Some New Recurrence Relations Concerning Jacobi Functions

Abstract

The aim of this paper is to obtain some new recurrence relations for the “modified” Jacobi functions Φ^𝛼,𝛽_𝜆. 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

It is well known that the Jacobi function 𝑡→𝜑_𝜆^𝛼,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 𝐿_𝛼,𝛽𝑢(𝑡)=−𝜆²+𝛼+𝛽+1²𝑢(𝑡),𝑢(0)=1,𝑢^′(0)=0, (1) where 𝐿_𝛼,𝛽 is the Jacobi operator given by 𝐿_𝛼,𝛽=𝑑²𝑑𝑡²+)𝑑(2𝛼+1coth𝑡+2𝛽+1tanh𝑡.𝑑𝑡 (2)

In this paper some new summation formulas for the “modified” Jacobi functions 𝑡↦Φ_𝜆^𝛼,𝛽(𝑡)=Γ𝛼+𝛽+1+𝑖𝜆/2Γ𝛼+𝛽+1−𝑖𝜆/22^−𝛽+1Γ()×()1+𝛼sinh𝑡^𝛼𝜑_𝜆^𝛼,𝛽(𝑡) (3) are proved. We establish, under some conditions, that Φ_𝜆^{𝛼−1,𝛽+1}(𝑡)−Φ_𝜆^{𝛼+1,𝛽+1}(𝑡)=2𝑑cosh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}𝑡),2(()𝑑sinh2𝑡𝑑𝑡^𝑚(sinh𝑡)^𝛼Φ_𝜆^𝛼,𝛽𝑡)=()(sinh𝑡^𝛼−𝑚Φ_𝜆^{𝛼−𝑚,𝛽+𝑚}(𝑡),(sinh𝑡)^𝑚Φ_𝜆^{𝛼−𝑚,𝛽+𝑚}(𝑡)−(−1)^𝑚Φ_𝜆^{𝛼+𝑚,𝛽+𝑚}(𝑡=_{𝑚−1𝑗=0})(−1)^𝑗𝑏^𝑗_𝑚(𝛼)(sinh𝑡)^𝑗Φ_𝜆^{𝛼+𝑗,𝛽+𝑗}(𝑡), (4) where 𝑏^𝑗_𝑚(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. 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 function 𝜑_𝜆^𝛼,𝛽(𝛼,𝛽,𝜆∈ℂ,𝛼≠−1,−2,−3,...) (cf. [1, 6]) can be expressed as a hypergeometric function under form 𝜑_𝜆^𝛼,𝛽(𝑡)=₂𝐹₁𝜌+𝑖𝜆2,𝜌−𝑖𝜆2;𝛼+1;−sinh²𝑡∶𝜌=𝛼+𝛽+1. (5) This function satisfies the following properties.(i) For every 𝑡∈[0,+∞[, (Γ(𝛼+1))⁻¹𝜑_𝜆^𝛼,𝛽(𝑡) is an entire function of 𝛼,𝛽, and 𝜆.(ii) For each 𝛼,𝛽∈ℂ and for each nonnegative integer 𝑛 there exists a positive constant 𝑘 such that Γ(∀𝑡≥0,∀𝜆∈ℂ,(𝛼+1))⁻¹𝑑𝑑𝑡^𝑛𝜑_𝜆^𝛼,𝛽(𝑡(|𝜆)≤𝑘1+|)^𝑛+𝑘(1+𝑥)𝑒^{(|Im𝜆|−Re(𝜌))𝑡}, (6) where 𝑘=0 if Re(𝛼)>−1/2 and 𝑘=[(1/2)−Re(𝛼)] if Re(𝛼)≤−1/2. (iii) The function 𝑡→𝜑_𝜆^𝛼,𝛽(𝑡) for Re(𝛼)>−1/2 possesses the Laplace type integral representation ∀𝑡>0,∀𝜆∈ℂ,𝜑_𝜆^𝛼,𝛽(𝑡)=^𝑡₀𝒦𝑡,𝑦cos𝜆𝑦𝑑𝑦, (7)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 function 𝜑_𝜆^𝛼,𝛽 possesses also an integral representation with respect to the dual variable [12].

