## Summability of a Jacobi Series by Lower Triangular Matrix Method

Tribhuvan University, Nepal### Abstract

The Jacobi polynomial P_{n}^{(}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

*American Journal of Mathematical Analysis*, 2013 1 (3),
pp 42-47.

DOI: 10.12691/ajma-1-3-4

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

**Copyright:**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Dhakal, Binod Prasad. "Summability of a Jacobi Series by Lower Triangular Matrix Method."
*American Journal of Mathematical Analysis*1.3 (2013): 42-47.

- Dhakal, B. P. (2013). Summability of a Jacobi Series by Lower Triangular Matrix Method.
*American Journal of Mathematical Analysis*,*1*(3), 42-47.

- Dhakal, Binod Prasad. "Summability of a Jacobi Series by Lower Triangular Matrix Method."
*American Journal of Mathematical Analysis*1, no. 3 (2013): 42-47.

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### 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 non-decreasing 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 non-negative and non-decreasing 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.

### References

[1] | Beohar, B. K. and Sharma, K. G.: On Nörlund summability, Indian J. Pure Appl. Math., 11, 1475-1481. 1980. | ||

In article | |||

[2] | Borwein, D.: On products of sequences, J. London Math. Soc., 33, 352-357. 1958. doi: 10.1112/jlms/s1-33.3.352 | ||

In article | CrossRef | ||

[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. | ||

In article | |||

[4] | Gupta, D. P.: D.Sc. Thesis, University of Allahabad, Allahabad, 1970. | ||

In article | |||

[5] | Khare, S. P. and Tripathi, S. K.: On (N,p,q) summability of Jacobi series, Indian J. Pure Appl. Math., 19( 4), 353-368. 1988. | ||

In article | |||

[6] | Lal, Shyam: On the degree of approximation of conjugate of function belonging to Weighted W (L^{p},ξ(t)) class by matrix summability means of conjugate series of a Fourier series, Tamkang J. Math., 31(4), 279-288. 2004. | ||

In article | |||

[7] | Nörlund, N. E : Sur une application des fonctions permutables, Lund. Universities Arsskrift, 16, 1-10. 1919. | ||

In article | |||

[8] | Obrechkoff, N.: Formules asymptotiques pour les polynômes de Jacobi et sur les séries suivant les memes polynômes, Ann. Univ. Sofia, Fac. Phys.-Math., 32, 39-135. 1936. | ||

In article | |||

[9] | Pandey, B. N.: On the summability of Jacobi series by (N,p_{n}) method, Indian J. Pure Appl. math., 12 (12), 1438 -1447. 1981. | ||

In article | |||

[10] | Pandey, G. S. and Beohar, B. K.: On Nörlund summability of Jacobi series, Indian J. Pure Appl. Math., 9(5), 501-509. 1978 | ||

In article | |||

[11] | Prasad, Rajendra and Saxena, Ashok: On the Nörlund summability of Fourier-Jacobi series, Indian J. Pure Appl. Math., 10(10), 1303-1311. 1979. | ||

In article | |||

[12] | Rau, H.: Über die Lebesgueschen Konstanten der Reihenentwicklungen nach Jacobischen Polynomen (German), J. f. M. , 161, 237-254. 1929. | ||

In article | |||

[13] | Szegö, Gabor: Orthogonal polynomials, Amer. Math. Soc. Colloquium Publ., 23. 1959. | ||

In article | |||

[14] | Chandra Satish: On double Nörlund summability of Fourier-Jacobi series, The Islamic University Journal, 15(2), 1-14, 2007. | ||

In article | |||

[15] | Szili L. and Weisz F. : Uniform CesàroSummability of Jacobi-Fourier series, Acta Math. Hunger, 127(1-2), 112-138, 2010. doi: 10.1007/s10474-009-9101-2 | ||

In article | CrossRef | ||

[16] | Thorpe, B.: Nörlund summability of Jacobi and Laguerre series, J. Reine Angew. Math., 276, 137-141. 1975. | ||

In article | |||

[17] | Töeplitz, O.: Über allgemeine lineare Mittelbildungen, Prace mat. - fiz., 22, 113 -119. 1913. | ||

In article | |||

[18] | Tripathi, L. M., Tripathi, V. N. and Yadav, S. J.: On Nörlund summability of Jacobi series, Math. Soc., 4, 183-193. 1988. | ||

In article | |||

[19] | Zygmund, A.: Trigonometric series, Cambridge University Press, 1959. | ||

In article | |||