Some New Recurrence Relations Concerning Jacobi Functions

Hindawi Publishing Corporation Journal of Mathematics Volume 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 1 2

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

Correspondence should be addressed to Hatem Mejjaoli; [email protected] Received 6 December 2012; Revised 15 January 2013; Accepted 29 January 2013 Academic Editor: Baoding Liu Copyright © 2013 H. Mejjaoli and A. 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. 𝛼,𝛽

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functions Φ𝜆 . 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

are proved. We establish, under some conditions, that 𝛼,𝛽

It is well known that the Jacobi function 𝑡 → 𝜑𝜆 (𝑡), 𝛼, 𝛽, 𝜆 ∈ C, 𝛼 ≠ − 1, −2, −3, . . ., is defined (cf. [1]) as the even 𝐶∞ function on R which satisfies the differential equation 2

𝐿 𝛼,𝛽 𝑢 (𝑡) = − (𝜆2 + (𝛼 + 𝛽 + 1) ) 𝑢 (𝑡) , 𝑢 (0) = 1,

󸀠

𝑢 (0) = 0,

𝐿 𝛼,𝛽 =

𝑑 𝑑 + [(2𝛼 + 1) coth 𝑡 + (2𝛽 + 1) tanh 𝑡] . 2 𝑑𝑡 𝑑𝑡

𝛼,𝛽

𝛼−𝑚,𝛽+𝑚

𝑚−1

(𝑡) ,

𝛼+𝑚,𝛽+𝑚

(𝑡) − (−1)𝑚 Φ𝜆

𝛼+𝑗,𝛽+𝑗

= ∑ (−1)𝑗 𝑏𝑚𝑗 (𝛼) (sinh 𝑡)𝑗 Φ𝜆

(4) (𝑡)]

(𝑡) ,

𝑗=0

(2)

𝛼,𝛽

× (sinh 𝑡)𝛼 𝜑𝜆 (𝑡)

2 𝑑 𝛼,𝛽 [Φ ] (𝑡) , cosh 𝑡 𝑑𝑡 𝜆

= (sinh 𝑡)𝛼−𝑚 Φ𝜆 𝛼−𝑚,𝛽+𝑚

𝑡 󳨃󳨀→ Φ𝜆 (𝑡) Γ ((𝛼 + 𝛽 + 1 + 𝑖𝜆) /2) Γ ((𝛼 + 𝛽 + 1 − 𝑖𝜆) /2) 2−𝛽+1 Γ (1 + 𝛼)

(𝑡) =

𝑑 𝑚 2 𝛼,𝛽 ] [(sinh 𝑡)𝛼 Φ𝜆 ] (𝑡) sinh (2𝑡) 𝑑𝑡

(sinh 𝑡)𝑚 [Φ𝜆

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

=

[

𝛼+1,𝛽+1

(𝑡) − Φ𝜆

(1)

where 𝐿 𝛼,𝛽 is the Jacobi operator given by 2

𝛼−1,𝛽+1

Φ𝜆

(3)

where 𝑏𝑚𝑗 (𝛼) 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

Journal of Mathematics

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]. 𝛼,𝛽

2.1. Jacobi Function. A Jacobi function 𝜑𝜆 (𝛼, 𝛽, 𝜆 ∈ C, 𝛼 ≠ − 1, −2, −3, ...) (cf. [1, 6]) can be expressed as a hypergeometric function under form 𝜌 + 𝑖𝜆 𝜌 − 𝑖𝜆 , ; 𝛼 + 1; −sinh2 𝑡) : (𝑡) = 2 𝐹1 ( 2 2

𝛼,𝛽 𝜑𝜆

can be exExample 1. The functions 𝜑𝜆1/2,1/2 and Φ1/2,1/2 𝜆 pressed in terms of elementary functions as ∀𝑡 > 0,

𝜑𝜆1/2,1/2 (𝑡) =

2 sin (𝜆𝑡) , 𝜆 sinh (2𝑡)

(10)

2 2 + 𝑖𝜆 2 − 𝑖𝜆 Φ1/2,1/2 )Γ( ) (𝑡) = √ Γ ( 𝜆 𝜋 2 2

∀𝑡 > 0,

×

sin (𝜆𝑡) 1 , 𝜆 √sinh 𝑡 cosh 𝑡

(11)

|Im 𝜆| < 2,

(5)

𝜌 = 𝛼 + 𝛽 + 1. This function satisfies the following properties. 𝛼,𝛽

(i) For every 𝑡 ∈ [0, +∞[, (Γ(𝛼 + 1))−1 𝜑𝜆 (𝑡) is an entire function of 𝛼, 𝛽, and 𝜆.