To finish this paragraph, we consider the “modified” Jacobi function 𝑡↦Φ_𝜆^𝛼,𝛽(𝑡)=Γ𝛼+𝛽+1+𝑖𝜆/2Γ𝛼+𝛽+1−𝑖𝜆/22^−𝛽+1Γ()×()1+𝛼sinh𝑡^𝛼𝜑_𝜆^𝛼,𝛽(𝑡), (8) which can be written (cf. [7], page 693) as follows: ∀𝑡>0,^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝐽_𝛼(𝑥sinh𝑡)𝑑𝑥=Φ_𝜆^𝛼,𝛽(𝑡),|Im𝜆|<Re𝛼+𝛽+1, (9) where 𝐽_𝛼 and 𝐾_𝛼 denote, respectively, the Bessel function of the first kind and Macdonald's function.

Example 1. The functions 𝜑_𝜆^1/2,1/2 and Φ_𝜆^1/2,1/2 can be expressed in terms of elementary functions as ∀𝑡>0,𝜑_𝜆^1/2,1/2(𝑡)=2sin(𝜆𝑡)(),𝜆sinh2𝑡∀𝑡>0,Φ_𝜆^1/2,1/2(𝑡)=2𝜋Γ2+𝑖𝜆2Γ2−𝑖𝜆2×1𝜆sin(𝜆𝑡)√,||sinh𝑡cosh𝑡Im𝜆<2, (10)(11)

2.2. Bessel Function of the First Kind

We recall that the Bessel function of the first kind and order 𝛼 denoted by 𝐽_𝛼 is defined as an analytic function on {𝑧∈ℂ∶|arg𝑧|<𝜋} by 𝐽_𝛼(𝑧)=^∞_𝑘=0(−1)^𝑘Γ()Γ()𝑧𝑘+1𝑘+𝛼+12^𝛼+2𝑘 (12) and possesses the following integral representation: 𝐽_𝛼(𝑧)=(𝑧/2)^𝛼√𝜋Γ(𝛼+1/2)¹₋₁1−𝑡^2𝛼−1/2()(𝛼)1cos𝑧𝑡𝑑𝑡,Re>−2. (13) It satisfies the following functional relations: 𝐽_𝛼−1(𝑧)−𝐽_𝛼+1(𝑧)=2𝐽^′_𝛼(𝑧),1𝑧𝑑𝑧𝑑𝑧^𝛼𝐽_𝛼(𝑧)=𝑧^𝛼−1𝐽_𝛼−1(𝑧). (14)(15) It also verifies [10] 𝑧^𝑚𝐽_𝛼−𝑚(𝑧)−(−1)^𝑚𝐽_𝛼+𝑚(𝑧)=^𝑚−1_𝑗=0(−1)^𝑗𝑏^𝑗_𝑚(𝛼)𝑧^𝑗𝐽_𝛼+𝑗(𝑧), (16) with 𝑏^𝑗_𝑚(𝛼)()2=Γ𝛼+1^𝑚−𝑗𝐶^𝑗_𝑚Γ,𝛼+𝑗−𝑚+1 (17) and satisfies 𝐽₀(𝑧)+2^∞_𝑗=1𝐽_2𝑗(𝑧)=1,_∞𝑗=0(−1)^𝑗𝐽_2𝑗+1(𝑧)=sin𝑧2,𝐽₀(𝑧)+2^∞_𝑗=1(−1)^𝑗𝐽_2𝑗(𝑧)=cos𝑧,_∞𝑗=1(−1)^𝑗+12𝑗²𝐽_2𝑗(𝑧)=𝑧sin𝑧2_∞𝑗=0,(−1)^𝑗2𝑗+1²𝐽_2𝑗+1(𝑧)=𝑧cos𝑧2. (18)(19)(20)(21)(22) It has, for 𝑧≥0 and 𝛼≥0, the following asymptotic representation: 𝐽_𝛼(𝑧)≍𝑧^𝛼2^𝛼Γ()𝐽𝛼+1,𝑧→0,_𝛼(𝑧)≍21𝜋𝑧cos𝑧−21𝛼𝜋−4𝜋,𝑧→∞. (23)

