773
Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 773-780
TRANSLATIVITY OF ROGOSlNSKI SUMMABILITY METHODS OF DIFFERENT ORDERS
AZMI K. AL-MADI
Department of Mathematics
U.A.E. Unlverslty AI-Ain
United Arab Emlrates
(Recelved June 15, 1988)
ABSTRACT.
methods
The paper deals with the problem of translatlvlty of Rogoslnskl summability
r),,
(R,
when r
(Rh, O)
is
<
of orders h, r; 0
[I
0, h e
neither
,[],
(Rh,0)
translatlve
to
the
I/2)
(R
neither
is
h 0
(0,
I/2)’ (Rh,r)
|/2,1], (R
h
results establish both,
to
the
I
right.
The
problem is
,r
left
with the conjecture that if
In this
is translatlve, and when
translatlve to the
neither
is
I/2
nor
translatlve to the right nor to the left.
paper we prove that when h-r e
h-r e
left
(0,
unsettled for the rest of the interval
h e (0,
It has been shown by the author [i] that
(Z-I)),
O, h e (0,
h-r < I.
is translatlve, and when r
right nor to the
These
left.
the open problem and its conjecture which have been given by
the author [I].
KEY WORDS AND PHRASES.
Translatlve, Rogoslnskl methods, (C,I) method, left and right
translatlvlty, equivalent to convergence.
Primary 40G05, Secondary 40D99.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
+ a +... of real or complex terms with its partial sums
0
+
a +...+ a is said to be summable by the Rogoslnskl method
0
n
);0 < h-r < I, if t +t, as n+% where
The series a
S
a
n
(R
h ,r
n
n
t
n
and
[ cos(k+r)Snak,
(1.1)
k=0
0 _-__! n
2(n+H)’
n
0,1,2
In the special case in which r
Bernsteln method (R
h)
which has
Agnew [2] and Petersen [3]).
if M
n
t
M as n
thls method will reduce to the Rogoslnskl-
The series is evaluable to t
[I],
(C,I) method,
(see Ai-Madl
the
M by
% where
S
M
0,
been the subject of many papers
n
0
+... +S n
+S
n
+
(.)
774
A. K. AL-MADI
.
For a summability method A
summable A to s, then so is
if A
T
Tiff
(translat[ve to the left) if, whenever
T
Tr
A
an-’’1
.
an
is
if the converse holds. A e T if, and only
r
Much work has already been done on translative summability methods (see AI-Madi
[1,4,5], Chowdhury [6], and Kuttner [7-10].
I/2 ,i],
The author [I] proved that if h e
(Rh, O)
then
interval
2.
T U
(0,
I/2 ).
Tr.
The
author
left
(Rh, 0)
e T, and if 0
problem unsettled
the
(0,
He conjectured that when h
I/2 ),
<
for
(Rh, O)
then
< I/2(--I),
h
the
T U
of
rest
the
Tr.
OBJECT OF THE PAPER.
The
oF
object
then (R
e T
paper
this
(0,
when h-r
and
prove
to
is
I/2 ),
then (R
T
that
U T
I/2
when h-r e
second
The
I],
result
r
h Dr
h ,r
establishes both the open problem and its conjecture which have been given in [I].
3.
MAIN RESULTS.
In this section we will prove the following three main results:
THEOREM 3.1.
If h-r
THEOREM 3.2.
If h-r
THEOREM 3.3.
If h-r
I/2
E
I]
i/2
g
(0,
then
I/2 ),
(R
h ,r
E
T.
(Rh,r) T.
then (Rh,r)
TIDTr.
The following result will be used in the proof of Theorems 3.1 and 3.2.
.
(Agnew [II]) If
THEOREM 3.4.
n
H
n
d
k=O
n
(3.1)
kSk
is regular and
llm
inf
n
[Id n,n
.
n-I
d
n k
I] >
(3 2)
0
then the transformation given by (3.1) is equivalent to convergence.
PROOF OF THEOREM 3.1.
(C,I)
are equivalent,
We ill
show
that
when h-r
E
(1_
I], then
and the result follows from the fact that (C,I)
T.