2.2. Bessel Function of the First Kind. We recall that the Bessel function of the first kind and order 𝛼 denoted by 𝐽𝛼 is defined as an analytic function on {𝑧 ∈ C : | arg 𝑧| < 𝜋} by ∞

𝑧 𝛼+2𝑘 (−1)𝑘 ( ) Γ (𝑘 + 1) Γ (𝑘 + 𝛼 + 1) 2 𝑘=0

𝐽𝛼 (𝑧) = ∑

(ii) For each 𝛼, 𝛽 ∈ C and for each nonnegative integer 𝑛 there exists a positive constant 𝑘 such that ∀𝑡 ≥ 0,

∀𝜆 ∈ C,

and possesses the following integral representation:

𝑛 󵄨󵄨 󵄨󵄨 𝛼,𝛽 󵄨 󵄨󵄨 −1 𝑑 󵄨󵄨(Γ (𝛼 + 1)) ( ) 𝜑𝜆 (𝑡)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑑𝑡

(6)

𝐽𝛼 (𝑧) =

1 𝛼−1/2 (𝑧/2)𝛼 cos (𝑧𝑡) 𝑑𝑡, ∫ (1 − 𝑡2 ) √𝜋Γ (𝛼 + 1/2) −1

≤ 𝑘(1 + |𝜆|)𝑛+𝑘 (1 + 𝑥) 𝑒(| Im 𝜆|−Re(𝜌))𝑡 , where 𝑘 = 0 if Re(𝛼) > −1/2 and 𝑘 = [(1/2) − Re(𝛼)] if Re(𝛼) ≤ −1/2.

1 Re (𝛼) > − . 2

> −1/2 possesses (iii) The function 𝑡 → the Laplace type integral representation 𝑡

𝛼,𝛽 𝜑𝜆 (𝑡) = ∫ K (𝑡, 𝑦) cos (𝜆𝑦) 𝑑𝑦, (7) 0

where K(𝑡, ⋅) is explicitly given in [8] and it is a positive function on (0, 𝑡) if 𝛼 > −1/2 and |𝛽| ≤ max{1/2, 𝛼}. 𝛼,𝛽 We note that the function 𝜑𝜆 possesses also an integral representation with respect to the dual variable [12]. To finish this paragraph, we consider the “modified” Jacobi function

(14)

1 𝑑 𝛼 (𝑧 𝐽𝛼 (𝑧)) = 𝑧𝛼−1 𝐽𝛼−1 (𝑧) . 𝑧 𝑑𝑧

(15)

𝑚−1

𝑧𝑚 (𝐽𝛼−𝑚 (𝑧) − (−1)𝑚 𝐽𝛼+𝑚 (𝑧)) = ∑ (−1)𝑗 𝑏𝑚𝑗 (𝛼) 𝑧𝑗 𝐽𝛼+𝑗 (𝑧) , 𝑗=0

(16) with

𝛼,𝛽

Γ ((𝛼 + 𝛽 + 1 + 𝑖𝜆) /2) Γ ((𝛼 + 𝛽 + 1 − 𝑖𝜆/2)) 2−𝛽+1 Γ (1 + 𝛼)

(17)

∞

𝐽0 (𝑧) + 2 ∑𝐽2𝑗 (𝑧) = 1,

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

𝑗=1

𝛼,𝛽

∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) 𝐽𝛼 (𝑥 sinh 𝑡) 𝑑𝑥 = Φ𝜆 (𝑡) , 0

𝑗 2𝑚−𝑗 𝐶𝑚 , Γ (𝛼 + 𝑗 − 𝑚 + 1)

and satisfies

𝛼,𝛽

∞

𝑏𝑚𝑗 (𝛼) = Γ (𝛼 + 1)

(8)

× (sinh 𝑡)𝛼 𝜑𝜆 (𝑡) ,

∀𝑡 > 0,

𝐽𝛼−1 (𝑧) − 𝐽𝛼+1 (𝑧) = 2𝐽𝛼󸀠 (𝑧) ,

It also verifies [10]

𝑡 󳨃󳨀→ Φ𝜆 (𝑡) =

(13)

It satisfies the following functional relations:

𝛼,𝛽 𝜑𝜆 (𝑡) for Re(𝛼)

∀𝑡 > 0, ∀𝜆 ∈ C,

(12)

∞

(9)

|Im 𝜆| < Re (𝛼 + 𝛽 + 1) , where 𝐽𝛼 and 𝐾𝛼 denote, respectively, the Bessel function of the first kind and Macdonald’s function.

∑ (−1)𝑗 𝐽2𝑗+1 (𝑧) =

𝑗=0

sin 𝑧 , 2

(18)

(19)

∞

𝐽0 (𝑧) + 2 ∑(−1)𝑗 𝐽2𝑗 (𝑧) = cos 𝑧, 𝑗=1

(20)

Journal of Mathematics

3

∞

2

∑(−1)𝑗+1 (2𝑗) 𝐽2𝑗 (𝑧) =

𝑗=1 ∞

𝑧 sin 𝑧 , 2

2

∑ (−1)𝑗 (2𝑗 + 1) 𝐽2𝑗+1 (𝑧) =

𝑗=0

𝑧 cos 𝑧 . 2

(21)

We recall also that the modified Bessel function of the first kind and Macdonald’s function have, for 𝑧 > 0 and 𝛼 ≥ 0, the following asymptotic representations ([9], page 136):

(22)

𝐼𝛼 (𝑧) ≍

𝑧𝛼 , 2𝛼 Γ (𝛼 + 1)

𝐼𝛼 (𝑧) ≍

𝑧 󳨀→ 0,

2 1 1 𝐽𝛼 (𝑧) ≍ √ cos (𝑧 − 𝛼𝜋 − 𝜋) , 𝜋𝑧 2 4

(23)

𝐾±1/2 (𝑧) = (

𝑧 󳨀→ 0

𝑧 󳨀→ ∞,

(30)

(31)

𝑧 󳨀→ ∞.

