A study on integral transforms of the generalized Lommel-Wright function

Исследование интегральных преобразований обобщенных функций Ломмеля-Райта

Студија о интегралним трансформацијама генерализоване функције Ломела и Рајта

Mohammad Saeed Khana drsaeed9@gmail.com

Sefako Makgatho Health Sciences University, South Africa

Sirazul Haqb sirajulhaq007@gmail.com

J.S.University, India

Moharram Ali Khanc mkhan91@gmail.com

Umaru Musa Yaradua University, Nigeria

Nicola Fabianod nicola.fabiano@gmail.com

University of Belgrade, Serbia

This work is licensed under Creative Commons Attribution 4.0 International.

The transform defined by the following integral equation

(1)

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

(2)

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):

(3)

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

(4)

can be slightly generalized (3) as given below.

(5)

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

(6)

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);

(7)

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:

(8)

(9)

(10)

(11)

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

(12)

(13)

(14)

(15)

(16)

(17)

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.

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

(18)

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

(19)

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

(20)

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

(21)

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

[22a]

[22b]

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

(23)

Now using (14) in the above equation we get

(24)

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

(25a)

(25b)

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

(26)

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

(27a)

(27b)

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

(28)

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

(29)

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

(30a)

(30b)

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:

(31)

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

(32)

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

(33)

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

(34)

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

(35)

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

(36)

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:

(37)

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

(38)

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

(39)

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

(40)

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

(41)

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

(42)

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

(43)

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

(44)

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

(45)

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

(46a)

(46b)

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

(47)

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

(48)

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:

(49)

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

(50a)

(50b)

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

(51)

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

(52)

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

(53)

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

(54a)

(54b)

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

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