Certain integrals associated with the generalized Bessel-Maitland function

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1 Available at htt://vau.edu/aa Al. Al. Math. ISSN: Vol. Issue (Deceber 7). - 6 Alications and Alied Matheatics: An International ournal (AAM) Certain integrals associated with the generalied Bessel-Maitland function D.L. Suthar and Hafte Asalu Deartent of Matheatics Wollo University Dessie P.O. Bo: 45 South Wollo Ahara Region Ethioia Eail- dlsuthar@gail.co hafasalu@gail.co Received: February 4 7 Acceted: Seteber 7 Abstract The ai of this aer is to establish two general finite integral forulas involving the generalied Bessel-Maitland functions ( ). The result given in ters of generalied (Wright s) hyergeoetric functions v and generalied hyergeoetric functions F. These results are obtained with the hel of finite integral due to Lavoie and Trottier. Soe interesting secial cases involving Bessel-Maitland function Struve s functions Bessel functions generalied Bessel functions Wright function generalied Mittag-Leffler functions are deduced. Keywords: Lavoie-Trottier integral forula Gaa function Hyergeoetric functions Bessel function Generalied Bessel-Maitland function Generalied Wright Hyergeoetric functions MSC No.: 6A33 33B5 33C 33C. Introduction and Preliinaries In Alied sciences any iortant functions are defined via iroer integrals or series (or finite roducts). These iortant functions are generally nown as secial functions. In secial functions one of the ost iortant functions (Bessel function) is widely used in hysics and engineering therefore they are of interest to hysicists and engineers as well as atheaticians. In recent years a rearably large nuber of integral forulas involving a variety of secial functions have been develoed by any authors: Brychov (8) Choi and Agarwal (3) Choi et al. (4) Agarwal et al. (4) Manaria et al. (4) Khan and Kashin (6) Parar and Purohit (6) Suthar and Haile (6) and Nisar et al. (6 7). We ai at resenting two generalied integral forulas involving the Bessel-

2 AAM: Intern.. Vol. Issue (Deceber 7) 3 Maitland function which are eressed in ters of the generalied (Wright s) hyergeoetric and generalied hyergeoetric functions. For our urose we begin by recalling soe nown functions and earlier wors. The Bessel- Maitland function is defined through a series reresentation by Marichev (983) as follows: v v v! C. (.) The generalied Bessel function of the for follows: v is defined by ain and Agarwal (5) as v (.) C v \ ( ] C. Further generaliation of the generalied Bessel-Maitland function v defined by Patha (966) is as follows: and v (.3)! C ( ) ( v ) ( ) and N is nown as generalied Pochhaer sybol defined by Mittag-Leffler (93). Fro the generaliation of the generalied Bessel-Maitland function (.3) it is ossible to find soe secial cases by giving articular values to the araeters and. ) If and v is relaced by v and is relaced by 4 in (.3) then we obtain v 4 v (.4)

3 4 D. L. Suthar and Hafte Asalu v denotes Bessel-Maitland function defined by Agarwal et al. (4). ) If we relace by and by v H H v v v in (.4) we obtain 3 Hv (.5) 4 denotes Struve s function defined by Erdẻlyi et al. (954). v. (.6) 3 3 v 3) If then (.3) reduces to v (.7) v is generalied Bessel function defined by Agarwal (5). 4) If and is relaced by 4 then (.3) reduces to 4 v v (.8) v is any non- v is called Bessel s function of the first ind and of order v negative constant. 5) If and v is relaced by v and is relaced by then (.3) reduces to v v (.9) v is nown as Wright function defined by Choi et al. (4). 6) If v is relaced by v and is relaced by then (.3) reduces to E v v (.) E v is generalied Mittag -Leffler function and was given by Shula and Prajaati (7). 7) If v is relaced by v and is relaced by then (.3) reduces to E v v (.)

4 AAM: Intern.. Vol. Issue (Deceber 7) 5 was introduced by Prabhaar (97). 8) If v is relaced by v and is relaced by (.3) reduces to E v v (.) ( ) ( v) and was studied by Wian (95). C 9) If v and is relaced by (.3) reduces to E. (.3) C ( ) and was introduced by Mittag-Leffler (93). Further another reresentation of the generalied Bessel Maitland function by Singh et al. (4): if N and N then (.3) reduces to v defined F (.4) v v F. is the generalied hyergeoetric function and the sybols tule and v is a tule The generaliation of the generalied hyergeoetric series F v is a v v 3 v. is due to Fo (98) and Wright (935 94((a) (b))) who studied the asytotic eansion of the generalied Wright hyergeoetric function defined by (see also (985)). The generalied Wright hyergeoetric function () Karlsson (985)) for i =... j =... ) is defined as below: (see for detail Srivastava and C cole a C and R ( i b j ( a ) i i ai i i= ( ) = (.5) bj j ( b )! j = j Introduced by Wright (935) the generalied Wright function and roved several theores on the asytotic eansion of () for all values of the arguent under the condition: j i j i j j = j i= >. i (.6)