(32)

𝜋 1/2 −𝑧 ) 𝑒 . 2𝑧

(33)

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

∞

(𝑧/2)𝛼+2𝑘 , Γ (𝑘 + 1) Γ (𝑘 + 𝛼 + 1) 𝑘=0

𝜋 −𝑧 𝑒 , 2𝑧

(29)

Note that in the special case 𝛼 = ±1/2, Macdonald’s function is given by

(i) If | Im 𝜆| < 1 + Re(𝛽), then we have

𝐼𝛼 (𝑧) = ∑

𝜋 [𝐼−𝛼 (𝑧) − 𝐼𝛼 (𝑧)] { { , if 𝛼 ∉ Z, {2 sin (𝜋𝛼) 𝐾𝛼 (𝑧) = { { 𝐾 (𝑧) , if 𝛼 ∈ Z. {]lim →𝛼 ] {

𝑒𝑧 , √2𝜋𝑧

𝐾𝛼 (𝑧) ≍ √

𝑧 󳨀→ ∞.

2.3. Modified Bessel Function of the First Kind and Macdonald’s Function. We recall that the modified Bessel function of the first kind 𝐼𝛼 of order 𝛼 ∈ C and Macdonald’s function 𝐾𝛼 of order 𝛼 ∈ C are defined as analytic functions on {𝑧 ∈ C : | arg 𝑧| < 𝜋} by the formulas

𝑧 󳨀→ 0,

2 log , if 𝛼 = 0 { { { 𝑧 𝐾𝛼 (𝑧) ≍ { 𝛼−1 Γ (𝛼) { , if 𝛼 > 0, {2 𝑧𝛼 {

It has, for 𝑧 ≥ 0 and 𝛼 ≥ 0, the following asymptotic representation:

𝐽𝛼 (𝑧) ≍

𝑧𝛼 , 2𝛼 Γ (𝛼 + 1)

∞

(24)

∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) 𝑑𝑥 = 2𝛽−1 Γ ( 0

1 + 𝛽 + 𝑖𝜆 1 + 𝛽 − 𝑖𝜆 )Γ( ). 2 2 (34)

(ii) If | Im 𝜆| < 1 + Re(𝛽) and 𝑡 > 0, then we have ∞

∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) cos (𝑥 sinh 𝑡) 𝑑𝑥 = √

The Macdonald’s function satisfies

0

𝜋 −1/2,𝛽+1/2 sinh 𝑡Φ𝜆 (𝑡) . 2 (35)

𝐾𝛼 (𝑧) = 𝐾−𝛼 (𝑧) ,

(25)

(iii) If | Im 𝜆| < 2 + Re(𝛽) and 𝑡 > 0, then we have

𝐾𝛼−1 (𝑧) + 𝐾𝛼+1 (𝑧) = −2𝐾𝛼󸀠 (𝑧) .

(26)

∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) sin (𝑥 sinh 𝑡) 𝑑𝑥 = √

∞

0

𝜋 1/2,𝛽+1/2 sinh 𝑡Φ𝜆 (𝑡) . 2 (36)

It also verifies (cf. [10]) 𝛼,𝛽

3. Recurrence Relations for Φ𝜆

𝑧𝑚 (𝐾𝛼+𝑚 (𝑧) − 𝐾𝛼−𝑚 (𝑧)) 𝑚−1

= ∑ (−1)𝑚−𝑗−1 𝑏𝑚𝑗 (𝛼) 𝑧𝑗 𝐾𝛼+𝑗 (𝑧) ,

(27)

3.1. Some Reccurrence Relations with Respect to Parameters 𝛼 and 𝛽 Proposition 2. Let 𝛼, 𝛽, 𝜆 ∈ C. If | Im 𝜆| < Re(𝜌), with 𝜌 = 𝛼 + 𝛽 + 1 and 𝑚 ∈ N, then one has

𝑗=0

where 𝑏𝑚𝑗 (𝛼) is the constant defined by (17). For Re 𝑧 > 0 and 𝛼 ∈ C, it can be written under the representation integral ([9], page 119)

𝛼−𝑚,𝛽+𝑚

(sinh 𝑡)𝑚 [Φ𝜆 𝑚−1

𝛼+𝑚,𝛽+𝑚

(𝑡) − (−1)𝑚 Φ𝜆

𝛼+𝑗,𝛽+𝑗

= ∑ (−1)𝑗 𝑏𝑚𝑗 (𝛼) (sinh 𝑡)𝑗 Φ𝜆

(𝑡)]

(𝑡) ,

𝑗=0

∞

𝐾𝛼 (𝑧) = ∫ 𝑒−𝑧 cosh 𝑢 cosh (𝛼𝑢) 𝑑𝑢. 0

(28)

where 𝑏𝑚𝑗 (𝛼) stands for the constant given by (17).

