Keywords: summability, jacobi series, triangular matrix
American Journal of Mathematical Analysis, 2013 1 (3),
pp 4247.
DOI: 10.12691/ajma134
Received October 13, 2013; Revised October 30, 2013; Accepted November 03, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
The Nörlund summability (N, p_{n}) on Jacobi series has been studied by a number of researchers like Gupta ^{[4]}, Choudhary ^{[3]}, Thorpe ^{[16]}, Pandey and Beohar ^{[10]}, Prasad and Saxena ^{[11]}, Beohar and Sharma ^{[1]}, Pandey ^{[9]}, Tripathi et al. ^{[18]} and Chandra ^{[14]}. After quite a good amount of work in the ordinary Nörlund summability of Jacobi series at the point x =1, Khare and Tripathi ^{[5]} discussed the generalized Nörlund summability (N, p, q) of Jacobi series. The (N, p, q) summability reduces to the (N, p_{n}) summability for q_{n} = 1. The Cesàro Summability of Jacobi series has been studied by Szili &Weisz ^{[15]}. The Cesàro Summability, Nörlund Summability, generalized Nörlund Summability are special cases of The matrix Summability method. In this paper a more general result than those Gupta ^{[4]}, Choudhary ^{[3]}, Khare and Tripathi ^{[5]} has been obtained so that their results come out as particular cases.
2. Definitions and Notations
Let f (x) be defined in closed interval [1, 1] such that the function
The Jacobi series corresponding to this function is
 (2.1) 
where
and are Jacobi polynomials.
Let be an infinite lower triangular matrix method T satisfying the Silverman Töeplitz ^{[17]} conditions of regularity i.e.
, for k > n and where M is a finite positive constant.
Let be an infinite series whose n ^{th} partial sum is given by
The sequence  to  sequence transformation
defines the sequence {t_{n}}of matrix means of sequence {s_{n}}, generated by the sequence of coefficient (a_{n,k}).
If
then the series or sequence is said to be summable by matrix method to s. It is denoted by
We use the following notations:
 (2.2) 
A being fixed constant.
3. Main Theorem
The purpose of this paper is to establish a theorem under a very general condition so that it generalizes all the known results for Nörlund summability (N,p_{n}) of Jacobi series in this direction. In fact, we prove the following:
Theorem: Let T = (a_{n,k}) be an infinite lower triangular regular matrix such that the element (a_{n,k}) is positive, monotonic increasing in k for 0 ≤ k ≤ n, and
If
 (3.1) 
then the Jacobi series (2.1) is summable (T) to the sum A at x = 1 provided is positive monotonic nondecreasing function of t such that
 (3.2) 
and the antipole condition
 (3.3) 
is satisfied.
4. Lemmas
The following lemmas are required for the proof of the theorem:
Lemma 4.1. (Szegö, ^{[13]}): If α > 1, β > 1 then as
where
Lemma 4.2. (Gupta, ^{[4]}): The antipole condition (.3.3) includes
 (4.2.1) 
b fixed, and
 (4.2.2) 
Lemma 4.3 Condition (3.1) is equivalent to
 (4.3.1) 
Proof:
Conversely
Lemma 4.4 If ( a_{n,k}) is nonnegative and nondecreasing with 0 ≤ k ≤ n, then, for 0 ≤ a < b≤ ∞, 0 ≤ t ≤ π and for any n,
 (4.4.1) 
where .
Lemma 4.4 may be proved by the following technique of Lemma 4.1 in Lal ^{[6]}.
Lemma 4.5 Under the condition of the theorem on (a_{n,k}), for large n, uniformly in, ,
 (4.5.1) 
where
Proof:
by Abel’s Lemma.
Lemma 4.6 Under the hypothesis of the theorem,
 (4.6.1) 
Proof:
since Also, and putting this in the above gives the result
Lemma 4.7 Let
where
then for and if a_{n, k }satisfies the hypothesis of the theorem,
Proof: For
For
For
5. Proof of the Theorem
Following the Obrechkoff ^{[8]}, the n^{th} partial sum of the Jacobi series (2.1) at the point x =1 is given by
where denotes the nth partial sum of the series
where
Rau ^{[12]} has shown that
Therefore
where is defined as in Lemma 4.7
The matrix mean of the Jacobi series (2.1) at x =1, is given by
In order to prove the theorem, we have to show that
Let us denotes
 (5.1) 
δ being a suitable constant.
 (5.2) 
In order of to estimate I_{2}, we employ the asymptotic relation given in 4.7.3),
thus
 (5.3) 
Now, for I_{2.1},_{ }given є> 0 choose δ such that if 0 <t ≤ δ, then
 (5.4) 
We have, I_{2.1.1}
 (5.5) 
Again, for I_{2.1.2}
and using the change of variables , we get ( assuming that δ< 1),
 (5.6) 
If m is the integers with, then
 (5.7) 
Now, for I_{2.1.2.2,}
 (5.8) 
Collecting (5.3) – (5.8), we get
 (5.9) 
Considering I_{3}, we have
Finally, we consider I_{4,}
Collecting (5.1), (5.2), (5.9), (5.10) and (5.11) we get
Thus, theorem is completely established.
6. Applications
The following particular cases are obtained:
(1) The result of Gupta ^{[4]} becomes particular case of our main theorem if,
(2) The result of Chaudhary ^{[3]} becomes particular case of our theorem if,
(3) The result of Khare and Tripathi ^{[5]} becomes particular case of our main theorem if,
7. Conclusion
Cesàro, Nörlund, generalized Nörlund Summability methods are the particular cases of matrix Summablity method. In this paper matrix Summability method taken with a condition (3.1) on the Jacobi series (2.1) so that series (2.1) is summable at x=1 to sum A. The result of Gupta ^{[4]}, Chaudhary ^{[3]} and Khare and Tripathi ^{[5]} are particular cases of my result.
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