(Rh,r)
Observe
that since
a
n
it follows form
(n+l)M
n
2nM n-1 + (n-l)M n-2
n
0 1,2,
M_I
M _2
0
(3.3)
(I.I) that
tn
=k=O An’ kMk’
and
(3.4)
TRANSLATIVITY OF ROGOSINSKI SUMMABILITY METHODS OF DIFFERENT ORDERS
775
w1e re
A
A
A
(3.5)
(n+l)sin(h-r)O
n,n
n
1/20nCOS I/2 (2h-2r+l)On
n[2sln
n,n-1
-4(k+l)sin
n,k
2
n]
I/20cos(k+r+l)O
n
n
sin(h-r)O
(3.6)
n-2
(3.7)
0
k
and
A
n,k
0
>
k
(3.8)
n.
Using the same technique as Agnew [2; p.
(Rh,r)
(C,I).
is equivalent to
PROOF OF THEOREM
translative
summability
to (R
the
in
To prove
3.2.
case
method,
in
544-545],
the
result,
to
show
and
which h-r
=I
h
sequence method Q given by the transformation
n
Qn
where
Mn
is
2n---l
+ n+l
Mn-I
we see that if h-r is in
it
that
For this,
enough
is
I],
consider
consider
to
method
this
we
some
equivalent
is
the
sequence-to-
(3.9)
Mn’
Using (1.3) and (3.9) to obtain
given by (1.3).
(I/3
This completes the proof.
Qn
in terms
Qn’
of
the
result is
Qn+l 2n+____1_
2n+3 Qn"
(3.10)
This implies that Q E T.
Next
A
n,k
put h-r
=I
-2(k+l)sec
and write A
n,k
given in (3 7) in the form
1/20nsin 2 I/2 Ontcos(2k+2r+l) I/2 On+
and use (3.9) to obtain the transformation
(Rh,)(Q)
r
cos(Zk+2r+3)
I/2 On
(3.11)
-1
n
t
I
n
k=O
where
Bn, n
8n, k
(3.12)
nkQk
(2n+l)cos
-2(2k+l)sec
(n+r)n,
(3.13)
I/20nsln 2 I/20nCOS(2k+2r+l) I/2 On
0
(
k
n-l.
(3.14)
It is easily seen that the transformation given by (3.12) is regular. Hence applying
Theorem 3.4 to the matrix given in (3.12), one can easily show that the transformation
given by (3.12) is equivalent to convergence, and
is equivalent to Q.
This
(Rh,r)
completes the proof.
A.K. AL-MADI
776
PROOF OF THEOREM 3.3. We will show first that If h-r
For this we consider the sequence {V
V
S
n
+ S
0
+
n
+ S
E
(0,
I/2
then (R
h r
T
given by
(3.15)
n’
where
an Vn
2Vn_l + Vn-2’ (V_k=
0 for positive integer k).
(3.16)
Using (3.16), it follows from (l.l) that
n
t
=k =0 Cn kVk
n
where
C
sin(h+r) 8,
nn
C
(3.17)
(3.18)
n
sin(l+h+r)O-
n,n-I
(3 19)
2sin(h+r)On,
n
and
C
Let
-4sin
n,k
(tn} {tn}
terms of
be
2
I/2 8_cos(k+r+l)8_
respectively
the
(0
(Rh,r)
transforms
(3 20)
n-2)
k
{Sn }’ {Sn-l}"
of
Obtain
tn
in
The result is
n
n
(3.21)
Fn,ktk+l
n
where
n
F
and C
of C
n,v
n,v
is
n,k
given
=v!
C
k
(3.18)
by
(0
,vDv+l ,k+l
n
(3.20),
where D
and
(3.22)
n)
k
is
n,v
the
reciprocal
matrix
Write
C
Gn, v Cn+l
v+l
C
n,______v_
_nn
Cn+l, v+l
_],
(0
Cn+l, n+l
n)
v
(3.23)
and use the fact that
n-I
Dn+l,k+l
to obtain from
Cn+l,n+l
n,n-j
:u
G
=I
We will show that if 0
1/2m
j
(3.24)
(3.22) that
F
and n
[ Cn+l,v+iDv+l,k+l,
v=k
n
z
F
n,
<
(0
n-uDn-u+l ,n-j+l
h-r
<I/2
and 0
<
z
j
<I/4,
n).