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

We recall that the modified Bessel function of the first kind 𝐼_𝛼 of order 𝛼∈ℂ and Macdonald's function 𝐾_𝛼 of order 𝛼∈ℂ are defined as analytic functions on {𝑧∈ℂ∶|arg𝑧|<𝜋} by the formulas 𝐼_𝛼(𝑧)=^∞_𝑘=0(𝑧/2)^𝛼+2𝑘Γ()Γ(),𝐾𝑘+1𝑘+𝛼+1_𝛼(𝑧)=𝜋2𝐼_−𝛼(𝑧)−𝐼_𝛼(𝑧)sin(𝜋𝛼),if𝛼∉ℤ,lim_𝜈→𝛼𝐾_𝜈(𝑧),if𝛼∈ℤ. (24) The Macdonald's function satisfies 𝐾_𝛼(𝑧)=𝐾_−𝛼(𝑧),𝐾_𝛼−1(𝑧)+𝐾_𝛼+1(𝑧)=−2𝐾^′_𝛼(𝑧). (25)(26) It also verifies (cf. [10]) 𝑧^𝑚𝐾_𝛼+𝑚(𝑧)−𝐾_𝛼−𝑚(𝑧=_{𝑚−1𝑗=0})(−1)^{𝑚−𝑗−1}𝑏^𝑗_𝑚(𝛼)𝑧^𝑗𝐾_𝛼+𝑗(𝑧), (27) where 𝑏^𝑗_𝑚(𝛼) is the constant defined by (17).

For Re𝑧>0 and 𝛼∈ℂ, it can be written under the representation integral ([9], page 119) 𝐾_𝛼(𝑧)=^∞₀𝑒^{−𝑧cosh𝑢}cosh(𝛼𝑢)𝑑𝑢. (28) 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): 𝐼_𝛼(𝑧)≍𝑧^𝛼2^𝛼Γ()𝐾𝛼+1,𝑧→0,_𝛼(𝑧)≍2log𝑧2,if𝛼=0_𝛼−1Γ(𝛼)𝑧^𝛼𝐼,if𝛼>0,𝑧→0_𝛼(𝑧)≍𝑒^𝑧√𝐾2𝜋𝑧,𝑧→∞,_𝛼(𝑧)≍𝜋𝑒2𝑧^−𝑧,𝑧→∞. (29)(30)(31)(32) Note that in the special case 𝛼=±1/2, Macdonald's function is given by 𝐾_±1/2(𝑧)=𝜋2𝑧^1/2𝑒^−𝑧. (33)

To complete this paragraph, we recall the following (cf. [7], page 684 and page 747).(i)If |Im𝜆|<1+Re(𝛽), then we have ^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑𝑥=2^𝛽−1Γ1+𝛽+𝑖𝜆2Γ1+𝛽−𝑖𝜆2. (34)(ii)If |Im𝜆|<1+Re(𝛽) and 𝑡>0, then we have ^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)cos(𝑥sinh𝑡)𝑑𝑥=𝜋2sinh𝑡Φ_𝜆^{−1/2,𝛽+1/2}(𝑡). (35)(iii)If |Im𝜆|<2+Re(𝛽) and 𝑡>0, then we have ^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)sin(𝑥sinh𝑡)𝑑𝑥=𝜋2sinh𝑡Φ_𝜆^1/2,𝛽+1/2(𝑡). (36)

3. Recurrence Relations for Φ_𝜆^𝛼,𝛽

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 (sinh𝑡)^𝑚Φ_𝜆^{𝛼−𝑚,𝛽+𝑚}(𝑡)−(−1)^𝑚Φ_𝜆^{𝛼+𝑚,𝛽+𝑚}(𝑡=_{𝑚−1𝑗=0})(−1)^𝑗𝑏^𝑗_𝑚(𝛼)(sinh𝑡)^𝑗Φ_𝜆^{𝛼+𝑗,𝛽+𝑗}(𝑡), (37) where 𝑏^𝑗_𝑚(𝛼) stands for the constant given by (17).

