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Introduction

The transform defined by the following integral equation

is called the k transform with p as a complex parameter and Kν(px) is called the Modified Bessel function of the third kind or the Macdonald function, see (Mathai et al, 2010, p.53). The Hankel transform of a function f(x), denoted by g(p, ν) is defined as

where Jν(px) is called the Bessel-Maitland function or the Maitland-Bessel function (Mathai et al, 2010, p.22 and p.56).

The Wright hypergeometric function defined by the series (Srivastava & Manocha, 1984):

where the coefficients A₁, ....A_p and B₁, ....B_q are positive real numbers such that

can be slightly generalized (3) as given below.

where pFq is the generalized hypergeometric function defined by (Srivastava & Manocha, 1984; Rainville, 1960)

where (λ)n is the well known Pochhammer symbol (Srivastava & Manocha, 1984).

The series representation of the generalized Lommel Wright function as (Kachhia & Prajapati, 2016);

Also, we have the following relations of the generalized Lommel Wright functions with trigonometric functions and the generalized Bessel function and the Struve function as follows:

The following known results of Mathai and Saxena (Mathai & Saxena, 1973):

Various generalizations and cases of the Lommel-Wright function have been investigated. For details, see (Paneva-Konovska, 2007; Menaria et al, 2016; Mondal & Nisar, 2017; Srivastava & Daoust, 1969; Kiryakova, 2000).

Integral formulas involving the Lommel-Wright functions have been developed by many authors. See e.g., (Choi & Agarwal, 2013; Choi et al, 2014; Jain et al, 2016; Chaurasia & Pandey, 2010). In this sequel, here, we aim at establishing a certain new generalized integral formula involving the generalized Lommel-Wright function interesting integral formulas which are derived as special cases.

Main results

This section deals with the evaluation of integrals formulas involving the Lommel-Wright function defined in (7) and the integrals involving the product of the Bessel function of first kind, Kelvin‘s function and the Whittaker function (Whittaker & Watson, 2013) with the generalized Lommel-Wright function.

THEOREM 1. Let Then the Hankel transform of the generalized Lommel-Wright function defined in (7) is given by

Proof. On using (7) in the integrand of (1) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (12) in the above equation we get

THEOREM 2. Let Then the K-Transform of the generalized Lommel-Wright function defined in (7) is given by

Proof. On using (7) in the integrand of (2) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (13) in the above equation we get

THEOREM 3. Then the K-Transform of the generalized Lommel-Wright function defined in (7) is given by

Proof. On using (7) in the integrand of (3) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (14) in the above equation we get

THEOREM 4. Then the product of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

Proof. Putting az = x, adz = dx as z → 0, x → 0 and z → +∞, x → +∞ and using (7) in the integrand of (4) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (15) in the above equation we get

THEOREM 5. Let Then the product of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

Proof. Putting az = x, adz = dx as z → 0, x → 0 and z → +∞, x → +∞ and using (7) in the integrand of (5) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (16) in the above equation we get

THEOREM 6. Let Then the product of the Whittaker function and the generalized Lommel-Wright function defined in (7) is given by

Proof. Putting az = x, adz = dx as z → 0, x → 0 and z → +∞, x → +∞ and using (7) in the integrand of (6) which is verified by uniform convergence of the involved series under the given conditions, we get

Now using (17) in the above equation we get

Special cases

In this section, we get some integral formulas involving a trigonometric function and the generalized Lommel-Wright function as follows:

COROLLARY 1. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (1) and then by using (8), we derive the following integral formula:

COROLLARY 2. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (2) and then by using (8), we obtain:

COROLLARY 3. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (3) and then by using (8), we obtain:

COROLLARY 4. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (4) and then by using (8), we obtain:

COROLLARY 5. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (5) and then by using (8), we obtain:

COROLLARY 6. If we take m = 1, µ = 1, λ = 0 and ν = 1/2 in (6) and then by using (8), we obtain:

COROLLARY 7. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (1) and then by using (9), we derive the following integral formula:

COROLLARY 8. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (2) and then by using (9), we obtain:

COROLLARY 9. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (3) and then by using (9), we obtain:

COROLLARY 10. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (4) and then by using (9), we obtain:

COROLLARY 11. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (5) and then by using (9), we obtain:

COROLLARY 12. If we take m = 1, µ = 1, λ = 0 and ν = −1/2 in (6) and then by using (9), we obtain:

COROLLARY 13. If we take m = 1 in (1) and then by using (10), we derive the following integral formula:

COROLLARY 14. If we take m = 1 in (2) and then by using (10), we obtain:

COROLLARY 15. If we take m = 1 in (3) and then by using (10), we obtain:

