Journal of Mathematics

Volume 2013 (2013), Article ID 162429, 7 pages

http://dx.doi.org/10.1155/2013/162429

Research Article

Some New Recurrence Relations Concerning Jacobi Functions

Hatem Mejjaoli¹ and Ahmedou Ould Ahmed Salem²

¹Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah AL Munawara, Saudi Arabia

²Department of Mathematics, College of Sciences, King Khalid University, Mohayil, Saudi Arabia

Received 6 December 2012; Revised 15 January 2013; Accepted 29 January 2013

Academic Editor: Baoding Liu

Copyright © 2013 Hatem Mejjaoli and Ahmedou Ould Ahmed Salem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functions . Based on an asymptotic relationship between the Jacobi function and the Bessel function, the expression of Bessel function in terms of elementary functions follows as particular cases.

1. Introduction

It is well known that the Jacobi function , , , is defined (cf. [1]) as the even -function on which satisfies the differential equation

where is the Jacobi operator given by

In this paper some new summation formulas for the “modified” Jacobi functions

are proved. We establish, under some conditions, that

where stands for a constant that will be determined. A same of the last formula with respect to the dual variable is also shown, another formulas are proven and some examples are treated. We note that the subject for the generalization of recurrence relations concerning special functions was studied by many authors (cf. [2–5]).

The remaining part of the paper is organized as follows. In Section 2 we recall the main results about some necessary notions related to Jacobi functions, Bessel functions, and Macdonald's functions. Section 3 is devoted to establishing some new formulas concerning “modified” Jacobi functions and some examples are given.

2. Preliminaries

This section gives an introduction to the Jacobi function, Bessel function of the first kind, and Macdonald's function. Main references are [1, 6–11].

2.1. Jacobi Function

A Jacobi function (cf. [1, 6]) can be expressed as a hypergeometric function under form

This function satisfies the following properties.

(i) For every , is an entire function of , and .(ii) For each and for each nonnegative integer there exists a positive constant such that

where if and if . (iii) The function for possesses the Laplace type integral representation

where is explicitly given in [8] and it is a positive function on if and .

We note that the function possesses also an integral representation with respect to the dual variable [12].

To finish this paragraph, we consider the “modified” Jacobi function

which can be written (cf. [7], page 693) as follows:

where and denote, respectively, the Bessel function of the first kind and Macdonald's function.

Example 1. The functions and can be expressed in terms of elementary functions as

2.2. Bessel Function of the First Kind

We recall that the Bessel function of the first kind and order denoted by is defined as an analytic function on by

and possesses the following integral representation:

It satisfies the following functional relations:

It also verifies [10]

with

and satisfies

It has, for and , the following asymptotic representation:

2.3. Modified Bessel Function of the First Kind and Macdonald's Function

We recall that the modified Bessel function of the first kind of order and Macdonald's function of order are defined as analytic functions on by the formulas

The Macdonald's function satisfies

It also verifies (cf. [10])

where is the constant defined by (17).

For and , it can be written under the representation integral ([9], page 119)

We recall also that the modified Bessel function of the first kind and Macdonald's function have, for and , the following asymptotic representations ([9], page 136):

Note that in the special case , Macdonald's function is given by

To complete this paragraph, we recall the following (cf. [7], page 684 and page 747).

(i)If , then we have

(ii)If and , then we have

(iii)If and , then we have

3. Recurrence Relations for

3.1. Some Reccurrence Relations with Respect to Parameters and

Proposition 2. Let . If , with and , then one has

where stands for the constant given by (17).

Proof. Formula (16) asserts that

We multiply both sides of the last identity by and integrate with respect to from to . Then use formula (9) to obtain the required formula.

Proposition 3. For such that , one has

Proof. At first, we suppose that and . With the help of formulas (14) and (9), we obtain

On the other hand, by using formulas (14), (28), and (13), we can deduce that

where is a positive constant.

Taking account of the asymptotic formulas of Macdonald's function (30) and (32) (or formula (34)) and taking in any compact of , we can deduce the result in the case . According to principle of analytic continuation, the restriction used can be dropped.

Using Propositions 2 and 3, we obtain the following corollary.

Corollary 4. If , then one has

Example 5. Using Example 1 and relation (42) for and , we obtain

Example 6. Using Example 1 and relation (43) for and , we obtain

Example 7. Using Example 5 and relation (42) for and , we obtain

Example 8. Using Example 7 and relation (42) for and , we obtain

Example 9. Using Example 5 and relation (43) for and , we obtain

Using the previous relation and relation (42) for and , we obtain

Proposition 10. For such that , one has

Proof. According to formula (15), we obtain

and consequently

By using formulas (14), (28), (13), and (34), we deduce the result.

Corollary 11. For such that and one has

Proof. The result follows by induction argument.

3.2. Some Reccurrence Relations with Respect to the Dual Variable

We give now some recurrence relations with respect to the dual variable in the next proposition.

Proposition 12. If and , then one has

where is the constant given by (17). For all , we have

Proof. The first equality is an immediate consequence of formulas (27) and (9) while the second is proved by using (26) and the asymptotic representations of and .

As a consequence of this proposition, we have the following corollary.

Corollary 13. If and , one has

Example 14. By using formulas (5.32) (in [1], page 44) and (10), we can see that

and therefore

Using now the last corollary, we can get

which coincided with the value of already calculated at and we have

Remark 15. By using (33) and the fact that (cf., e.g., [7], page 707)

we can get

As an application of the last corollary, we obtain

3.3. Some Summations Intervening the “Modified” Jacobi Functions

Proposition 16. If and , then one has

Proof. Using formulas (18), (34), and (9), we obtain

With the help of formulas (13), (28), and (34), we can see that

which permits to conclude the convergence of series

and consequently we obtain the first equality. The second identity is obtained by using formulas (20), (9), and (35).

Remark 17. With the help of formulas (62) and (65), we deduce that

Using now Corollary 13, we see that

Proposition 18. If then one has

Proof. The first identity deduced from (21), (9), and (36). The second follows from (22), (9), and (35).

Remark 19. By using the fact

formula (62), and the last proposition, we can see that

Remark 20. It is well known that the hypergeometric function tends to the confluent hypergeometric function as and such a way that . Consequently, as ,

tends to the normalized Bessel function

Using this remark, the expression of Bessel function in terms of elementary functions for some particular values of the parameters follows as particular cases of our findings; we get the results given in [7, 9, 11].

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