Proof. Formula (16) asserts that 𝑥^𝛽(𝑥sinh𝑡)^𝑚𝐽_𝛼−𝑚(𝑥sinh𝑡)−(−1)^𝑚𝐽_𝛼+𝑚()=𝑥sinh𝑡_{𝑚−1𝑗=0}(−1)^𝑗𝑏_𝑚𝑗𝑥^𝛽(𝑥sinh𝑡)^𝑗𝐽_𝛼+𝑗(𝑥sinh𝑡). (38) We multiply both sides of the last identity by 𝐾_𝑖𝜆(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 Φ_𝜆^{𝛼−1,𝛽+1}(𝑡)−Φ_𝜆^{𝛼+1,𝛽+1}(𝑡)=2𝑑cosh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}(𝑡). (39)

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 Φ_𝜆^{𝛼−1,𝛽+1}(𝑡)−Φ_𝜆^{𝛼+1,𝛽+1}(𝑡)=2cosh𝑡^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑𝐽𝑑𝑡^𝛼(𝑥sinh𝑡)𝑑𝑥. (40) On the other hand, by using formulas (14), (28), and (13), we can deduce that 𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑𝐽𝑑𝑡^𝛼((𝛼)𝑥sinh𝑡)≤𝐶cosh𝑡⋅𝐾_|Im(𝜆)|(𝑥)𝑥^Re(𝛽)+1×(𝑥sinh𝑡)^Re(𝛼)−1+(𝑥sinh𝑡)^Re(𝛼)+1, (41) 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 Φ_𝜆^{𝛼+1,𝛽+1}(𝑡)=𝛼Φsinh𝑡^{𝜆𝛼,𝛽}(𝑡)−1𝑑cosh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}𝑡),Φ_{𝜆𝛼−1,𝛽+1}((𝑡)=𝛼Φsinh𝑡^{𝜆𝛼,𝛽}(𝑡)+1𝑑cosh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}(𝑡). (42)(43)

Example 5. Using Example 1 and relation (42) for 𝛼=1/2 and 𝛽=1/2, we obtain Φ_𝜆^3/2,3/2(𝑡)=1√Γ2𝜋2+𝑖𝜆2Γ2−𝑖𝜆21𝜆×2sin(𝜆𝑡)cosh(2𝑡)−𝜆cos(𝜆𝑡)sinh(2𝑡)(sinh𝑡)^3/2(cosh𝑡)³,𝜑_𝜆^3/2,3/2(𝑡)=12𝜆4+𝜆²×2sin(𝜆𝑡)cosh(2𝑡)−𝜆cos(𝜆𝑡)sinh(2𝑡)(sinh(2𝑡))³. (44)

Example 6. Using Example 1 and relation (43) for 𝛼=1/2 and 𝛽=1/2, we obtain Φ_𝜆^−1/2,3/2(𝑡)=2𝜋Γ2+𝑖𝜆2Γ2−𝑖𝜆21𝜆×𝜆cos(𝜆𝑡)cosh𝑡−sin(𝜆𝑡)sinh𝑡√sinh𝑡(cosh𝑡)³,𝜑_𝜆^−1/2,3/2(𝑡)=𝜆cos(𝜆𝑡)cosh𝑡−sin(𝜆𝑡)sinh𝑡sinh𝑡(cosh𝑡)³. (45)

Example 7. Using Example 5 and relation (42) for 𝛼=1 and 𝛽=1, we obtain Φ_𝜆^5/2,5/2(𝑡)=1√Γ2𝜋2+𝑖𝜆2Γ2−𝑖𝜆21×4𝜆16+𝜆²−𝜆²(()()()×()−8cosh4𝑡)sin𝜆𝑡−6𝜆cos𝜆𝑡sinh4𝑡sinh𝑡^5/2(cosh𝑡)⁵⁻¹,𝜑_𝜆^5/2,5/2(𝑡)=158𝜆16+𝜆²4+𝜆²×16+𝜆²−𝜆²(()()()×)−8cosh4𝑡)sin𝜆𝑡−6𝜆cos𝜆𝑡sinh4𝑡(sinh𝑡⁵(cosh𝑡)⁵⁻¹. (46)

