# Some New Recurrence Relations Concerning Jacobi Functions

Authors: Hatem Mejjaoli; Ahmedou Ould Ahmed Salem
Journal: Journal of Mathematics
 (2013)

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## Abstract

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functionsΦλα,β. 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.


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## Body

## 1. Introduction

It is well known that the Jacobi functiont→φλα,β(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
(1)Lα,βu(t)=-(λ2+(α+β+1)2)u(t),u(0)=1,u′(0)=0,
where Lα,β is the Jacobi operator given by
(2)Lα,β=d2dt2+[(2α+1)cotht+(2β+1)tanht]ddt.In this paper some new summation formulas for the “modified” Jacobi functions(3)t↦Φλα,β(t)=Γ((α+β+1+iλ)/2)Γ((α+β+1-iλ)/2)2-β+1Γ(1+α)×(sinht)αφλα,β(t)
are proved. We establish, under some conditions, that
(4)Φλα-1,β+1(t)-Φλα+1,β+1(t)=2coshtddt[Φλα,β](t),[2sinh(2t)ddt]m[(sinht)αΦλα,β](t)=(sinht)α-mΦλα-m,β+m(t),(sinht)m[Φλα-m,β+m(t)-(-1)mΦλα+m,β+m(t)]=∑j=0m-1‍(-1)jbmj(α)(sinht)jΦλα+j,β+j(t),
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 Section2 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 functionφλα,β(α,β,λ∈ℂ,α≠-1,-2,-3,...) (cf. [1, 6]) can be expressed as a hypergeometric function under form
(5)φλα,β(t)=F21(ρ+iλ2,ρ-iλ2;α+1;-sinh2t):ρ=α+β+1.
This function satisfies the following properties.(i)
For everyt∈[0,+∞[, (Γ(α+1))-1φλα,β(t) is an entire function of α,β, and λ.(ii)
For eachα,β∈ℂ and for each nonnegative integer n there exists a positive constant k such that
(6)∀t≥0,∀λ∈ℂ,|(Γ(α+1))-1(ddt)nφλα,β(t)|≤k(1+|λ|)n+k(1+x)e(|Imλ|-Re(ρ))t,
where k=0 if Re(α)>-1/2 and k=[(1/2)-Re(α)] if Re(α)≤-1/2.(iii)
The functiont→φλα,β(t) for Re(α)>-1/2 possesses the Laplace type integral representation
(7)∀t>0,∀λ∈ℂ,φλα,β(t)=∫0t‍𝒦(t,y)cos(λy)dy,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}.We note that the functionφλα,β possesses also an integral representation with respect to the dual variable [12].To finish this paragraph, we consider the “modified” Jacobi function(8)t↦Φλα,β(t)=Γ((α+β+1+iλ)/2)Γ((α+β+1-iλ/2))2-β+1Γ(1+α)×(sinht)αφλα,β(t),
which can be written (cf. [7], page 693) as follows:
(9)∀t>0,∫0∞‍xβKiλ(x)Jα(xsinht)dx=Φλα,β(t),|Imλ|<Re(α+β+1),
where Jα and Kα denote, respectively, the Bessel function of the first kind and Macdonald's function.Example 1.
The functionsφλ1/2,1/2 and Φλ1/2,1/2 can be expressed in terms of elementary functions as
(10)∀t>0,φλ1/2,1/2(t)=2sin(λt)λsinh(2t),(11)∀t>0,Φλ1/2,1/2(t)=2πΓ(2+iλ2)Γ(2-iλ2)×1λsin(λt)sinhtcosht,|Imλ|<2,


### 2.2. Bessel Function of the First Kind

We recall that the Bessel function of the first kind and orderα denoted by Jα is defined as an analytic function on {z∈ℂ:|argz|<π} by
(12)Jα(z)=∑k=0∞‍(-1)kΓ(k+1)Γ(k+α+1)(z2)α+2k
and possesses the following integral representation:
(13)Jα(z)=(z/2)απΓ(α+1/2)∫-11‍(1-t2)α-1/2cos(zt)dt,Re(α)>-12.
It satisfies the following functional relations:
(14)Jα-1(z)-Jα+1(z)=2Jα′(z),(15)1zddz(zαJα(z))=zα-1Jα-1(z).
It also verifies [10]
(16)zm(Jα-m(z)-(-1)mJα+m(z))=∑j=0m-1‍(-1)jbmj(α)zjJα+j(z),
with
(17)bmj(α)=Γ(α+1)2m-jCmjΓ(α+j-m+1),
and satisfies
(18)J0(z)+2∑j=1∞‍J2j(z)=1,(19)∑j=0∞‍(-1)jJ2j+1(z)=sinz2,(20)J0(z)+2∑j=1∞‍(-1)jJ2j(z)=cosz,(21)∑j=1∞‍(-1)j+1(2j)2J2j(z)=zsinz2,(22)∑j=0∞‍(-1)j(2j+1)2J2j+1(z)=zcosz2.
It has, for z≥0 and α≥0, the following asymptotic representation:
(23)Jα(z)≍zα2αΓ(α+1),z→0,Jα(z)≍2πzcos(z-12απ-14π),z→∞.


