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.

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  1/2,1/2  and Φ 1/2,1/2  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 { ∈ C : | arg | < } by and possesses the following integral representation: It satisfies the following functional relations: It also verifies [10]   ( − () − (−1) with and satisfies It has, for  ≥ 0 and  ≥ 0, the following asymptotic representation: ,   → 0, The Macdonald's function satisfies It also verifies (cf.[10]) where    () is the constant defined by (17).For Re  > 0 and  ∈ C, 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  > 0 and  ≥ 0, the following asymptotic representations ( [9], page 136): ,   → 0, (29) Note that in the special case  = ±1/2, Macdonald's function is given by To complete this paragraph, we recall the following (cf.[7], page 684 and page 747).
Proof.Formula (16) asserts that We multiply both sides of the last identity by   () and integrate with respect to  from 0 to ∞.Then use formula (9) to obtain the required formula.
Using Propositions 2 and 3, we obtain the following corollary.
Corollary 11.For , ,  ∈ C such that | Im | < Re(++1) and  ∈ N one has Proof.The result follows by induction argument.

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.(55) 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   .
Remark 17.With the help of formulas (62) and (65), we deduce that