(37)

4

Journal of Mathematics Example 5. Using Example 1 and relation (42) for 𝛼 = 1/2 and 𝛽 = 1/2, we obtain

Proof. Formula (16) asserts that 𝑥𝛽 (𝑥 sinh 𝑡)𝑚 (𝐽𝛼−𝑚 (𝑥 sinh 𝑡) − (−1)𝑚 𝐽𝛼+𝑚 (𝑥 sinh 𝑡)) 𝑚−1

= ∑ (−1)𝑗 𝑏𝑚 (𝑗) 𝑥𝛽 (𝑥 sinh 𝑡)𝑗 𝐽𝛼+𝑗 (𝑥 sinh 𝑡) .

(38)

𝑗=0

We multiply both sides of the last identity by 𝐾𝑖𝜆 (𝑥) and integrate with respect to 𝑥 from 0 to ∞. Then use formula (9) to obtain the required formula.

Φ3/2,3/2 (𝑡) = 𝜆

× 𝜑𝜆3/2,3/2 (𝑡) =

Proposition 3. For 𝛼, 𝛽, 𝜆 ∈ C such that | Im 𝜆| < Re(𝛼 + 𝛽 + 1), one has 𝛼−1,𝛽+1

𝛼+1,𝛽+1

Φ𝜆

(𝑡) − Φ𝜆

(𝑡) =

2 𝑑 𝛼,𝛽 [Φ ] (𝑡) . cosh 𝑡 𝑑𝑡 𝜆

𝛼−1,𝛽+1

𝛼+1,𝛽+1

(𝑡) − Φ𝜆

(𝑡)

∞

𝑑 2 = ∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) [𝐽𝛼 (𝑥 sinh 𝑡)] 𝑑𝑥. cosh 𝑡 0 𝑑𝑡

(40)

×

(41)

× ((𝑥 sinh 𝑡)

Re(𝛼)−1

+ (𝑥 sinh 𝑡)

Re(𝛼)+1

),

where 𝐶(𝛼) is a positive constant. Taking account of the asymptotic formulas of Macdonald’s function (30) and (32) (or formula (34)) and taking 𝑡 in any compact of (0, ∞), we can deduce the result in the case Re(𝛼) > −1/2. According to principle of analytic continuation, the restriction Re(𝛼) > −1/2 used can be dropped.

,

2 sin (𝜆𝑡) cosh (2𝑡) − 𝜆 cos (𝜆𝑡) sinh (2𝑡) . (sinh (2𝑡))3 (44)

2 2 + 𝑖𝜆 2 − 𝑖𝜆 1 Φ−1/2,3/2 )Γ( ) (𝑡) = √ Γ ( 𝜆 𝜋 2 2 𝜆 × 𝜑𝜆−1/2,3/2 (𝑡) =

≤ 𝐶 (𝛼) cosh 𝑡 ⋅ 𝐾| Im(𝜆)| (𝑥) 𝑥 Re(𝛽)+1

(sinh 𝑡)3/2 (cosh 𝑡)3

Example 6. Using Example 1 and relation (43) for 𝛼 = 1/2 and 𝛽 = 1/2, we obtain

On the other hand, by using formulas (14), (28), and (13), we can deduce that 󵄨󵄨 󵄨󵄨 𝑑 󵄨 󵄨󵄨 𝛽 󵄨󵄨𝑥 𝐾𝑖𝜆 (𝑥) [𝐽𝛼 (𝑥 sinh 𝑡)]󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑑𝑡

2 sin (𝜆𝑡) cosh (2𝑡) − 𝜆 cos (𝜆𝑡) sinh (2𝑡)

12 𝜆 (4 + 𝜆2 )

(39)

Proof. At first, we suppose that Re(𝛼) > −1/2 and | Im 𝜆| < Re(𝛼+𝛽+1). With the help of formulas (14) and (9), we obtain Φ𝜆

2 + 𝑖𝜆 2 − 𝑖𝜆 1 1 Γ( )Γ( ) √2𝜋 2 2 𝜆

𝜆 cos (𝜆𝑡) cosh 𝑡 − sin (𝜆𝑡) sinh 𝑡 , √sinh 𝑡(cosh 𝑡)3

(45)

