Abstract
Two integral operators involving Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind is expressed in terms of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function, are also discussed.
1. Introduction
Let , , , , and ; then the generalized fractional integral operators involving Appell’s functions or Horn’s function are defined as follows: with . The generalized fractional integral operators of the types (1) and (2) have been introduced by Marichev [1] and later extended and studied by Saigo and Maeda [2]. These operators together are known as the Marichev-Saigo-Maeda operator.
The fractional integral operator has many interesting applications in various subfields in applicable mathematical analysis; for example, [3], it has applications related to a certain class of complex analytic functions. The results given in [4–6] can be referred to for some basic results on fractional calculus.
The purpose of this work is to investigate compositions of integral transforms (1) and (2) with the generalized Bessel function of the first kind defined for complex and by where . More details related to the function and its particular cases can be found in [7, 8] and references therein. It is worth mentioning that is Bessel function of order and is modified Bessel function of order . Also, is spherical Bessel function of order and is modified spherical Bessel function of order . Thus the study of the integral transform of will give far reaching results than the result in [9, 10].
The present paper is organized as follows. In Sections 2 and 3, composition of integral transforms (1) and (2) with generalized Bessel function (3) is given in terms of generalized Wright functions and generalized hypergeometric functions, respectively. Special cases like () of give the composition of (1) and (2) with cosine and hyperbolic cosine (sine and hyperbolic sine) functions, which are discussed in Section 4. Some concluding remarks and comparison with earlier known work are mentioned in Section 5.
The following two results given by Saigo et al. [2, 11] are needed in sequel.
Lemma 1. Let be such that and Then there exists the relation where
Lemma 2. Let be such that and Then there exists the relation
2. Representations in terms of Generalized Wright Functions
In this section composition of integral transforms (1) and (2) with generalized Bessel function (3) is given in terms of the generalized Wright hypergeometric function which is defined by the series Here , and (; ). Asymptotic behavior of this function for large values of argument of was studied in [12] and under the condition in [13, 14]. Properties of this generalized Wright function were investigated in [9, 15, 16]. In particular, it was proved [15] that , , is an entire function under the condition (10). Interesting results related to generalized Wright functions are also given in [17].
Theorem 3. Let such that . Suppose that and . Then
Proof. An application of integral transform (1) to the generalized Bessel function (3) leads to the formula Now changing the order of integration and summation in right-hand side of (12) yields Note that, for all , Replacing by in Lemma 1 and using (5), we obtain
Interpreting the right-hand side of (15), the equality (11) can be obtained from (6) and then by using the definition of generalized Wright function.
Theorem 4. Let such that . Suppose that and . Then
Proof. Using (2) and (3) and then changing the order of integration and summation, which is justified under the conditions with Theorem 4, yield Note that, for all , Hence replacing by in Lemma 2 and using (8), we obtainNow (6), (9), and (19) together imply thatand this completes the proof.
3. Representation in terms of Generalized Hypergeometric Series
The generalized hypergeometric function ,, is given by the representation where none of the denominator parameters is zero or a negative integer. Here or are allowed to be zero. The series (21) is convergent for all finite if , while, for , it is convergent for and divergent for .
Results obtained in this section demonstrate the image formula for the generalized Bessel functions under the operators (1) and (2) in terms of generalized hypergeometric functions. The well-known Legendre duplication formulas [18] given by are required for this purpose.
Theorem 5. Let such that . Suppose that and . Then the following formula holds:
Proof. It is known that . ThusNow apply (3.3) on the right-hand side of the above equation and then the result follows from (15). This completes the proof.
By adopting a similar method the next result can be obtained from (19); we omit the details.
Theorem 6. Let such that . Suppose that and . Then
4. Fractional Integration of Trigonometric Functions
4.1. Cosine and Hyperbolic Cosine Functions
For all , if , then the generalized Bessel function has the form Hence the following results are a consequence of Theorems 3 and 4, respectively.
Corollary 7. Let such that and Then
Proof. On setting and replacing by into (11) and using (37), we haveThis implies thatThe identity (39) follows from (30) by replacing by .
Similarly, the identity (4.7) can be obtained from (11) by setting and replacing by .
Corollary 8. Let be such that , and Then
The next statements show that the image formulas for cosine and hyperbolic cosine under Saigo-Maeda fractional integral operators can also be represented in terms of the generalized hypergeometric series. This result follows from Theorems 5 and 6 with taking and replacing by or , respectively.
Corollary 9. Let . Suppose that and Then the following formula holds:
Corollary 10. Let . Suppose that and Then
4.2. Sine and Hyperbolic Sine Functions
For all , if , then the generalized Bessel function has the form Thus, the composition of Saigo-Maeda fractional integral operators with sine and hyperbolic sine functions can be obtained from Theorems 3 and 4, respectively.
Corollary 11. Let such that and Then
The next result follows from Theorem 4 by setting and replacing by or , respectively.
Corollary 12. Let be such that and Then
The following result can be obtained from Theorems 5 and 6 with taking and replacing by or , respectively.
Corollary 13. Let . Suppose that and Then the following formula holds:
Corollary 14. Let . Suppose that and . Then
5. Concluding Observations
In this section some consequences of the main result derived in previous sections are given in detail. Also comparison with other known results from the literature is listed.(1)We remark that all the results given by Purohit et al. [10] are followed from the results derived in this paper by setting .(2)The results in Sections 2 and 3 also provide the Marichev-Saigo-Maeda fractional integration of modified Bessel function and spherical Bessel functions.(3)Set in the operators (1) and (2). Then due to identities given by Saxena and Saigo [11, P. 93], it follows that The generalized integral transforms that appear in the right-hand side of the above equations are due to Saigo [19] and defined as follows: where is the Euler gamma function [20] and is the Gauss hypergeometric function. The above fact helps us to conclude that all the results given in [9, 21] can also be obtained from the results in this paper by setting .(4)Note that Riemann-Liouville and Weyl and Erdélyi-Kober fractional calculus [22] are special case of Saigo’s operator (46). Thus this paper is also useful to derive certain composition formula involving Riemann-Liouville and Weyl and Erdléyi-Kober fractional calculus and Bessel, modified Bessel, and spherical Bessel function of the first kind.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research work presented here was supported in part by the Deanship of Scientific Research (DSR) at Salman bin Abdulaziz University for K. S. Nisar.