k=0
where m
k
K + n +
next constructs a
it follows (cf:
ao
ek
Akxkyk
k, K
(i
+
maxk> 0
Newton’s polygon (cf:
nk(X))
_(a
k
0
ao
and
nk
+
0 as X
+
0.
One
[33], [47; 67-72]) with points (k,mk) and
Knob [25,27]) that the negative slopes of the sides of the polygon
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
427
give the orders of the solutions and that the negative of the slope of the side
through (0,m
o)
is the order of the solution of maximal growth.
a -s
k
o
> i
this slope is in fact i +
maxk> 0
REMARKS.
(i)
The negative of
k
Note that the condition (4.7) implies (4.9).
We require (4.7)
for the concluding part of the proof of Theorem (4.9).
(ii)
The following example shows that the hypothesis, in Theorem 4.9, that all
solutions are entire is necessary.
EXAMPLE 4.10.
[40].
zw" +
Here (4.7) is satisfied.
(z2
1/2)w’
z
(z
Wl(Z)
One solution
2
e
1
)w
z
0
is entire but the second solution
(4.9) does not hold.
is not entire, and the conclusion of Theorem
We now give two more examples.
EXAMPLE 4.11. [5, p. 61-62]).
(2z
2
l)w"’ + (-8z
2z
+ 6z 2 + 2z + 3)w" + (8z 4
3
3
4
+ (-8z + 8z + 2z
2
9)w
2z
10z
2
+ 2z + 7)w’
0
Here all three solutions are entire functions:
w
e
I
2
z -z
w
z2+z
e
2
w
e
3
z
Here
a
o
2
i
3
a2
4
4
a3
m
o
So
ak
max
k
k>0
5
ml
I
2
3
m2
i
m3
0
EXAMPLE 4.12.
w"
2zw’ + 2nw
0
Here both solutions are entire functions, one a polynomial (Hermite polynomial,
when normalized) and the second, a transcendental function of order
max
Here m
(iii)
o
3, mI
i, m
2
{i,0}
2
i.
Frank and Frank and Mues ([6-8];
to define a function of b.i.
see also
[40]) introduce
a function
l(r,f)
Consider the Taylor expansion of an entire function
428
G.H. FRICKE, R. ROY and S.M. SHAH
f about a point a:
f(z)
f(n)
7.
n=O
and let k
(z
a)
n
a
be the largest nonnegative integer such that
a
If (ka)
(a)
If (n)
>
Define l(r,f)
sUP’a’-<rl
SUPr_ l(r,f)
lent to (i.i).
<
for
0 1 2
n
n!
(ka)
If lim
(a)
n!
k
a
then f is said to be of b.i.
This definition is equiva-
Frank and Mues [8] showed that if f is an entire function of finite
order p, then
+ l(r,f)
log
max(O -I) < llm sup
]o= r
r-o
<
We now state an extension of this theorem.
THEOREM 4.13.
(a) Suppose the hypothesis of Theorem 4.9 is satisfied.
there is a solution w of order
%1
Then
The index l(r,w) of this solution
given by (4.8).
w satisfies
+
lim
(b)
If deg
Po
->
maxk> 0
deg
Pk’
log
l(.r,w)
X
log r
1
1
then
max
o
k
k>0
k
<_i
and any entire solution of (4.2) satisfies
lim l(r,w) <
rWe omit the proof of (a) which is similar to the proof of Theorem 2 of [7].
second part (b) is a
(iv)
restat’ement
The
of Theorem 4.8.
Heath considers vector-valued entire functions of b.i. and proves a result,
similar to Theorem 4.2, for vector equations.
THEOREM 4.14.
A
[rij
(Heath [18]).
If F is an entire solution of F’
AF + Q where
is a matrix whose entries are rational functions which are bounded at
infinity and Q is a vector whose entries are rational functions which are bounded
at infinity, then F is a function of bounded index.
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
429
5. BOUNDED INDEX AND SUMMABILITY [ETHODS.
A sequence X
We begin with definitions and notations.
bers is an entire sequence if
if f(z)
that is
Ek= 0
entire sequences by
k=0
z
k
z
E. An
k
k
lk= 0 Ilq
{}o
of complex
Hum-
converges for every positive integer q,
We will denote the set of
function.
Is an entire
{
entire sequence X
o
is of bounded index if
is of b.i., and we denote the set of sequences of b.i. by
f(z)
B.