5 6 D. L. Suthar and Hafte Asalu It is noted that the generalied (Wright) hyergeoetric function in (.5) whose asytotic eansion was investigated by Fo (98) and Wright is an interesting further generaliation of the generalied hyergeoetric series as follows: = j = j = j j F (.7) F is the generalied hyergeoetric series defined by (see : () Section.5) F = n = ( ) ( ) ( ) ( ) n n n n n n! = F ( ) (.8) For our resent investigation we also need to recall the following Lavoie-Trottier integral forula (969): ( ) ( ) d = (.9) ( ) rovided ( ) ( ).. Main Results In this section we established two generalied integral forulas which are eressed in ters of generalied (Wright) hyergeoetric functions by inserting the generalied Bessel- Maitland function (.3) with suitable arguent in to the integrand of (.9). Theore.. The following integral forula holds true for C with Re( v) Re( ) Re( ) Re( ) and > we have d y 3 v y. (.) Proof: Now alying (.3) to the integrand of (.) and then interchanging the order of integration and suation which is verified by unifor convergence of the involved series under the given conditions in Theore. we get

6 AAM: Intern.. Vol. Issue (Deceber 7) 7 d y v y! ( ) 3 4 d By considering the condition given in Theore. since Re( v) N and alying (.9) v y! 3 3 ( ) v y! which uon using the definition (.7) we get the desired result (.). Theore.. Re( ) Re( ) The following integral forula holds true: for C with Re( v) Re( ) Re( ) Re( ) and > we have y d v. 9 (.) By siilar anner as in roof of Theore. we can also rove the integral forula (.). Net we consider other variations of Theore. and Theore.. In fact we establish soe integral forulas for the generalied Bessel-Maitland function eressed in ters of the generalied hyergeoetric function. To do this we recall the well-nown Legendre dulication forula (see ()) as N. (.3) Corollary.3. Assuing the condition of Theore. is satisfied and relacing by in the generalied v Bessel-Maitland function in (.) and using (.4) it can be shown that the following integral forula holds true:

7 8 D. L. Suthar and Hafte Asalu d y y v F 3 v (.4) ( ) ( ) is a tule and v is a tule v v v 3 v. Corollary.4. Let the condition of Theore. be satisfied and relacing Bessel-Maitland function forula holds true: v y d by in the generalied in (.) and using (.4). Then the following integral 3 v F v 9. (.5) Proof: By writing the right hand side of (.) in the original suation forula after a little silification we find that when the last resulting suation is eressed in the ter of hyergeoetric in the relation (.4) this coletes the roof of Corollary.3. Siilarly it is easy to see that a siilar arguent as in roof of Corollary.3 will established the integral forula (.4). Therefore we oit the details of the roof of the Corollary Secial Cases In this section we reresent certain cases of generalied for of Bessel-Maitland function (.3). On setting and is relaced by ( / 4) in Theore. and Theore. and aing use of the relation (.4) then the generalied Bessel-Maitland function will have the following relation with Bessel-Maitland function as follows:

8 AAM: Intern.. Vol. Issue (Deceber 7) 9 Corollary 3.. Let the conditions of C following integral forula holds true: ( ) and ( ) be satisfied then the 3 d y y v Corollary v v v y. 4 (3.) Let the conditions of C ( ) and ( ) be satisfied then the following integral forula holds true: y d y v 9 3 v v v. 8 (3.) On setting / in (3.) and (3.) and using the relation (.5) then we get the integral forulas involving the Struve s function H () as follows: Corollary 3.3. Let the conditions of forula holds true: C and ( ) be satisfied then the following integral v d H y 3 y v 3 v v 3 v 3 y. 4 (3.3) Corollary 3.4. Let the conditions of forula holds true: C and ( ) be satisfied then the following integral j H v y d 3 4 3

9 D. L. Suthar and Hafte Asalu 3 ( v ) y v 3 v v 3 v 3. 8 (3.4) On setting in Theore. and Theore. and aing use of the relation (.7) then the generalied Bessel Maitland function Bessel-Maitland function Corollary 3.5. v as follows: v will have the following relation with Let the conditions of C ( ) integral forula holds true: and be satisfied then the following v d y 3 v y. (3.5) Corollary 3.6. Let the conditions of C ( ) integral forula holds true: and v y d be satisfied then the following 3 v. 9 (3.6) Corollary 3.7. Let the conditions of C ( ) integral forula holds true: and v d y be satisfied then the following 3 ( ) v v F y. (3.7) is ositive integer.