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

We recall that the modified Bessel function of the first kindIα of order α∈ℂ and Macdonald's function Kα of order α∈ℂ are defined as analytic functions on {z∈ℂ:|argz|<π} by the formulas
(24)Iα(z)=∑k=0∞‍(z/2)α+2kΓ(k+1)Γ(k+α+1),Kα(z)={π2[I-α(z)-Iα(z)]sin(πα),ifα∉ℤ,limν→αKν(z),ifα∈ℤ.
The Macdonald's function satisfies
(25)Kα(z)=K-α(z),(26)Kα-1(z)+Kα+1(z)=-2Kα′(z).
It also verifies (cf. [10])
(27)zm(Kα+m(z)-Kα-m(z))=∑j=0m-1‍(-1)m-j-1bmj(α)zjKα+j(z),
where bmj(α) is the constant defined by (17).ForRez>0 and α∈ℂ, it can be written under the representation integral ([9], page 119)
(28)Kα(z)=∫0∞‍e-zcoshucosh(αu)du.
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):
(29)Iα(z)≍zα2αΓ(α+1),z→0,(30)Kα(z)≍{log2z,ifα=02α-1Γ(α)zα,ifα>0,z→0(31)Iα(z)≍ez2πz,z→∞,(32)Kα(z)≍π2ze-z,z→∞.
Note that in the special case α=±1/2, Macdonald's function is given by
(33)K±1/2(z)=(π2z)1/2e-z.To complete this paragraph, we recall the following (cf. [7], page 684 and page 747).(i)
If|Imλ|<1+Re(β), then we have
(34)∫0∞‍xβKiλ(x)dx=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).(ii)
If|Imλ|<1+Re(β) and t>0, then we have
(35)∫0∞‍xβKiλ(x)cos(xsinht)dx=π2sinhtΦλ-1/2,β+1/2(t).(iii)
If|Imλ|<2+Re(β) and t>0, then we have
(36)∫0∞‍xβKiλ(x)sin(xsinht)dx=π2sinhtΦλ1/2,β+1/2(t).


## 2.1. Jacobi Function

A Jacobi functionφλα,β(α,β,λ∈ℂ,α≠-1,-2,-3,...) (cf. [1, 6]) can be expressed as a hypergeometric function under form
(5)φλα,β(t)=F21(ρ+iλ2,ρ-iλ2;α+1;-sinh2t):ρ=α+β+1.
This function satisfies the following properties.(i)
For everyt∈[0,+∞[, (Γ(α+1))-1φλα,β(t) is an entire function of α,β, and λ.(ii)
For eachα,β∈ℂ and for each nonnegative integer n there exists a positive constant k such that
(6)∀t≥0,∀λ∈ℂ,|(Γ(α+1))-1(ddt)nφλα,β(t)|≤k(1+|λ|)n+k(1+x)e(|Imλ|-Re(ρ))t,
where k=0 if Re(α)>-1/2 and k=[(1/2)-Re(α)] if Re(α)≤-1/2.(iii)
The functiont→φλα,β(t) for Re(α)>-1/2 possesses the Laplace type integral representation
(7)∀t>0,∀λ∈ℂ,φλα,β(t)=∫0t‍𝒦(t,y)cos(λy)dy,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}.We note that the functionφλα,β possesses also an integral representation with respect to the dual variable [12].To finish this paragraph, we consider the “modified” Jacobi function(8)t↦Φλα,β(t)=Γ((α+β+1+iλ)/2)Γ((α+β+1-iλ/2))2-β+1Γ(1+α)×(sinht)αφλα,β(t),
which can be written (cf. [7], page 693) as follows:
(9)∀t>0,∫0∞‍xβKiλ(x)Jα(xsinht)dx=Φλα,β(t),|Imλ|<Re(α+β+1),
where Jα and Kα denote, respectively, the Bessel function of the first kind and Macdonald's function.Example 1.
The functionsφλ1/2,1/2 and Φλ1/2,1/2 can be expressed in terms of elementary functions as
(10)∀t>0,φλ1/2,1/2(t)=2sin(λt)λsinh(2t),(11)∀t>0,Φλ1/2,1/2(t)=2πΓ(2+iλ2)Γ(2-iλ2)×1λsin(λt)sinhtcosht,|Imλ|<2,