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

Φ_𝜆^7/2,7/2(𝑡)=1√Γ2𝜋2+𝑖𝜆2Γ2−𝑖𝜆21×8𝜆1216+𝜆²−𝜆²−4cosh(4𝑡)cosh(2𝑡)sin(𝜆𝑡)(sinh𝑡)^7/2(cosh𝑡)⁷+𝜆−76−𝜆²+𝜆²−44cosh(4𝑡)sinh(2𝑡)cos(𝜆𝑡)(sinh𝑡)^7/2(cosh𝑡)⁷,𝜑_𝜆^7/2,7/2(𝑡)=10516𝜆4+𝜆²16+𝜆²36+𝜆²×1216+𝜆²−𝜆²−4cosh(4𝑡)cosh(2𝑡)sin(𝜆𝑡)(sinh𝑡)⁷(cosh𝑡)⁷+𝜆−76−𝜆²+𝜆²−44cosh(4𝑡)sinh(2𝑡)cos(𝜆𝑡)(sinh𝑡)⁷(cosh𝑡)⁷. (47)

Example 9. Using Example 5 and relation (43) for 𝛼=3/2 and 𝛽=3/2, we obtain Φ_𝜆^1/2,5/2(𝑡)=1√Γ2𝜋2+𝑖𝜆2Γ2−𝑖𝜆21𝜆×4+𝜆²+𝜆²(()()()×()−2cosh2𝑡)sin𝜆𝑡+3𝜆cos𝜆𝑡sinh2𝑡sinh𝑡^1/2(cosh𝑡)⁵⁻¹. (48)
Using the previous relation and relation (42) for 𝛼=1/2 and 𝛽=5/2, we obtain

Φ_𝜆^3/2,7/2(𝑡)=1√Γ2𝜋2+𝑖𝜆2Γ2−𝑖𝜆21×4𝜆16+𝜆²(−3+4cosh(2𝑡))+7𝜆²−8cosh(4𝑡)sin(𝜆𝑡)(sinh𝑡)^3/2(cosh𝑡)⁷+2𝜆16+𝜆²+𝜆²−14cosh(2𝑡)sinh(2𝑡)cos(𝜆𝑡)(sinh𝑡)^3/2(cosh𝑡)⁷. (49)

Proposition 10. For 𝛼,𝛽,𝜆∈ℂ such that |Im𝜆|<Re(𝛼+𝛽+1), one has 2∀𝑡>0,()𝑑sinh2𝑡)𝑑𝑡(sinh𝑡^𝛼Φ_𝜆^𝛼,𝛽𝑡)=()(sinh𝑡^𝛼−1Φ_𝜆^{𝛼−1,𝛽+1}(𝑡). (50)

Proof. According to formula (15), we obtain 1(𝑡)(𝑡)𝑑sinhcosh)𝑑𝑡(sinh𝑡^𝛼𝐽_𝛼(()𝑥sinh𝑡)=𝑥sinh𝑡^𝛼−1𝐽_𝛼−1(𝑥sinh𝑡) (51) and consequently (sinh𝑡)^𝛼−1Φ_𝜆^{𝛼−1,𝛽+1}(𝑡)=2sinh(2𝑡)^∞₀𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑)𝑑𝑡(sinh𝑡^𝛼𝐽_𝛼(𝑥sinh𝑡)𝑑𝑥. (52) 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 2()𝑑sinh2𝑡𝑑𝑡^𝑚(sinh𝑡)^𝛼Φ_𝜆^𝛼,𝛽𝑡)=()(sinh𝑡^𝛼−𝑚Φ_𝜆^{𝛼−𝑚,𝛽+𝑚}(𝑡),Φ_𝜆^{𝛼+𝑚,𝛽+𝑚}=(−1)^𝑚(sinh𝑡)^𝛼−𝑚2()𝑑sinh2𝑡𝑑𝑡^𝑚(sinh𝑡)^𝛼Φ_𝜆^𝛼,𝛽+(−1)^𝑚+1(sinh𝑡)^𝑚𝑚−1_𝑗=0(−1)^𝑗𝑏_𝑚𝑗(sinh𝑡)^𝑗Φ_𝜆^{𝛼+𝑗,𝛽+𝑗}. (53)