𝜆 cos (𝜆𝑡) cosh 𝑡 − sin (𝜆𝑡) sinh 𝑡 . sinh 𝑡(cosh 𝑡)3

Example 7. Using Example 5 and relation (42) for 𝛼 = 1 and 𝛽 = 1, we obtain

Φ5/2,5/2 (𝑡) = 𝜆

1 2 + 𝑖𝜆 2 − 𝑖𝜆 1 Γ( )Γ( ) √2𝜋 2 2 4𝜆 × ([16 + 𝜆2 − (𝜆2 − 8) cosh (4𝑡)] sin (𝜆𝑡) −6𝜆 cos (𝜆𝑡) sinh (4𝑡)) −1

× ((sinh 𝑡)5/2 (cosh 𝑡)5 ) , Using Propositions 2 and 3, we obtain the following corollary. Corollary 4. If | Im 𝜆| < Re(𝛼 + 𝛽 + 1), then one has

15 8𝜆 (16 + 𝜆2 ) (4 + 𝜆2 ) × ([16 + 𝜆2 − (𝜆2 − 8) cosh (4𝑡)] sin (𝜆𝑡)

Φ𝜆

(𝑡) =

𝛼 1 𝑑 𝛼,𝛽 𝛼,𝛽 Φ (𝑡) − [Φ ] (𝑡) , sinh 𝑡 𝜆 cosh 𝑡 𝑑𝑡 𝜆

(42)

𝛼−1,𝛽+1 Φ𝜆

𝛼 1 𝑑 𝛼,𝛽 𝛼,𝛽 Φ𝜆 (𝑡) + [Φ ] (𝑡) . (𝑡) = sinh 𝑡 cosh 𝑡 𝑑𝑡 𝜆

(43)

𝛼+1,𝛽+1

𝜑𝜆5/2,5/2 (𝑡) =

−6𝜆 cos (𝜆𝑡) sinh (4𝑡)) −1

× ((sinh 𝑡)5 (cosh 𝑡)5 ) . (46)

Journal of Mathematics

5

Example 8. Using Example 7 and relation (42) for 𝛼 = 1 and 𝛽 = 1, we obtain

Proof. According to formula (15), we obtain 𝑑 1 [(sinh 𝑡)𝛼 𝐽𝛼 (𝑥 sinh 𝑡)] sinh (𝑡) cosh (𝑡) 𝑑𝑡

Φ7/2,7/2 (𝑡) 𝜆

= 𝑥(sinh 𝑡)

1 2 − 𝑖𝜆 1 2 + 𝑖𝜆 = )Γ( ) Γ( √2𝜋 2 2 8𝜆 ×(

(sinh 𝑡) 2

+

𝛼−1,𝛽+1

(sinh 𝑡)𝛼−1 Φ𝜆

7

(cosh 𝑡)

=

2

𝜆 [−76 − 𝜆 + (𝜆 − 44) cosh (4𝑡)] sinh (2𝑡) cos (𝜆𝑡) (sinh 𝑡)7/2 (cosh 𝑡)7

),

𝜑𝜆7/2,7/2 (𝑡) =

×(

(sinh 𝑡)7 (cosh 𝑡)7 𝜆 [−76 − 𝜆2 + (𝜆2 − 44) cosh (4𝑡)] sinh (2𝑡) cos (𝜆𝑡) (sinh 𝑡)7 (cosh 𝑡)7

[ ).

Example 9. Using Example 5 and relation (43) for 𝛼 = 3/2 and 𝛽 = 3/2, we obtain 1 2 + 𝑖𝜆 2 − 𝑖𝜆 1 Γ( )Γ( ) √2𝜋 2 2 𝜆 2

𝛼−𝑚,𝛽+𝑚

= (sinh 𝑡)𝛼−𝑚 Φ𝜆 𝛼+𝑚,𝛽+𝑚

2 𝑑 𝑚 (−1)𝑚 𝛼,𝛽 = ] [(sinh 𝑡)𝛼 Φ𝜆 ] 𝛼−𝑚 [ sinh 𝑑𝑡 (2𝑡) (sinh 𝑡)

(48)

+3𝜆 cos (𝜆𝑡) sinh (2𝑡))

(𝑡) ,

Φ𝜆

+

2

× ([4 + 𝜆 + (𝜆 − 2) cosh (2𝑡)] sin (𝜆𝑡)

(53)

(−1)𝑚+1 𝑚−1 𝛼+𝑗,𝛽+𝑗 . ∑ (−1)𝑗 𝑏𝑚 (𝑗) (sinh 𝑡)𝑗 Φ𝜆 (sinh 𝑡)𝑚 𝑗=0

Proof. The result follows by induction argument.