## 2.2. Bessel Function of the First Kind

We recall that the Bessel function of the first kind and orderα denoted by Jα is defined as an analytic function on {z∈ℂ:|argz|<π} by
(12)Jα(z)=∑k=0∞‍(-1)kΓ(k+1)Γ(k+α+1)(z2)α+2k
and possesses the following integral representation:
(13)Jα(z)=(z/2)απΓ(α+1/2)∫-11‍(1-t2)α-1/2cos(zt)dt,Re(α)>-12.
It satisfies the following functional relations:
(14)Jα-1(z)-Jα+1(z)=2Jα′(z),(15)1zddz(zαJα(z))=zα-1Jα-1(z).
It also verifies [10]
(16)zm(Jα-m(z)-(-1)mJα+m(z))=∑j=0m-1‍(-1)jbmj(α)zjJα+j(z),
with
(17)bmj(α)=Γ(α+1)2m-jCmjΓ(α+j-m+1),
and satisfies
(18)J0(z)+2∑j=1∞‍J2j(z)=1,(19)∑j=0∞‍(-1)jJ2j+1(z)=sinz2,(20)J0(z)+2∑j=1∞‍(-1)jJ2j(z)=cosz,(21)∑j=1∞‍(-1)j+1(2j)2J2j(z)=zsinz2,(22)∑j=0∞‍(-1)j(2j+1)2J2j+1(z)=zcosz2.
It has, for z≥0 and α≥0, the following asymptotic representation:
(23)Jα(z)≍zα2αΓ(α+1),z→0,Jα(z)≍2πzcos(z-12απ-14π),z→∞.


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

We recall that the modified Bessel function of the first kindIα of order α∈ℂ and Macdonald's function Kα of order α∈ℂ are defined as analytic functions on {z∈ℂ:|argz|<π} by the formulas
(24)Iα(z)=∑k=0∞‍(z/2)α+2kΓ(k+1)Γ(k+α+1),Kα(z)={π2[I-α(z)-Iα(z)]sin(πα),ifα∉ℤ,limν→αKν(z),ifα∈ℤ.
The Macdonald's function satisfies
(25)Kα(z)=K-α(z),(26)Kα-1(z)+Kα+1(z)=-2Kα′(z).
It also verifies (cf. [10])
(27)zm(Kα+m(z)-Kα-m(z))=∑j=0m-1‍(-1)m-j-1bmj(α)zjKα+j(z),
where bmj(α) is the constant defined by (17).ForRez>0 and α∈ℂ, it can be written under the representation integral ([9], page 119)
(28)Kα(z)=∫0∞‍e-zcoshucosh(αu)du.
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):
(29)Iα(z)≍zα2αΓ(α+1),z→0,(30)Kα(z)≍{log2z,ifα=02α-1Γ(α)zα,ifα>0,z→0(31)Iα(z)≍ez2πz,z→∞,(32)Kα(z)≍π2ze-z,z→∞.
Note that in the special case α=±1/2, Macdonald's function is given by
(33)K±1/2(z)=(π2z)1/2e-z.To complete this paragraph, we recall the following (cf. [7], page 684 and page 747).(i)
If|Imλ|<1+Re(β), then we have
(34)∫0∞‍xβKiλ(x)dx=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).(ii)
If|Imλ|<1+Re(β) and t>0, then we have
(35)∫0∞‍xβKiλ(x)cos(xsinht)dx=π2sinhtΦλ-1/2,β+1/2(t).(iii)
If|Imλ|<2+Re(β) and t>0, then we have
(36)∫0∞‍xβKiλ(x)sin(xsinht)dx=π2sinhtΦλ1/2,β+1/2(t).


