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Summability of a Jacobi Series by Lower Triangular Matrix Method

Binod Prasad Dhakal

Tribhuvan University, Nepal

Abstract: The Jacobi polynomial Pn⁽α,β⁾(x) which is obtained from Jacobi differential equation is an orthogonal polynomial over the interval [-1, 1] with respect to weight function (1-x)α(1+x)β, α>-1, β>-1. Here Jacobi series has been taken and established a theorem on lower triangular matrix summability of a Jacobi series.

Keywords: summability, jacobi series, triangular matrix

Received October 13, 2013; Revised October 30, 2013; Accepted November 03, 2013

Citation: Binod Prasad Dhakal, Summability of a Jacobi Series by Lower Triangular Matrix Method. American Journal of Mathematical Analysis, 1 (3). 42-47. January 2013. doi:10.12691/ajma-1-3-4

1. Introduction

The Nörlund summability (N, pn) 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, pn) summability for qn = 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

[picture] | (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 {tn}of matrix means of sequence {sn}, generated by the sequence of coefficient (an,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:

[picture] | (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,pn) of Jacobi series in this direction. In fact, we prove the following:

Theorem: Let T = (an,k) be an infinite lower triangular regular matrix such that the element (an,k) is positive, monotonic increasing in k for 0 ≤ k ≤ n, and

If

[picture] | (3.1)

then the Jacobi series (2.1) is summable (T) to the sum A at x = 1 provided is positive monotonic non-decreasing function of t such that

[picture] | (3.2)

and the antipole condition

[picture] | (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

[picture] | (4.2.1)

b fixed, and

[picture] | (4.2.2)

Lemma 4.3 Condition (3.1) is equivalent to

[picture] | (4.3.1)

Proof:

Conversely

Lemma 4.4 If ( an,k) is non-negative and non-decreasing with 0 ≤ k ≤ n, then, for 0 ≤ a < b≤ ∞, 0 ≤ t ≤ π and for any n,

[picture] | (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 (an,k), for large n, uniformly in, ,

[picture] | (4.5.1)

where

Proof:

by Abel’s Lemma.

Lemma 4.6 Under the hypothesis of the theorem,

[picture] | (4.6.1)

Proof:

since Also, and putting this in the above gives the result

Lemma 4.7 Let

where

then for and if an, k satisfies the hypothesis of the theorem,

Proof: For

For

For

5. Proof of the Theorem

Following the Obrechkoff [8], the nth 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

[picture] | (5.1)

δ being a suitable constant.

[picture] | (5.2)

In order of to estimate I₂, we employ the asymptotic relation given in 4.7.3),

thus

[picture] | (5.3)

Now, for I2.1, given є> 0 choose δ such that if 0 <t ≤ δ, then

[picture] | (5.4)

We have, I2.1.1

[picture] | (5.5)

Again, for I2.1.2

and using the change of variables , we get ( assuming that δ< 1),

[picture] | (5.6)

If m is the integers with, then

[picture] | (5.7)

Now, for I2.1.2.2,

[picture] | (5.8)

Collecting (5.3) – (5.8), we get

[picture] | (5.9)

Considering I₃, we have

Finally, we consider I4,

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.

References

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[2]   | Borwein, D.: On products of sequences, J. London Math. Soc., 33, 352-357. 1958.
[3]   | Choudhary, R. S.: On Nörlund summability of Jacobi series, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 52, 644-652. 1972.
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[12]   | Rau, H.: Über die Lebesgueschen Konstanten der Reihenentwicklungen nach Jacobischen Polynomen (German), J. f. M. , 161, 237-254. 1929.
[13]   | Szegö, Gabor: Orthogonal polynomials, Amer. Math. Soc. Colloquium Publ., 23. 1959.
[14]   | Chandra Satish: On double Nörlund summability of Fourier-Jacobi series, The Islamic University Journal, 15(2), 1-14, 2007.
[15]   | Szili L. and Weisz F. : Uniform CesàroSummability of Jacobi-Fourier series, Acta Math. Hunger, 127(1-2), 112-138, 2010.
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[17]   | Töeplitz, O.: Über allgemeine lineare Mittelbildungen, Prace mat. - fiz., 22, 113 -119. 1913.
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