−1

× ((sinh 𝑡)1/2 (cosh 𝑡)5 ) . Using the previous relation and relation (42) for 𝛼 = 1/2 and 𝛽 = 5/2, we obtain

3.2. Some Reccurrence Relations with Respect to the Dual Variable. We give now some recurrence relations with respect to the dual variable 𝜆 in the next proposition. Proposition 12. If | Im 𝜆| < Re(𝛼 + 𝛽 + 1) and 𝑚 ∈ N, then one has

Φ3/2,7/2 (𝑡) 𝜆

∀𝑡 > 0,

1 2 − 𝑖𝜆 1 2 + 𝑖𝜆 = )Γ( ) Γ( √2𝜋 2 2 4𝜆 ×(

∞ 𝑑 2 ∫ 𝑥𝛽 𝐾𝑖𝜆 (𝑥) [(sinh 𝑡)𝛼 𝐽𝛼 (𝑥 sinh 𝑡)] 𝑑𝑥. sinh (2𝑡) 0 𝑑𝑡 (52)

𝑑 𝑚 2 𝛼,𝛽 ] [(sinh 𝑡)𝛼 Φ𝜆 ] (𝑡) sinh (2𝑡) 𝑑𝑡

(47)

Φ1/2,5/2 (𝑡) = 𝜆

(𝑡)

Corollary 11. For 𝛼, 𝛽, 𝜆 ∈ C such that | Im 𝜆| < Re(𝛼+𝛽+1) and 𝑚 ∈ N one has

2

12 [16 + 𝜆 − (𝜆 − 4) cosh (4𝑡)] cosh (2𝑡) sin (𝜆𝑡)

+

𝐽𝛼−1 (𝑥 sinh 𝑡)

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

105 16𝜆 (4 + 𝜆2 ) (16 + 𝜆2 ) (36 + 𝜆2 ) 2

(51)

and consequently

12 [16 + 𝜆2 − (𝜆2 − 4) cosh (4𝑡)] cosh (2𝑡) sin (𝜆𝑡) 7/2

𝛼−1

𝛼,𝛽+𝑚

[(16 + 𝜆2 ) (−3 + 4 cosh (2𝑡)) + (7𝜆2 − 8) cosh (4𝑡)] sin (𝜆𝑡)

𝑚−1

2𝜆 [16 + 𝜆2 + (𝜆2 − 14) cosh (2𝑡)] sinh (2𝑡) cos (𝜆𝑡) (sinh 𝑡)3/2 (cosh 𝑡)7

).

Proposition 10. For 𝛼, 𝛽, 𝜆 ∈ C such that | Im 𝜆| < Re(𝛼 + 𝛽 + 1), one has 𝑑 2 𝛼,𝛽 [(sinh 𝑡)𝛼 Φ𝜆 ] (𝑡) sinh (2𝑡) 𝑑𝑡 𝛼−1,𝛽+1

= (sinh 𝑡)𝛼−1 Φ𝜆

(𝑡) .

𝛼,𝛽+𝑘

𝑘=0

(49)

∀𝑡 > 0,

(54)

= ∑ (−1)𝑚−𝑘−1 𝑏𝑚𝑘 (𝑖𝜆) Φ𝜆−𝑖𝑘 (𝑡) ,

(sinh 𝑡)3/2 (cosh 𝑡)7 +

𝛼,𝛽+𝑚

Φ𝜆−𝑖𝑚 (𝑡) − Φ𝜆+𝑖𝑚 (𝑡)

(50)

where 𝑏𝑚𝑘 (𝛼) is the constant given by (17). For all 𝑡 > 0, we have 𝛼,𝛽+1

𝛼,𝛽+1

Φ𝜆+𝑖 (𝑡) + Φ𝜆−𝑖 (𝑡) 𝛼,𝛽

= 2 (𝛽 + 1) Φ𝜆 (𝑡) + 2 tanh 𝑡

𝑑 𝛼,𝛽 Φ (𝑡) . 𝑑𝑡 𝜆

(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 𝐽𝛼 and 𝐾𝛼 .

6

Journal of Mathematics

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

3.3. Some Summations Intervening the “Modified” Jacobi Functions

Corollary 13. If | Im 𝜆| < Re(𝛼 + 𝛽 + 1) and 𝑡 > 0, one has

Proposition 16. If | Im 𝜆| < Re(1 + 𝛽) and 𝑡 ∈ (0, arg sinh 1), then one has

𝛼,𝛽+1

𝛼,𝛽

Φ𝜆−𝑖 (𝑡) = (𝛽 + 1 + 𝑖𝜆) Φ𝜆 (𝑡) + tanh 𝑡 𝛼,𝛽+1

𝛼,𝛽

Φ𝜆+𝑖 (𝑡) = (𝛽 + 1 − 𝑖𝜆) Φ𝜆 (𝑡) + tanh 𝑡

𝑑 𝛼,𝛽 Φ (𝑡) , 𝑑𝑡 𝜆 𝑑 𝛼,𝛽 Φ (𝑡) . 𝑑𝑡 𝜆

(56)

sin (𝜆𝑡) , (𝑡) = 𝜆 sinh 𝑡

∞

0,𝛽

𝛽−1

=2

=

(𝜆 + 𝑖) sin ((𝜆 + 𝑖) 𝑡) cosh 𝑡 + cos ((𝜆 + 𝑖) 𝑡) sinh 𝑡 . √sinh 𝑡cosh3 𝑡 (60)