## 3. Recurrence Relations forΦλα,β

### 3.1. Some Reccurrence Relations with Respect to Parametersα and β

Proposition 2.
Letα,β,λ∈ℂ. If |Imλ|<Re(ρ), with ρ=α+β+1 and m∈ℕ, then one has
(37)(sinht)m[Φλα-m,β+m(t)-(-1)mΦλα+m,β+m(t)]=∑j=0m-1‍(-1)jbmj(α)(sinht)jΦλα+j,β+j(t),
where bmj(α) stands for the constant given by (17).Proof.
Formula (16) asserts that
(38)xβ(xsinht)m(Jα-m(xsinht)-(-1)mJα+m(xsinht))=∑j=0m-1‍(-1)jbm(j)xβ(xsinht)jJα+j(xsinht).
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.Proposition 3.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1), one has
(39)Φλα-1,β+1(t)-Φλα+1,β+1(t)=2coshtddt[Φλα,β](t).Proof.
At first, we suppose thatRe(α)>-1/2 and |Imλ|<Re(α+β+1). With the help of formulas (14) and (9), we obtain
(40)Φλα-1,β+1(t)-Φλα+1,β+1(t)=2cosht∫0∞‍xβKiλ(x)ddt[Jα(xsinht)]dx.
On the other hand, by using formulas (14), (28), and (13), we can deduce that
(41)|xβKiλ(x)ddt[Jα(xsinht)]|≤C(α)cosht·K|Im(λ)|(x)xRe(β)+1×((xsinht)Re(α)-1+(xsinht)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.Using Propositions2 and 3, we obtain the following corollary.Corollary 4.
If|Imλ|<Re(α+β+1), then one has
(42)Φλα+1,β+1(t)=αsinhtΦλα,β(t)-1coshtddt[Φλα,β](t),(43)Φλα-1,β+1(t)=αsinhtΦλα,β(t)+1coshtddt[Φλα,β](t).Example 5.
Using Example1 and relation (42) for α=1/2 and β=1/2, we obtain
(44)Φλ3/2,3/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)1λ×2sin(λt)cosh(2t)-λcos(λt)sinh(2t)(sinht)3/2(cosht)3,φλ3/2,3/2(t)=12λ(4+λ2)×2sin(λt)cosh(2t)-λcos(λt)sinh(2t)(sinh(2t))3.Example 6.
Using Example1 and relation (43) for α=1/2 and β=1/2, we obtain
(45)Φλ-1/2,3/2(t)=2πΓ(2+iλ2)Γ(2-iλ2)1λ×λcos(λt)cosht-sin(λt)sinhtsinht(cosht)3,φλ-1/2,3/2(t)=λcos(λt)cosht-sin(λt)sinhtsinht(cosht)3.Example 7.
Using Example5 and relation (42) for α=1 and β=1, we obtain
(46)Φλ5/2,5/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)14λ×([16+λ2-(λ2-8)cosh(4t)]sin(λt)-6λcos(λt)sinh(4t)[16+λ2-(λ2-8)cosh(4t)])×((sinht)5/2(cosht)5)-1,φλ5/2,5/2(t)=158λ(16+λ2)(4+λ2)×([16+λ2-(λ2-8)cosh(4t)]sin(λt)-6λcos(λt)sinh(4t)[16+λ2-(λ2-8)cosh(4t)])×((sinht)5(cosht)5)-1.Example 8.
Using Example7 and relation (42) for α=1 and β=1, we obtain(47)
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.Example 9.
Using Example5 and relation (43) for α=3/2 and β=3/2, we obtain
(48)Φλ1/2,5/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)1λ×([4+λ2+(λ2-2)cosh(2t)]sin(λt)+3λcos(λt)sinh(2t)[4+λ2+(λ2-2)cosh(2t)])×((sinht)1/2(cosht)5)-1.
Using the previous relation and relation (42) for α=1/2 and β=5/2, we obtain(49)
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.Proposition 10.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1), one has
(50)∀t>0,2sinh(2t)ddt[(sinht)αΦλα,β](t)=(sinht)α-1Φλα-1,β+1(t).Proof.
According to formula (15), we obtain
(51)1sinh(t)cosh(t)ddt[(sinht)αJα(xsinht)]=x(sinht)α-1Jα-1(xsinht)
and consequently
(52)(sinht)α-1Φλα-1,β+1(t)=2sinh(2t)∫0∞‍xβKiλ(x)ddt[(sinht)αJα(xsinht)]dx.
By using formulas (14), (28), (13), and (34), we deduce the result.Corollary 11.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1) and m∈ℕ one has
(53)[2sinh(2t)ddt]m[(sinht)αΦλα,β](t)=(sinht)α-mΦλα-m,β+m(t),Φλα+m,β+m=(-1)m(sinht)α-m[2sinh(2t)ddt]m[(sinht)αΦλα,β]+(-1)m+1(sinht)m∑j=0m-1‍(-1)jbm(j)(sinht)jΦλα+j,β+j.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 variableλ in the next proposition.Proposition 12.
If|Imλ|<Re(α+β+1) and m∈ℕ, then one has
(54)∀t>0,Φλ-imα,β+m(t)-Φλ+imα,β+m(t)=∑k=0m-1‍(-1)m-k-1bmk(iλ)Φλ-ikα,β+k(t),
where bmk(α) is the constant given by (17). For all t>0, we have
(55)Φλ+iα,β+1(t)+Φλ-iα,β+1(t)=2(β+1)Φλα,β(t)+2tanhtddtΦλα,β(t).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.As a consequence of this proposition, we have the following corollary.Corollary 13.
If|Imλ|<Re(α+β+1) and t>0, one has
(56)Φλ-iα,β+1(t)=(β+1+iλ)Φλα,β(t)+tanhtddtΦλα,β(t),Φλ+iα,β+1(t)=(β+1-iλ)Φλα,β(t)+tanhtddtΦλα,β(t).Example 14.
By using formulas (5.32) (in [1], page 44) and (10), we can see that
(57)φλ1/2,-1/2(t)=sin(λt)λsinht,
and therefore
(58)Φλ1/2,-1/2(t)=Γ((1+iλ)/2)Γ((1-iλ)/2)2πsin(λt)λsinht.
Using now the last corollary, we can get
(59)Φλ+i1/2,1/2(t)=-iΓ((1+iλ)/2)Γ((1-iλ)/2)2πsin((i+λ)t)coshtsinht,
which coincided with the value of ΦΛ1/2,1/2(t) already calculated at Λ=i+λ and we have(60)
Φ
λ
+
i
1
/
2
,
3
/
2
(
t
)
=
-
2
π
Γ
(
2
+
i
λ
2
)
Γ
(
2
-
i
λ
2
)
i
λ
×
(
λ
+
i
)
sin
(
(
λ
+
i
)
t
)
cosh
t
+
cos
(
(
λ
+
i
)
t
)
sinh
t
sinh
t
cosh
3
t
.Remark 15.
By using (33) and the fact that (cf., e.g., [7], page 707)
(61)∫0∞‍e-xJα(xsinht)dx=1cosht(cosht-1sinht)α,t>0,Re(α)>-1,
we can get
(62)Φi/2α,1/2(t)=π21cosht(cosht-1sinht)α=π21coshttanhα(t2),t>0,Re(α)>-1.
As an application of the last corollary, we obtain
(63)Φ3(i/2)α,3/2(t)=π41cosh3ttanhα(t2)(3+2αcosht+cosh(2t)),Φ-i/2α,3/2(t)=π21cosh3ttanhα(t2)(1+αcosht).