(3.25)
then for sufficiently large n
bounded.
This implies that the transformation
n,n-j
given in (3.21) is not regular, and consequently, when h-r E (0,
),
z
To prove this, we will show that for sufficiently large n and 2
n
the terms of
is
not
I
(Rh, r) Tr.
777
TRANSLATIVITY OF ROGOSINSKI SUMMABILITY METHODS OF DIFFERENT ORDERS
the sum (3.25) alternate in sign,
and then we will show tIat the limit of F
is
n,n-j
unbounded.
The inversion formula o[ (3.17) gives
Cj+ 1,j
Cj+!,j+I
0
0
0
Cj+2,
Cj+2,j+
Cj+2,j+ 2
0
0
c n,j
c n,n-1
D
C
Using (1.2) and (3.18)
(3.20),
(-1) u-j
lira
n+
C
n,n n-l,n-l’’"
D
,j
(-I) n-j
(3 26)
Cj ,j
we have from (3.26) that
n-u,n-_ wU-J
2
n
(h-r
(3.27)
f’
-h+r
h-r
where w
(3.28)
z,
Next, we will show that when
( n
u (
G
O.
This together with
n ,n-u <
(3.27) imply that the terms of the sum given in (3.25) alternate in sign for all
sufficiently large n uniformly in 2
u
n
j
Using (3.18) and (3.19), we have
from (3.23) that
z.
Gn,n_l
sin(h-r)@n[f(On
f(%+l )1’
(3.29)
where
sin(l+h-r) 0
n
f(6))
Differentiate,
x in
n
and
(%+I,%)
(3.30)
stn(h-r) O
n
use
the
mean
value
theorem
to
see
that
there
Is
some
such that
sin(h-r)O
Gn
n
stn2(h_r)x [On_ On+l
n-1
[sin(l+h-r)xeos(h-r)x-(l+h-r)sin]
(3.31)
sin(h-r)O
n
sin
<
2
(h-r)x
,
n
[8n
On+l][sin(l+h-r)x
(l+h-r)stnx]
(3.32)
(3.33)
A.K. AL-MADI
778
where q is some positive constant.
Next,
u
G
n ,n-u
(3.18)
using
Z
n
-4s n
(3.19),
and
have
we
(3.23)
from
and
(3.20)
that
t1en
2
I/2 O-n+
cos
sin
sin
2
2
n-u+r +
n+l
1/2 9n cos(n-u+r+l)9n
I/2 @n+ICOS(n-u+r+2) n+
sin(h-r)9
n
-d(h-- r
n+ o].
(3.34)
Observe that the quantity inside the square brackets of (3.34) is greater than
sin2 I/2 @n
sin2 I/2 @n+l
sin(h-r) 9
9
n
n
sinCh-r)
9
[__n__
I]
%+I n+l
n+
>
0
(3.35)
Using (3.20) and (3.35), we have from (3.34) that
G
n,n-u
This together with
O
Hence,
<
0
2
u
j
Z
< n
u
j
n
(3.36)
(3.33), imply that
n,n-u
<
0
g
(3.37)
it follows from (3.27) and (3.37) that the terms of the sum (3.25) alternate in
sign for sufficiently large n uniformly in 2
z
when n
n
then
u
j
n
Finally we will show that
1/2z
lim
Using (3.18)
IF n,
.
n-j
(3.38)
(3.20),
and noting that
sin O
3
9--.. + 95 );
0),
(9
(3.39)
we have from (3.23) that
3
G
(l+h-r) (l+2h-2r)
n,n-I
24n
4
(3.40)
and from (3.34) that
3
G
n, n-u
(u + h
4n
r
1)
(3.41)
779
TRANSLATIVITY OF ROGOSINSKI SUMMABILITY METHODS OF DIFFERENT ORDERS
for sufficiently large n uniformly in 2 < u (J
n
z
Using (3.27), (3.40) and (3.41),
we have from (3.25) that
2
F
n,n-j
2(h_r)n
3
[(_l)J-I
[l+h-r][l+2h-2r]
6
j -u
[
+
(u +h-r- )w J
wJ-I
-u
3.42
u=2
w
2
[A + B]
(3 43)
say.