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

𝑡 > 0, Re (𝛼) > −1,

∞

=√

𝜋 1 cosh 𝑡 − 1 𝛼 ( ) 2 cosh 𝑡 sinh 𝑡 𝜋 1 𝑡 tanh𝛼 ( ) , 2 cosh 𝑡 2

𝑡 > 0, Re (𝛼) > −1. (62)

√𝜋 1 𝑡 tanh𝛼 ( ) (3+2𝛼 cosh 𝑡 +cosh (2𝑡)) , (𝑡) = 4 cosh3 𝑡 2

Φ𝛼,3/2 −𝑖/2

1 + 𝛽 + 𝑖𝜆 1 + 𝛽 − 𝑖𝜆 Γ( )Γ( ). 2 2

(66)

∞

((𝑥 sinh 𝑡) /2)2𝑘 Re(𝛽) 𝑥 𝐾| Im(𝜆)| (𝑥) 𝑑𝑥 (2𝑘)!

((sinh 𝑡) /2)2𝑘 2𝑘+ Re(𝛽)−1 2 (2𝑘)!

(67)

× Γ(

1 + 2𝑘 + Re (𝛽) − |Im (𝜆)| ) 2

× Γ(

1 + 2𝑘 + Re (𝛽) + |Im (𝜆)| ), 2

󵄨 󵄨󵄨 󵄨󵄨𝐽2𝑘 (𝑥 sinh 𝑡) 𝑥𝛽 𝐾𝑖𝜆 (𝑥)󵄨󵄨󵄨 𝑑𝑥, 󵄨 󵄨

∀𝑡 ∈ (0, arg sinh 1) , (68)

As an application of the last corollary, we obtain Φ𝛼,3/2 3(𝑖/2)

∞

𝑘=1 0

√ Φ𝛼,1/2 𝑖/2 (𝑡) =

𝑘=1

which permits to conclude the convergence of series ∑∫

we can get

0

∞ 󵄨 󵄨 ∫ 󵄨󵄨󵄨󵄨𝐽2𝑘 (𝑥 sinh 𝑡) 𝑥𝛽 𝐾𝑖𝜆 (𝑥)󵄨󵄨󵄨󵄨 𝑑𝑥 0

0

(61)

∞

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

2 2 + 𝑖𝜆 2 − 𝑖𝜆 𝑖 Φ1/2,3/2 )Γ( ) (𝑡) = − √ Γ ( 𝜆+𝑖 𝜋 2 2 𝜆

cosh 𝑡 − 1 1 ( ) , cosh 𝑡 sinh 𝑡

∞

0,𝛽

≤∫

0

𝜋 −1/2,𝛽+1/2 sinh 𝑡Φ𝜆 (𝑡) . 2 (65)

Φ𝜆 (𝑡) + 2 ∫ ( ∑ 𝐽2𝑘 (𝑥 sinh 𝑡)) 𝑥𝛽 𝐾𝑖𝜆 (𝑥) 𝑑𝑥

(𝑡) already calcuwhich coincided with the value of Φ1/2,1/2 Λ lated at Λ = 𝑖 + 𝜆 and we have

∞

(𝑡) = √

𝑘=1

Γ ((1 + 𝑖𝜆) /2) Γ ((1 − 𝑖𝜆) /2) sin (𝜆𝑡) . (58) √2𝜋 𝜆√sinh 𝑡

∫ 𝑒−𝑥 𝐽𝛼 (𝑥 sinh 𝑡) 𝑑𝑥 =

1 + 𝛽 + 𝑖𝜆 1 + 𝛽 − 𝑖𝜆 )Γ( ). 2 2 (64)

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

Γ ((1 + 𝑖𝜆) /2) Γ ((1 − 𝑖𝜆) /2) sin ((𝑖 + 𝜆) 𝑡) , (𝑡) = −𝑖 √2𝜋 cosh 𝑡√sinh 𝑡 (59)

×

2𝑘,𝛽

Φ𝜆 (𝑡)+2 ∑ (−1)𝑘 Φ𝜆

Using now the last corollary, we can get Φ1/2,1/2 𝜆+𝑖

(𝑡) = 2𝛽−1 Γ (

(57)

and therefore Φ1/2,−1/2 (𝑡) = 𝜆

2𝑘,𝛽

𝑘=1

Example 14. By using formulas (5.32) (in [1], page 44) and (10), we can see that 𝜑𝜆1/2,−1/2

∞

0,𝛽

Φ𝜆 (𝑡)+2 ∑ Φ𝜆

√𝜋 1 𝑡 tanh𝛼 ( ) (1 + 𝛼 cosh 𝑡) . (𝑡) = 2 cosh3 𝑡 2

(63)

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 Φ−1/2,2 −𝑖/2 (𝑡) =

1 − sinh2 𝑡 . √sinh 𝑡cosh4 𝑡

(69)

