3.1. Some Reccurrence Relations with Respect to Parameters α and β
Proposition 2.
Let α,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.
For α,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 that Re(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)x Re(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 Propositions 2 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 Example 1 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 Example 1 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 Example 5 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 Example 7 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
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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 Example 5 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.
For α,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.
For α,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 variable λ 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,arg sinh1), 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)!x Re(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,arg sinh1),
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,arg sinh1),∑k=0∞(-1)k(2k+1)2Φλ2k+1,β(t) =π2(sinht)3/2Φλ-1/2,β+3/2(t), ∀t∈(0,arg sinh1).
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 function 2F1(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].