### 3.3. Some Summations Intervening the “Modified” Jacobi Functions

Proposition 16.
If|Imλ|<Re(1+β) and t∈(0,argsinh1), then one has
(64)Φλ0,β(t)+2∑k=1∞‍Φλ2k,β(t)=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).(65)Φλ0,β(t)+2∑k=1∞‍(-1)kΦλ2k,β(t)=π2sinhtΦλ-1/2,β+1/2(t).Proof.
Using formulas (18), (34), and (9), we obtain
(66)Φλ0,β(t)+2∫0∞‍(∑k=1∞‍J2k(xsinht))xβKiλ(x)dx=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).
With the help of formulas (13), (28), and (34), we can see that
(67)∫0∞‍|J2k(xsinht)xβKiλ(x)|dx≤∫0∞‍((xsinht)/2)2k(2k)!xRe(β)K|Im(λ)|(x)dx=((sinht)/2)2k(2k)!22k+Re(β)-1×Γ(1+2k+Re(β)-|Im(λ)|2)×Γ(1+2k+Re(β)+|Im(λ)|2),
which permits to conclude the convergence of series
(68)∑k=1∞‍∫0∞‍|J2k(xsinht)xβKiλ(x)|dx,∀t∈(0,argsinh1),
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
(69)Φ-i/2-1/2,2(t)=1-sinh2tsinhtcosh4t.
Using now Corollary 13, we see that
(70)Φ(3/2)i-1/2,2(t)=2sinhtcosh4t.Proposition 18.
If|Imλ|<2+Re(β) then one has
(71)∑k=1∞‍(-1)k+1(2k)2Φλ2k,β(t)=π(sinht)3/2Φλ1/2,β+3/2(t),∀t∈(0,argsinh1),∑k=0∞‍(-1)k(2k+1)2Φλ2k+1,β(t)=π2(sinht)3/2Φλ-1/2,β+3/2(t),∀t∈(0,argsinh1).Proof.
The first identity deduced from (21), (9), and (36). The second follows from (22), (9), and (35).Remark 19.
By using the fact(72)∑k=1∞‍(-1)k+1(2k)2tanh2k(t2)=sinh2tcosh3t,∑k=0∞‍(-1)k(2k+1)2tanh2k+1(t2)=(3-cosh(2t))4sinhtcosh3t,
formula (62), and the last proposition, we can see that
(73)Φi/21/2,2(t)=12sinhtcosh4t,Φi/2-1/2,2(t)=14(3-cosh(2t))sinhtcosh4t.Remark 20.
It is well known that the hypergeometric function2F1(a,b;c;z) tends to the confluent hypergeometric function 0F1(c;Z) as a,b→∞ and z→∞ such a way that abz→Z. Consequently, as ε→0,
(74)φλ/εα,α(εx)=F21(α+12+iλ2ε,α+12-iλ2ε;α+1;-sinh2(εx))
tends to the normalized Bessel function
(75)jα(λx)=0F1(α+1;-(λx2)2)=Γ(α+1)∑k=0∞‍(-1)kΓ(k+1)Γ(k+α+1)(λx2)2k.
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].