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 Φ∀𝑡>0,_{𝛼,𝛽+𝑚𝜆−𝑖𝑚}(𝑡)−Φ^{𝛼,𝛽+𝑚}_{𝜆+𝑖𝑚}(𝑡)=^𝑚−1_𝑘=0(−1)^{𝑚−𝑘−1}𝑏^𝑘_𝑚(𝑖𝜆)Φ^{𝛼,𝛽+𝑘}_{𝜆−𝑖𝑘}(𝑡), (54) where 𝑏^𝑘_𝑚(𝛼) is the constant given by (17). For all 𝑡>0, we have Φ^𝛼,𝛽+1_𝜆+𝑖(𝑡)+Φ^𝛼,𝛽+1_𝜆−𝑖(𝑡)=2𝛽+1Φ_𝜆^𝛼,𝛽(𝑡)𝑑+2tanh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}(𝑡). (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 𝐽_𝛼 and 𝐾_𝛼.

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

Corollary 13. If |Im𝜆|<Re(𝛼+𝛽+1) and 𝑡>0, one has Φ^𝛼,𝛽+1_𝜆−𝑖(𝑡)=𝛽+1+𝑖𝜆Φ_𝜆^𝛼,𝛽(𝑡)𝑑+tanh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}(𝑡),Φ^𝛼,𝛽+1_𝜆+𝑖(𝑡)=𝛽+1−𝑖𝜆Φ_𝜆^𝛼,𝛽(𝑡)𝑑+tanh𝑡Φ𝑑𝑡^{𝜆𝛼,𝛽}(𝑡). (56)

Example 14. By using formulas (5.32) (in [1], page 44) and (10), we can see that 𝜑_𝜆^1/2,−1/2(𝑡)=sin(𝜆𝑡),𝜆sinh𝑡 (57) and therefore Φ_𝜆^1/2,−1/2(𝑡)=Γ((1+𝑖𝜆)/2)Γ((1−𝑖𝜆)/2)√()2𝜋sin𝜆𝑡𝜆√.sinh𝑡 (58) Using now the last corollary, we can get Φ^1/2,1/2_𝜆+𝑖(𝑡)Γ))Γ))=−𝑖((1+𝑖𝜆/2((1−𝑖𝜆/2√)𝑡)2𝜋sin((𝑖+𝜆cosh𝑡√,sinh𝑡 (59) which coincided with the value of Φ_Λ^1/2,1/2(𝑡) already calculated at Λ=𝑖+𝜆 and we have

Φ^1/2,3/2_𝜆+𝑖(𝑡)=−2𝜋Γ2+𝑖𝜆2Γ2−𝑖𝜆2𝑖𝜆×(𝜆+𝑖)sin((𝜆+𝑖)𝑡)cosh𝑡+cos((𝜆+𝑖)𝑡)sinh𝑡√sinh𝑡cosh³𝑡. (60)

Remark 15. By using (33) and the fact that (cf., e.g., [7], page 707) ^∞₀𝑒^−𝑥𝐽_𝛼()1𝑥sinh𝑡𝑑𝑥=cosh𝑡cosh𝑡−1sinh𝑡^𝛼,𝑡>0,Re(𝛼)>−1, (61) we can get Φ^𝛼,1/2_𝑖/2(𝑡)=𝜋21cosh𝑡cosh𝑡−1sinh𝑡^𝛼=𝜋21cosh𝑡tanh^𝛼𝑡2,𝑡>0,Re(𝛼)>−1. (62) As an application of the last corollary, we obtain Φ₃₍₎^𝛼,3/2_𝑖/2(𝑡)=√𝜋41cosh³𝑡tanh^𝛼𝑡2((,Φ3+2𝛼cosh𝑡+cosh2𝑡))_{𝛼,3/2−𝑖/2}(𝑡)=√𝜋21cosh³𝑡tanh^𝛼𝑡2(1+𝛼cosh𝑡). (63)