Furthermore, let c o be the set of null sequences, c the set of convergent
sequences,
be the set of absolutely convergent sequences, that is
{
{X
o
k=0
and let
i
is bounded
and
i
as
/0
k +=}
Then X can be regarded as the collection of functions f(z)
Ek= 0
z
k
of
exponential type of order i and type 0.
If R and S are collections of sequences, then a matrix A
(a
n k
is an R-S
method if it maps sequences of R to sequences in S.
(an, k
The Taylor matrix T(r)
is defined by
k) (l_r)n+l k-n
for k -> n
0
otherwise
r
n
n,k
THEOREM 5.1.
[35, p. 60]
(Fricke and Powell [12]).
The Taylor matrix is a
B-B method,
that is, maps sequences of b.i. to sequences of b.i. for any complex number r.
For an entire function f(z) and a sequence
the matrix method A(f,z
i)
(an, k
f(z)
a
k=0
n,
{zi}
of complex numbers, define
by
k(Z
z
n)
k
for n
0,1,2,
We can express the Silverman-Toeplitz conditions for regularity as follows (cf:
[35, p. 23]):
G.H. FRICKE, R. ROY and S.M. SHAH
430
(i)
lim
f(k)
n-oo
(ii)
lim f(z
(iii)
k=O
n
(z)
n
0
+ I)
1
for k
0,i
,and
an,kl
<
(Fricke and Powell [12]).
THEOREM 5.2.
regular for any sequence
0 1
M for some M > 0 and all n
If f is of b.i.o then A(f,z.) is not
1
{z.}0
The proof relies on the fact that if f is a function of b.i. and {a
sequence of complex numbers such that lim
n-o
then for any r > 0,
Let A’(f,z i)
limn_> maXlz_an [=r{If
(bn, k)
We then have the following:
THEOREM 5.3.
sup
n
(ii)
-
z
k)
A’ (f,z i)
(z)
n
(a)
n
I}
is an
k
is a
0,I
0,I
0 for k
A(f,zi),
n
that is,
0,i
Let f be of b.i.
method if and only if
{If (k) (Zn) I}
is an
0 for all k
for
(Fricke and Powell [12]).
A’(f,z i)
(i)
f
denote the transpose of
n=O7’ bn,k(Z
f(z)
(k)
(k)
for
<
-E method
k
0,i
if and only if for each integer n
>- 0
there exist an integer p > 0 and a constant M > 0 such that
If(n) (Zk)
-<
pi
for
k
0,i
The part (ii) of this theorem does not necessarily hold for functions of
exponential type and unbounded index.
THEOREM 5.4.
or
A’(f,z i)
is an
-
(Fricke and Powell [12]).
method then A’(f,z
i)
Let f be of b.i.
is an
E-E
If either A(f,z i)
method.
In a recent paper and its corregendum ([43]) Sridhar further examines the
A(f,z i) matrix transformation and obtains results which can be summarized as follows.
THEOREM 5.5.
(Sridhar [43]).
Let f be of b.i.
valent.
(i)
A(f z i) is a
c0
method
(ii)
A(f,z i)
Co-X
method.
is a
Then the following are equi-
ENTIRE SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
(iii)
(iv)
A(f,zi) is
f(k) (Zn)
THEOREM 5.6.
c-E
method.
(n)
as
(Sridhar [43]).
if and only if
for all
n +
Let f be of b.i.
k
Then
0,i
A(f,zi)
is a
c-E* method
I
f(k)
6.
a
i
431
n
O(n)
as
n
for all
/
k
0,i,
FUNCTIONS DEFINED BY DIRICHLET SERIES AND FUNCTIONS OF SEVERAL VARIABLES.
(i)
Let
f(s)
r.
n--O
a
n
exp(s% n)
%
o
> 0
be absolutely convergent everywhere and such that lim
%n+l
> %
n
infn_>(%n+I
n
>
O.
Azpeitia [i] considers entire functions f(s) and proves that if f(s) is of bounded
index N, then it reduces to an exponential polynomial.
Bajpai [2] replaces the
condition of b.i. of f(s) by four conditions and proves that if any one of these
four conditions is satisfied then f(s) reduces to an exponential polynomial.
Gross
[15] and Shah and Sisarcick [41] have considered similar conditions for functions
defined by Taylor series.
(ii)
In [32] Salmassi considers functions f(z)
f(zl, z 2)
and proves the following:
THEOREM 6.1.
(Salmassi [32]).
Let f(z) be of b.i. and a
.
of two variables
Then g(z)
f(az) is also of b.i.
He also obtains a necessary and sufficient condition for f(z) to be of b.i.
A similar theorem for a function of one variable is due to Fricke [9].
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