Some New Recurrence Relations Concerning Jacobi Functions
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.
It is well known that the Jacobi function , , , is defined (cf. ) 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.
2.1. 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  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 .
To finish this paragraph, we consider the “modified” Jacobi function which can be written (cf. , 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  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. ) where is the constant defined by (17).
For and , it can be written under the representation integral (, 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 (, page 136): Note that in the special case , Macdonald's function is given by
To complete this paragraph, we recall the following (cf. , 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).
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.
Corollary 4. If , then one has
Proposition 10. For such that , one has
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
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 , 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
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).
Proposition 18. If then one has
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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