3.3. Some Summations Intervening the “Modified” Jacobi Functions

Proposition 16. If |Im𝜆|<Re(1+𝛽) and 𝑡∈(0,argsinh1), then one has Φ_𝜆^0,𝛽(𝑡)+2^∞_𝑘=1Φ_𝜆^2𝑘,𝛽(𝑡)=2^𝛽−1Γ1+𝛽+𝑖𝜆2Γ1+𝛽−𝑖𝜆2.Φ_𝜆^0,𝛽(𝑡)+2^∞_𝑘=1(−1)^𝑘Φ_𝜆^2𝑘,𝛽(𝑡)=𝜋2sinh𝑡Φ_𝜆^{−1/2,𝛽+1/2}(𝑡). (64)(65)

Proof. Using formulas (18), (34), and (9), we obtain Φ_𝜆^0,𝛽(𝑡)+2^∞₀^∞_𝑘=1𝐽_2𝑘(𝑥sinh𝑡)𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑𝑥=2^𝛽−1Γ1+𝛽+𝑖𝜆2Γ1+𝛽−𝑖𝜆2. (66) With the help of formulas (13), (28), and (34), we can see that ^∞₀𝐽_2𝑘(𝑥sinh𝑡)𝑥^𝛽𝐾_𝑖𝜆(𝑥≤)𝑑𝑥_∞0((𝑥sinh𝑡)/2)^2𝑘()!𝑥2𝑘^Re(𝛽)𝐾_|Im(𝜆)|(𝑥)=))𝑑𝑥((sinh𝑡/2^2𝑘()!22𝑘^{2𝑘+Re(𝛽)−1}𝛽−|(𝜆×Γ1+2𝑘+ReIm)|2𝛽+|(𝜆×Γ1+2𝑘+ReIm)|2, (67) which permits to conclude the convergence of series ^∞_𝑘=1^∞₀𝐽_2𝑘(𝑥sinh𝑡)𝑥^𝛽𝐾_𝑖𝜆(𝑥)𝑑𝑥,∀𝑡∈0,argsinh1, (68) 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−sinh²𝑡√sinh𝑡cosh⁴𝑡. (69) Using now Corollary 13, we see that Φ_()𝑖^−1/2,2_3/2(𝑡)=2√sinh𝑡cosh⁴𝑡. (70)

Proposition 18. If |Im𝜆|<2+Re(𝛽) then one has ^∞_𝑘=1(−1)^𝑘+1(2𝑘)²Φ_𝜆^2𝑘,𝛽(𝑡)=√𝜋(sinh𝑡)^3/2Φ_𝜆^1/2,𝛽+3/2(𝑡),∀𝑡∈0,argsinh1_∞𝑘=0,(−1)^𝑘(2𝑘+1)²Φ_𝜆^2𝑘+1,𝛽(𝑡)=𝜋2(sinh𝑡)^3/2Φ_𝜆^{−1/2,𝛽+3/2}(𝑡),∀𝑡∈0,argsinh1. (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 ^∞_𝑘=1(−1)^𝑘+1(2𝑘)²tanh^2𝑘𝑡2=sinh²𝑡cosh³𝑡_∞𝑘=0,(−1)^𝑘(2𝑘+1)²tanh^2𝑘+1𝑡2=(3−cosh(2𝑡))4sinh𝑡cosh³𝑡, (72) formula (62), and the last proposition, we can see that Φ^1/2,2_𝑖/2(𝑡)=1√2√sinh𝑡cosh⁴𝑡,Φ^−1/2,2_𝑖/2(𝑡)=14(3−cosh(2𝑡))√sinh𝑡cosh⁴𝑡. (73)

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, 𝜑^𝛼,𝛼_𝜆/𝜀(𝜀𝑥)=₂𝐹₁1𝛼+2+𝑖𝜆12𝜀,𝛼+2−𝑖𝜆2𝜀;𝛼+1;−sinh²(𝜀𝑥) (74) tends to the normalized Bessel function 𝑗_𝛼(𝜆𝑥)=₀𝐹₁𝛼+1;−𝜆𝑥2²=Γ(𝛼+1)^∞_𝑘=0(−1)^𝑘Γ(𝑘+1)Γ(𝑘+𝛼+1)𝜆𝑥2^2𝑘. (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].