Using now Corollary 13, we see that Φ−1/2,2 (3/2)𝑖 (𝑡) =

2 . √sinh 𝑡cosh4 𝑡

(70)

Journal of Mathematics

7

Proposition 18. If | Im 𝜆| < 2 + Re(𝛽) then one has ∞

2𝑘,𝛽

∑ (−1)𝑘+1 (2𝑘)2 Φ𝜆

(𝑡)

[2]

𝑘=1

1/2,𝛽+3/2

= √𝜋(sinh 𝑡)3/2 Φ𝜆 ∞

2𝑘+1,𝛽

∑ (−1)𝑘 (2𝑘 + 1)2 Φ𝜆

(𝑡) ,

∀𝑡 ∈ (0, arg sinh 1) ,

[3]

(𝑡)

𝑘=0

[4]

𝜋 −1/2,𝛽+3/2 = √ (sinh 𝑡)3/2 Φ𝜆 (𝑡) , 2

∀𝑡 ∈ (0, arg sinh 1) . (71)

Proof. The first identity deduced from (21), (9), and (36). The second follows from (22), (9), and (35). Remark 19. By using the fact 𝑡 sinh2 𝑡 , ∑ (−1)𝑘+1 (2𝑘)2 tanh2𝑘 ( ) = 2 cosh3 𝑡 𝑘=1 ∞ 𝑡 (3 − cosh (2𝑡)) sinh 𝑡 , ∑ (−1)𝑘 (2𝑘 + 1)2 tanh2𝑘+1 ( ) = 2 4 cosh3 𝑡 𝑘=0 (72)

[10]

1 √sinh 𝑡 , (𝑡) = √2 cosh4 𝑡

(73)

1 (3 − cosh (2𝑡)) . (𝑡) = 4 √sinh 𝑡cosh4 𝑡

1 𝑖𝜆 1 𝑖𝜆 + , 𝛼 + − ; 𝛼 + 1; −sinh2 (𝜀𝑥)) 2 2𝜀 2 2𝜀 (74)

tends to the normalized Bessel function 𝑗𝛼 (𝜆𝑥) =

0 𝐹1 (𝛼 + 1; −( ∞

[11] [12]

Remark 20. It is well known that the hypergeometric function 2 𝐹1 (𝑎, 𝑏; 𝑐; 𝑧) tends to the confluent hypergeometric function 0 𝐹1 (𝑐; 𝑍) as 𝑎, 𝑏 → ∞ and 𝑧 → ∞ such a way that 𝑎𝑏𝑧 → 𝑍. Consequently, as 𝜀 → 0, 𝛼,𝛼 𝜑𝜆/𝜀 (𝜀𝑥) = 2 𝐹1 (𝛼 +

[8]

[9]

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

Φ−1/2,2 𝑖/2

[6]

[7]

∞

Φ1/2,2 𝑖/2

[5]

𝜆𝑥 2 )) 2 2𝑘

𝜆𝑥 (−1)𝑘 = Γ (𝛼 + 1) ∑ ( ) . Γ (𝑘 + 1) Γ (𝑘 + 𝛼 + 1) 2 𝑘=0

(75)

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].

References [1] T. H. Koornwinder, “Jacobi functions and analysis on noncompact semi simple Lie groups,” in Special Functions: Group

Theoretical Aspects and ApplicaTions, R. A. Askey, T. H. Koornwinder, and W. Schempp, Eds., D. Reidel, Dordrecht, The Netherlands, 1984. H. Exton, “Some new summation formulae for the generalised hypergeometric function of higher order,” Journal of Computational and Applied Mathematics, vol. 79, no. 2, pp. 183–187, 1997. H. Exton, “Summation formulae involving the Laguerre polynomial,” Journal of Computational and Applied Mathematics, vol. 100, no. 2, pp. 225–227, 1998. M. Neher, “A note on a sum associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1527–1534, 2007. R. Tremblay and B. J. Fug`ere, “Products of two restricted hypergeometric functions,” Journal of Mathematical Analysis and Applications, vol. 198, no. 3, pp. 844–852, 1996. M. Flensted-Jensen and T. H. Koornwinder, “Jacobi functions: the addition formula and the positivity of the dual convolution structure,” Arkiv f¨or Matematik, vol. 17, no. 1, pp. 139–151, 1979. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ce˘ıtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York, NY, USA, 1965. T. Koornwinder, “A new proof of a Paley-Wiener type theorem for the Jacobi transform,” Arkiv f¨or Matematik, vol. 13, pp. 145– 159, 1975. N. N. Lebedev, Special Functions and Their Applications, Dover, New York, NY, USA, 1972. S. G. Samko, “On some index relations for Bessel functions,” Integral Transforms and Special Functions, vol. 11, no. 4, pp. 397– 402, 2001. G. N. Watson, A Treatise on the Theory of Bessel Functions, Merchant Books, 2008. N. Ben Salem and K. Trim`eche, “Mehler integral transforms associated with Jacobi functions with respect to the dual variable,” Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 691–720, 1997.

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014