10 AAM: Intern.. Vol. Issue (Deceber 7) Corollary 3.8. Let the conditions of integral forula holds true: and C ( ) be satisfied then the following v y d v v F. 9 (3.8) On setting and is relaced by ( / 4) in Theore. and Theore. and aing use of the relation (.8) we obtain the following integral forulas involving the ordinary Bessel function as follows: Corollary 3.9. Let the conditions of C ( ) integral forula holds true: and be satisfied then the following v d y 3 y v v y. 4 (3.9) Corollary 3.. Let the conditions of C ( ) integral forula holds true: and v y d be satisfied then the following 3 ( ) y v v. 8 (3.) Rear In (3.9) and (3.) by aing soe suitable adjustents of the araeters we arrive at the nown result given by Agarwal et al. ((4). 3 euations (.) and (.3)). On setting and v is relaced by v and is relaced by in Theore. and Theore. and aing use of the relation (.9) we obtain the following integral forulas involving the Weight function as follows:

11 D. L. Suthar and Hafte Asalu Corollary 3.. Let the conditions of C ( ) following integral forula holds true: and be satisfied then the d v y 3 y. (3.) Corollary 3.. Let the conditions of C ( ) following integral forula holds true: and be satisfied then the v y d (3.) On setting by and is relaced by in Theore. and Theore. and aing use of the relation (.) we obtain the following integral forulas involving the generalied Mittag- Leffler function as follows: Corollary 3.3. Let the conditions of C ( ) ( ) ( ) ( ) ( ) and ( ) N be satisfied then the following integral forula holds true: v d E y 3 y. (3.3) Corollary 3.4. Let the conditions of C ( ) ( ) ( ) ( ) ( ) and ( ) N be satisfied then the following integral forula holds true: E v y d 3 4 3

12 AAM: Intern.. Vol. Issue (Deceber 7) (3.4) Rear In (3.3) and (3.4) by aing soe suitable adjustents of the araeters we arrive at the nown result given by Manaria et al.((6). 5 e.(.3) and (.4)). On setting by and is relaced by in Theore. and Theore. and aing use of the relation (.) we obtain the following integral forulas involving the generalied Mittag- Leffler function as follows: Corollary 3.5. Let the conditions of C ( ) ( ) be satisfied then the following integral forula holds true: ( ) ( ) and ( ) v d E y 3 y. (3.5) Corollary 3.6. Let the conditions of C ( ) ( ) ( ) ( ) and ( ) be satisfied then the following integral forula holds true: E v y d (3.6) On setting by and is relaced by in Theore. and Theore. and aing use of the relation (.) we obtain the following integral forulas involving the generalied Mittag- Leffler function as follows: Corollary 3.7. Let the conditions of vc ( ) ( ) ( ) and ( ) be satisfied then the following integral forula holds true: v d E y

13 4 D. L. Suthar and Hafte Asalu 3 y. (3.7) Corollary 3.8. Let the conditions of vc ( ) ( ) then the following integral forula holds true: ( ) and ( ) be satisfied E v y d (3.8) On setting is relaced by in Theore. and Theore. and aing use of the relation (.3) we obtain the following integral forulas involving the Mittag- Leffler function as follows: Corollary 3.9. and Let the conditions of vc ( ) ( ) and ( ) be satisfied then the following integral forula holds true: d E y 3 y. (3.9) Corollary 3.. Let the conditions of vc ( ) ( ) and ( ) be satisfied then the following integral forula holds true: E y d Concluding rears 3. 9 (3.) In the resent aer we investigate new integrals involving the generalied Bessel-Maitland function ( ) in ters of generalied (Wright) hyergeoetric functions and generalied v

