Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 274093, 11 pages

http://dx.doi.org/10.1155/2014/274093

Research Article

Marichev-Saigo-Maeda Fractional Integration Operators Involving Generalized Bessel Functions

Saiful R. Mondal¹ and K. S. Nisar²

¹Department of Mathematics & Statistics, College of Science, King Faisal University, P.O. Box 400, Hofuf, Al-Ahsa 31982, Saudi Arabia

²Department of Mathematics, College of Arts and Science, Salman bin Abdulaziz University, P.O. Box 54, Wadi Al-Dawaser 11991, Saudi Arabia

Received 12 February 2014; Accepted 4 March 2014; Published 8 April 2014

Academic Editor: Santanu Saha Ray

Copyright © 2014 Saiful R. Mondal and K. S. Nisar. 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.

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 obtain

Now (6), (9), and (19) together imply that

and 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 . Thus

Now 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 have

This implies that

The 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.

References

O. I. Marichev, “Volterra equation of Mellin convolution type with a Horn function in the kernel,” Izvestiya Akademii Nauk BSSR. Seriya Fiziko-Matematicheskikh Nauk, vol. 1, pp. 128–129, 1974 (Russian).

M. Saigo and N. Maeda, “More generalization of fractional calculus,” in Transform Methods & Special Functions, Varna '96, pp. 386–400, Bulgarian Academy of Sciences, Bulgaria, Sofia, 1998.

Y. C. Kim, K. S. Lee, and H. M. Srivastava, “Some applications of fractional integral operators and Ruscheweyh derivatives,” Journal of Mathematical Analysis and Applications, vol. 197, no. 2, pp. 505–517, 1996.

V. Kiryakova, “All the special functions are fractional differintegrals of elementary functions,” Journal of Physics A: Mathematical and General, vol. 30, no. 14, pp. 5085–5103, 1997.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

H. M. Srivastava, S.-D. Lin, and P.-Y. Wang, “Some fractional-calculus results for the -function associated with a class of Feynman integrals,” Russian Journal of Mathematical Physics, vol. 13, no. 1, pp. 94–100, 2006.

Á. Baricz, “Geometric properties of generalized Bessel functions,” Publicationes Mathematicae, vol. 73, no. 1-2, pp. 155–178, 2008.

S. R. Mondal and A. Swaminathan, “Geometric properties of generalized Bessel functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 1, pp. 179–194, 2012.

A. A. Kilbas and N. Sebastian, “Generalized fractional integration of Bessel function of the first kind,” Integral Transforms and Special Functions, vol. 19, no. 11-12, pp. 869–883, 2008.

S. D. Purohit, D. L. Suthar, and S. L. Kalla, “Marichev-Saigo-Maeda fractional integration operators of the Bessel functions,” Le Matematiche, vol. 67, no. 1, pp. 21–32, 2012.

R. K. Saxena and M. Saigo, “Generalized fractional calculus of the -function associated with the Appell function ,” Journal of Fractional Calculus, vol. 19, pp. 89–104, 2001.

C. Fox, “The asymptotic expansion of generalized hypergeometric functions,” Proceedings of the London Mathematical Society, vol. s2-27, no. 1, pp. 389–400.

E. M. Wright, “The asymptotic expansion of integral functions defined by Taylor series,” Philosophical Transactions of the Royal Society of London A: Mathematical and Physical Sciences, vol. 238, pp. 423–451, 1940.

E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,” Proceedings of the London Mathematical Society, vol. 46, pp. 389–408, 1940.

A. A. Kilbas, M. Saigo, and J. J. Trujillo, “On the generalized Wright function,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 437–460, 2002.

A. A. Kilbas and N. Sebastian, “Fractional integration of the product of Bessel functions of the first kind,” Fractional Calculus & Applied Analysis, vol. 13, no. 2, pp. 159–175, 2010.

H. M. Srivastava, “Some Fox-Wright generalized hypergeometric functions and associated families of convolution operators,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 1, pp. 56–71, 2007.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. I, II, McGraw-Hill Book Company, New York, NY, USA, 1953.

M. Saigo, “A remark on integral operators involving the Gauss hypergeometric functions,” Mathematical Reports of College of General Education, vol. 11, no. 2, pp. 135–143, 1978.

E. D. Rainville, Special Functions, The Macmillan, New York, NY, USA, 1960.

P. Malik, S. R. Mondal, and A. Swaminathan, “Fractional integration of generalized Bessel function of the first kind,” in Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE '11), Washington, Wash, USA, August 2011.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993, Translated from the 1987 Russian original.