COROLLARY 16. If we take m = 1 in (4) and then by using (10), we obtain:

COROLLARY 17. If we take m = 1 in (5) and then by using (10), we obtain:

COROLLARY 18. If we take m = 1 in (6) and then by using (10), we obtain:

COROLLARY 19. If we take m = 1, µ = 1 and λ = 1/2 in (1) and then by using (11), we derive the following integral formula:

COROLLARY 20. If we take m = 1, µ = 1 and λ = 1/2 in (2) and then by using (11), we obtain:

COROLLARY 21. If we take m = 1, µ = 1 and λ = 1/2 in (3) and then by using (11), we obtain:

COROLLARY 22. If we take m = 1, µ = 1 and λ = 1/2 in (4) and then by using (11), we obtain:

COROLLARY 23. If we take m = 1, µ = 1 and λ = 1/2 in (5) and then by using (11), we obtain:

COROLLARY 24. If we take m = 1, µ = 1 and λ = 1/2 in (6) and then by using (11), we obtain:

References

Chaurasia, V.B.L. & Pandey, S.C. 2010. On the fractional calculus of generalized Mittag-Leffler function. SCIENTIA Series A: Mathematical Sciences, 20, pp.113-122 [online]. Available at: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.399.5089&rep=rep1&type=pdf [Accessed: 9 February 2022].

Choi, J. & Agarwal, P. 2013. Certain unified integrals associated with Bessel functions. Boundary Value Problems, art.number:95. Available at: https://doi.org/10.1186/1687-2770-2013-95

Choi, J., Mathur, S. & Purohit, S.D. 2014. Certain new integral formulas involving the generalized Bessel functions. Bulletin of the Korean Mathematical Society, 51(4), pp.995-1003. Available at: https://doi.org/10.4134/BKMS.2014.51.4.995.

Jain, S., Choi, J., Agarwal, P. & Nisar, K.S. 2016. Integrals involving Laguerre type plynomials and Bessel functions. Far East Journal of Mathematical Sciences (FJMS), 100(6), pp.965-976. Available at: https://doi.org/10.17654/MS100060965.

Kachhia, K.B. & Prajapati, J.C. 2016. On generalized fractional kinetic equations involving generalized Lommel-Wright functions. Alexandria Engineering Journal, 55(3), pp.2953-2957. Available at: https://doi.org/10.1016/j.aej.2016.04.038.

Kiryakova, V.S. 2000. Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. Journal of Computational and Applied Mathematics, 118(1-2), pp.241-259. Available at: https://doi.org/10.1016/S0377-0427(00)00292-2.

Mathai, A.M., Saxena, R.K. & Haubold, H.J. 2010. The H-function: Theory and Applications. New York, NY: Springer. Aavailable at: https://doi.org/10.1007/978-1-4419-0916-9.

Mathai, A.M. & Saxena, R.K. 1973. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Berlin Heidelberg: Springer- Verlag. Available at: https://doi.org/10.1007/BFb0060468.

Menaria, N., Nisar, K.S. & Purohit, S.D. 2016. On a new class of integrals involving product of generalized Bessel function of the first kind and general class of polynomials. Acta Universitatis Apulensis, 46, pp.97-105 [online]. Available at: https://www.emis.de/journals/AUA/pdf/74_1366_aua_2841701.pdf [Accessed: 9

Mondal, S.R. & Nisar, K.S. 2017. Certain unified integral formulas involving the generalized modified k-Bessel function of first kind. Communications of the Korean Mathematical Society, 32(1), pp.47-53. Available at: https://doi.org/10.4134/CKMS.c160017.

Paneva-Konovska, J. 2007. Theorems on the convergence of series in generalized Lommel-Wright functions. Fractional Calculus and Applied Analysis, 10(1), pp.59-74 [online]. Available at: http://eudml.org/doc/11298 [Accessed: 9 February 2022].

Rainville, E.D. 1960. Special Functions. New York: The Macmillan Company.

Srivastava, H.M. & Daoust, M.C. 1969. Certain generalized Neumann expansions associated with Kampe-de-Feriet function. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Series A-Mathematical Sciences, 72(5), pp.449-457.

Srivastava, H.M. & Manocha, H.L. 1984. A treatise on generating functions. Chichester, West Sussex, England: E. Horwood & New York: Halsted Press. ISBN: 9780853125082.

Whittaker, E.T. & Watson, G.N. 2013. A Course of Modern Analysis, reprint of the fourth (1927) edition. Cambridge University Press, Cambridge Mathematical Library. Available at: https://doi.org/10.1017/CBO9780511608759.

Author notes

nicola.fabiano@gmail.com

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