## 3.1. Some Reccurrence Relations with Respect to Parametersα and β

Proposition 2.
Letα,β,λ∈ℂ. If |Imλ|<Re(ρ), with ρ=α+β+1 and m∈ℕ, then one has
(37)(sinht)m[Φλα-m,β+m(t)-(-1)mΦλα+m,β+m(t)]=∑j=0m-1‍(-1)jbmj(α)(sinht)jΦλα+j,β+j(t),
where bmj(α) stands for the constant given by (17).Proof.
Formula (16) asserts that
(38)xβ(xsinht)m(Jα-m(xsinht)-(-1)mJα+m(xsinht))=∑j=0m-1‍(-1)jbm(j)xβ(xsinht)jJα+j(xsinht).
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.Proposition 3.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1), one has
(39)Φλα-1,β+1(t)-Φλα+1,β+1(t)=2coshtddt[Φλα,β](t).Proof.
At first, we suppose thatRe(α)>-1/2 and |Imλ|<Re(α+β+1). With the help of formulas (14) and (9), we obtain
(40)Φλα-1,β+1(t)-Φλα+1,β+1(t)=2cosht∫0∞‍xβKiλ(x)ddt[Jα(xsinht)]dx.
On the other hand, by using formulas (14), (28), and (13), we can deduce that
(41)|xβKiλ(x)ddt[Jα(xsinht)]|≤C(α)cosht·K|Im(λ)|(x)xRe(β)+1×((xsinht)Re(α)-1+(xsinht)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.Using Propositions2 and 3, we obtain the following corollary.Corollary 4.
If|Imλ|<Re(α+β+1), then one has
(42)Φλα+1,β+1(t)=αsinhtΦλα,β(t)-1coshtddt[Φλα,β](t),(43)Φλα-1,β+1(t)=αsinhtΦλα,β(t)+1coshtddt[Φλα,β](t).Example 5.
Using Example1 and relation (42) for α=1/2 and β=1/2, we obtain
(44)Φλ3/2,3/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)1λ×2sin(λt)cosh(2t)-λcos(λt)sinh(2t)(sinht)3/2(cosht)3,φλ3/2,3/2(t)=12λ(4+λ2)×2sin(λt)cosh(2t)-λcos(λt)sinh(2t)(sinh(2t))3.Example 6.
Using Example1 and relation (43) for α=1/2 and β=1/2, we obtain
(45)Φλ-1/2,3/2(t)=2πΓ(2+iλ2)Γ(2-iλ2)1λ×λcos(λt)cosht-sin(λt)sinhtsinht(cosht)3,φλ-1/2,3/2(t)=λcos(λt)cosht-sin(λt)sinhtsinht(cosht)3.Example 7.
Using Example5 and relation (42) for α=1 and β=1, we obtain
(46)Φλ5/2,5/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)14λ×([16+λ2-(λ2-8)cosh(4t)]sin(λt)-6λcos(λt)sinh(4t)[16+λ2-(λ2-8)cosh(4t)])×((sinht)5/2(cosht)5)-1,φλ5/2,5/2(t)=158λ(16+λ2)(4+λ2)×([16+λ2-(λ2-8)cosh(4t)]sin(λt)-6λcos(λt)sinh(4t)[16+λ2-(λ2-8)cosh(4t)])×((sinht)5(cosht)5)-1.Example 8.
Using Example7 and relation (42) for α=1 and β=1, we obtain(47)
Φ
λ
7
/
2
,
7
/
2
(
t
)
=
1
2
π
Γ
(
2
+
i
λ
2
)
Γ
(
2
-
i
λ
2
)
1
8
λ
×
(
12
[
16
+
λ
2
-
(
λ
2
-
4
)
cosh
(
4
t
)
]
cosh
(
2
t
)
sin
(
λ
t
)
(
sinh
t
)
7
/
2
(
cosh
t
)
7
+
λ
[
-
76
-
λ
2
+
(
λ
2
-
44
)
cosh
(
4
t
)
]
sinh
(
2
t
)
cos
(
λ
t
)
(
sinh
t
)
7
/
2
(
cosh
t
)
7
)
,
φ
λ
7
/
2
,
7
/
2
(
t
)
=
105
16
λ
(
4
+
λ
2
)
(
16
+
λ
2
)
(
36
+
λ
2
)
×
(