14 AAM: Intern.. Vol. Issue (Deceber 7) 5 hyergeoetric function F. Certain secial cases of integrals involving generalied Bessel Maitland function have been investigated by the authors in the literature with different arguents. Therefore the results resented in this aer are easily converted in ters of a siilar tye of new interesting integrals with different arguents after soe suitable araetric relaceents. In this seuel one can obtain integral reresentation of ore generalied secial function which has uch alication in hysics and engineering science. Acnowledgent: The authors would lie to than the anonyous referees for their valuable coents and suggestions which have heled to irove the anuscrit. REFERENCES Agarwal P. (5). Pathway fractional integral forulas involving Bessel function of the first ind Adv. Stud. Conte. Math. Vol. 5 No Agarwal P. ain S. Agarwal S. and Nagal M. (4). On a new class of integrals involving Bessel functions of the first ind Coun. Nuer. Anal. Art. ID Agarwal P. ain S. Chand M. Dwivedi S.K. and Kuar S. (4). Bessel functions associated with Saigo-Maeda fractional derivatives oerators. Frac. Calc. Vol. 5 No.. -. Brycho Y.A. (8). Handboo of Secial Functions Derivatives Integrals Series and Other Forulas CRC Press Taylor and Francis Grou Boca Raton London and New Yor. Choi. Agarwal P. Mathur S. and Purohit S.D. (4). Certain new integral forulas involving the generalied Bessel functions Bull. Korean Math. Soc. Vol. 5 No Choi. and Agarwal P. (3). Certain unified integrals involving a roduct of Bessel functions of first ind Hona Matheatical. Vol. 35 No Choi. and Agarwal P. (3). Certain unified integrals associated with Bessel functions Bound. Value Probl. doi:.86/ Erdẻlyi A. Magnus W. Oberhettinger F. and Tricoi F. (954). Table of integral transfors Vol. II McGraw-Hill New Yor. Fo C. (98). The asytotic eansion of generalied hyergeoetric functions Proc. London Math. Soc. Vol. 7 No ain S. and Agarwal P. (5). A new class of integral relation involving general class of Polynoials and I- function Walaila. Sci. & Tech. Vol. No Khan N.U. and Kashin T. (6). Soe integrals for the generalied Bessel-Maitland functions Electron.. Math. Anal. Al. Vol. 4 No Lavoie.L. and Trottier G. (969). On the su of certain Aell s series Ganita Vol Mariche O.I. (983). Handboo of integral transfor and Higher transcendental functions Ellis Harwood chichester (ohn Wiley and Sons) New Yor. Manaria N. Nisar K.S. and Purohit S.D. (6). On a new class of integrals involving roduct of generalied Bessel function of the first ind and general class of olynoials Acta Univ. Aulensis Vol

15 6 D. L. Suthar and Hafte Asalu Menaria N. Baleanu D. and Purohit S.D. (6). Integral forulas involving roduct of general class of olynoials and generalied Bessel function Sohag. Math. Vol. 3 No Manaria N. Purohit S.D. and Parar R.K. (6). On a new class of integrals involving generalied Mittag-Leffler function Survey of atheatics and its alications Vol Mitag-Leffler G.M. (93). Sur la nouvelle function E_() C. R. Acad. Soc. Paris Vol Nisar K.S. Parar R.K. and Abusufian A.H. (6). Certain new unified integrals with the generalied -Bessel function Far East. Math. Sci. Vol Nisar K.S. Suthar D.L. Purohit S.D. and Aldhaifallah M. (7). Soe unified integral associated with the generalied Struve function Proc. angjeon Math. Soc. Vol. No Parar R.K. and Purohit S.D. (6).On a new class of integrals involving generalied hyergeoetric function Int. Bull. Math. Res. Vol. 3 No Patha R.S. (966). Certain convergence theores and asytotic roerties of a generaliation of Loel and Maitland transfor Proc. Nat. Acad. Sci. India Sect. A Vol Prabhaer T.R. (97). A singular integral euation with a generalied Mittag-Leffler function in the ernel Yoohaa Math.. Vol Shula A.K. and Prajaati.C. (7). On a generalied Mittag-Leffler function and its roerties. Math. Anal. Al. Vol Singh M. Khan M.A. and Khan A.H. (4). On soe roerties of a generaliation of Bessel-Maitland function ournal of Matheatics trends and Technology Vol. 4 No Srivastava H.M. and Choi. (). Zeta and -Zeta Functions and Associated Series and Integrals Elsevier Asterda. Srivastava H.M. and Karlsson P.W. (985). Multile Gaussian Hyergeoetric Series Halsted Press (Ellis Horwood Liited Chichester) ohn Wiley and Sons New Yor Chichester Brisbane and Toronto. Suthar D.L. and Habeno Haile (6). Integrals involving generalied Bessel-Maitland function. Sci. Arts Vol. 37 No Wian A. (95). Uber den Fundaentalsat in der Teorie der Funtionen Acta Math. Vol Wright E.M. (935). The asytotic eansion of the generalied hyergeoetric functions. London Math. Soc. Vol Wright E.M. (94(a)). The asytotic eansion of integral functions defined by Taylor series Philos. Trans. Roy. Soc. London A Vol Wright E.M. (94(b)). The asytotic eansion of the generalied hyergeoetric function II Proc. Lond. Math. Soc. Vol