2(h-r)n
Observe that for any x; (x # -I), we have
(3.44)
(j+h-r-l)k(x) -xk’(x), (with x instead of w),
B
where
x + x 2-
k(x)
Observe that w +
(h-r)
-I.
(3.45)
x) that
It follows from (3.44) and (3.45) (with w
(-t)J-lwJ-l+
(-h-r)
B
x3+...+ (-l)J-2xj-2.
jh-
(3.46)
Jr.
Using thls, it follows from (3.43) that for all sufficiently large n and 2
2
F
n, n-]
2 (h-r)n
Hence, when h-r e (0,
nl/2z ]
n
I/2);
3
[(-l)J-lwJ-I
that is when w
>
[1-h+rl[l-2h+2r] + (h-r) ].
j
6
I,
it follows from
]
n
z
(3.47)
(3.47) that when
z
n
and (3.38) is satisfied.
n ,n-J
This completes the proof of the first part.
part, we use (3.17) to obtain
n
in terms of t
n
For the second
The result is
n
t--n+l k__[0 Hn,ktk’
where
n
H
and where
(3.48)
n,k
=v[
k
Cn,k
and
C
[cn, nCn+l
nn
Dn,k
Cn+l v+lDv,k
(0
(3 49)
n)
k
are given in the proof of the first part.
v+l
Cn+l,n+lCn ,V
Cn+l n+l
C
n,n
G
n,v’
Using the identity
(3.50)
A.K. AL-MADI
780
where G
I,V
is given by
It
Cn+
C
n,n-]
Observe that since D
H
(3.23),
,n+l
Cn+
C
g
u=l
n,n
D
n-u, n-j
n,n-j
we have from
n.+
F
(3.51)
D
n,n-u n-u ,n-j
n-u+ I, n-j + I’
n,rl
(3.24) and (3.4q) that
we have from
(3.25) and (3.51) that
n,n-j
is not
Using (3. I8) and (3.47), it follows from (3.52) that when 0 < h-r < 1/2, H n
n-j
z
Z
This implies that the
n
bounded for all sufficiently large n and and n
I’-
transformation given by (3.48) is not regular; that is when h-r e
(R
h ,r
T
(0,
I/2
This completes the proof of leorem 3.
REFERENCES
I.
AL-MADI, A.., On Translativity of Rogosinski-Bernstein Trigonometric
Summab[lity Methods, Internat. J. Maths. and Math. Sci. (to appear).
2.
AGNEW, R.P., Rogosinski-Bernstein Trigonometric Summability Methods and
Modified Arithmetic Means, Annals. of Maths. 56(3) (1952), 537-59.
3.
PEFERSEN, G.M., A Note On Divergent Series, Can. J. of Maths. 4 (1952), 445-454.
4.
AL-MADI,
A.K.,
On
Translativity
of
the
Product
of
Riesz
Summability Methods,
Indian J. Pure and Applied Maths. Nov. (1980)
5.
6.
7.
II(Ii), 1144-57.
AL-MADI, A.K., On Translativity of the Product of Nrlund Summability Methods,
Jour. of the Indian Math. Soc. 44 (1980), 83-90.
CHOWDHURY, B., Semitranslative Summability Methods, Studia Sci. Math. and P.
Vermes Hunar
(1966), 403-410.
KUTTNER,
B.,
On
Translative
Summability Methods,
KUTTNER,
B.,
The
Problem
of
the
Ramanujan
(1970), 35-45.
Institute of Mathematics
8.
Publication
of
Translativity
For
Hausdorff
Summability,
Proc.
London Math. Soc. 6 (1957), 117-138.
B., On Translative Summability Methods (II), J. London Math. Soc. 4
(1971), 88-90.
i0. KUTTNER, B., On De La Vallee Poussin and Abel Summability, J. London Math.
Soc. (2), (5) (1972), 303-309.
II. AGNEW, R.P., On Equivalence of Methods of Evaluation of Sequences. Thoku
Mathematical Journal, 35, (1932), 244-252.
9.
KUTTNER,