12
[
16
+
λ
2
-
(
λ
2
-
4
)
cosh
(
4
t
)
]
cosh
(
2
t
)
sin
(
λ
t
)
(
sinh
t
)
7
(
cosh
t
)
7
+
λ
[
-
76
-
λ
2
+
(
λ
2
-
44
)
cosh
(
4
t
)
]
sinh
(
2
t
)
cos
(
λ
t
)
(
sinh
t
)
7
(
cosh
t
)
7
)
.Example 9.
Using Example5 and relation (43) for α=3/2 and β=3/2, we obtain
(48)Φλ1/2,5/2(t)=12πΓ(2+iλ2)Γ(2-iλ2)1λ×([4+λ2+(λ2-2)cosh(2t)]sin(λt)+3λcos(λt)sinh(2t)[4+λ2+(λ2-2)cosh(2t)])×((sinht)1/2(cosht)5)-1.
Using the previous relation and relation (42) for α=1/2 and β=5/2, we obtain(49)
Φ
λ
3
/
2
,
7
/
2
(
t
)
=
1
2
π
Γ
(
2
+
i
λ
2
)
Γ
(
2
-
i
λ
2
)
1
4
λ
×
(
[
(
16
+
λ
2
)
(
-
3
+
4
cosh
(
2
t
)
)
+
(
7
λ
2
-
8
)
cosh
(
4
t
)
]
sin
(
λ
t
)
(
sinh
t
)
3
/
2
(
cosh
t
)
7
+
2
λ
[
16
+
λ
2
+
(
λ
2
-
14
)
cosh
(
2
t
)
]
sinh
(
2
t
)
cos
(
λ
t
)
(
sinh
t
)
3
/
2
(
cosh
t
)
7
)
.Proposition 10.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1), one has
(50)∀t>0,2sinh(2t)ddt[(sinht)αΦλα,β](t)=(sinht)α-1Φλα-1,β+1(t).Proof.
According to formula (15), we obtain
(51)1sinh(t)cosh(t)ddt[(sinht)αJα(xsinht)]=x(sinht)α-1Jα-1(xsinht)
and consequently
(52)(sinht)α-1Φλα-1,β+1(t)=2sinh(2t)∫0∞‍xβKiλ(x)ddt[(sinht)αJα(xsinht)]dx.
By using formulas (14), (28), (13), and (34), we deduce the result.Corollary 11.
Forα,β,λ∈ℂ such that |Imλ|<Re(α+β+1) and m∈ℕ one has
(53)[2sinh(2t)ddt]m[(sinht)αΦλα,β](t)=(sinht)α-mΦλα-m,β+m(t),Φλα+m,β+m=(-1)m(sinht)α-m[2sinh(2t)ddt]m[(sinht)αΦλα,β]+(-1)m+1(sinht)m∑j=0m-1‍(-1)jbm(j)(sinht)jΦλα+j,β+j.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 variableλ in the next proposition.Proposition 12.
If|Imλ|<Re(α+β+1) and m∈ℕ, then one has
(54)∀t>0,Φλ-imα,β+m(t)-Φλ+imα,β+m(t)=∑k=0m-1‍(-1)m-k-1bmk(iλ)Φλ-ikα,β+k(t),
where bmk(α) is the constant given by (17). For all t>0, we have
(55)Φλ+iα,β+1(t)+Φλ-iα,β+1(t)=2(β+1)Φλα,β(t)+2tanhtddtΦλα,β(t).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.As a consequence of this proposition, we have the following corollary.Corollary 13.
If|Imλ|<Re(α+β+1) and t>0, one has
(56)Φλ-iα,β+1(t)=(β+1+iλ)Φλα,β(t)+tanhtddtΦλα,β(t),Φλ+iα,β+1(t)=(β+1-iλ)Φλα,β(t)+tanhtddtΦλα,β(t).Example 14.
By using formulas (5.32) (in [1], page 44) and (10), we can see that
(57)φλ1/2,-1/2(t)=sin(λt)λsinht,
and therefore
(58)Φλ1/2,-1/2(t)=Γ((1+iλ)/2)Γ((1-iλ)/2)2πsin(λt)λsinht.
Using now the last corollary, we can get
(59)Φλ+i1/2,1/2(t)=-iΓ((1+iλ)/2)Γ((1-iλ)/2)2πsin((i+λ)t)coshtsinht,
which coincided with the value of ΦΛ1/2,1/2(t) already calculated at Λ=i+λ and we have(60)
Φ
λ
+
i
1
/
2
,
3
/
2
(
t
)
=
-
2
π
Γ
(
2
+
i
λ
2
)
Γ
(
2
-
i
λ
2
)
i
λ
×
(
λ
+
i
)
sin
(
(
λ
+
i
)
t
)
cosh
t
+
cos
(
(
λ
+
i
)
t
)
sinh
t
sinh
t
cosh
3
t
.Remark 15.
By using (33) and the fact that (cf., e.g., [7], page 707)
(61)∫0∞‍e-xJα(xsinht)dx=1cosht(cosht-1sinht)α,t>0,Re(α)>-1,
we can get
(62)Φi/2α,1/2(t)=π21cosht(cosht-1sinht)α=π21coshttanhα(t2),t>0,Re(α)>-1.
As an application of the last corollary, we obtain
(63)Φ3(i/2)α,3/2(t)=π41cosh3ttanhα(t2)(3+2αcosht+cosh(2t)),Φ-i/2α,3/2(t)=π21cosh3ttanhα(t2)(1+αcosht).


## 3.3. Some Summations Intervening the “Modified” Jacobi Functions

Proposition 16.
If|Imλ|<Re(1+β) and t∈(0,argsinh1), then one has
(64)Φλ0,β(t)+2∑k=1∞‍Φλ2k,β(t)=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).(65)Φλ0,β(t)+2∑k=1∞‍(-1)kΦλ2k,β(t)=π2sinhtΦλ-1/2,β+1/2(t).Proof.
Using formulas (18), (34), and (9), we obtain
(66)Φλ0,β(t)+2∫0∞‍(∑k=1∞‍J2k(xsinht))xβKiλ(x)dx=2β-1Γ(1+β+iλ2)Γ(1+β-iλ2).
With the help of formulas (13), (28), and (34), we can see that
(67)∫0∞‍|J2k(xsinht)xβKiλ(x)|dx≤∫0∞‍((xsinht)/2)2k(2k)!xRe(β)K|Im(λ)|(x)dx=((sinht)/2)2k(2k)!22k+Re(β)-1×Γ(1+2k+Re(β)-|Im(λ)|2)×Γ(1+2k+Re(β)+|Im(λ)|2),
which permits to conclude the convergence of series
(68)∑k=1∞‍∫0∞‍|J2k(xsinht)xβKiλ(x)|dx,∀t∈(0,argsinh1),
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
(69)Φ-i/2-1/2,2(t)=1-sinh2tsinhtcosh4t.
Using now Corollary 13, we see that
(70)Φ(3/2)i-1/2,2(t)=2sinhtcosh4t.Proposition 18.
If|Imλ|<2+Re(β) then one has
(71)∑k=1∞‍(-1)k+1(2k)2Φλ2k,β(t)=π(sinht)3/2Φλ1/2,β+3/2(t),∀t∈(0,argsinh1),∑k=0∞‍(-1)k(2k+1)2Φλ2k+1,β(t)=π2(sinht)3/2Φλ-1/2,β+3/2(t),∀t∈(0,argsinh1).Proof.
The first identity deduced from (21), (9), and (36). The second follows from (22), (9), and (35).Remark 19.
By using the fact(72)∑k=1∞‍(-1)k+1(2k)2tanh2k(t2)=sinh2tcosh3t,∑k=0∞‍(-1)k(2k+1)2tanh2k+1(t2)=(3-cosh(2t))4sinhtcosh3t,
formula (62), and the last proposition, we can see that
(73)Φi/21/2,2(t)=12sinhtcosh4t,Φi/2-1/2,2(t)=14(3-cosh(2t))sinhtcosh4t.Remark 20.
It is well known that the hypergeometric function2F1(a,b;c;z) tends to the confluent hypergeometric function 0F1(c;Z) as a,b→∞ and z→∞ such a way that abz→Z. Consequently, as ε→0,
(74)φλ/εα,α(εx)=F21(α+12+iλ2ε,α+12-iλ2ε;α+1;-sinh2(εx))
tends to the normalized Bessel function
(75)jα(λx)=0F1(α+1;-(λx2)2)=Γ(α+1)∑k=0∞‍(-1)kΓ(k+1)Γ(k+α+1)(λx2)2k.
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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*Source: 162429-2013-03-19.xml*