CERTAIN INTEGRALS INVOLVING GENERALIZED MITTAG-LEFFLER TYPE FUNCTIONS

Introduction/purpose: Certain integrals involving the generalized Mittag-Leffler function with different types of polynomials are established. Methods: The properties of the generalized Mittag-Leffler function are used in conjunction with different kinds of polynomials such as Jacobi, Conclusions: The results obtained here are general in nature and could be useful to establish further integral formulae involving other kinds of polynomials.


Introduction
This paper follows the lines of the companion paper (Haq et al, 2019) involving the generalized Galuè-type Struve function in which the same topics are dealt here with the generalized Mittag-Leffler functions. As it is well known, a special function: and its general form are called Mittag-Leffler functions (Erdelyi et al, 1953a), C being the set of complex numbers. The former was established by Mittag-Leffler (Mittag-Leffler, 1903) in connection with his method of summation of some divergent series. Certain properties of this function were studied and investigated. The function defined by (2) appeared for the first time in the work of Wiman (Wiman, 1905). The functions given by equations (1) and (2) are entire functions of order µ = 1 υ and of type σ = 1 (see for example (Erdelyi et al, 1953b)). By means of the series representations, a generalization of the functions defined by equations (1) and (2) is introduced by Prabhakar (Prabhakar, 1971) as: where whenever Γ(ρ) is defined, (ρ) 0 = 1, ρ = 0. It is an entire function of order µ = (1/υ)[ℜ(υ) ℜ(υ) ] −1/υ . For various properties of this function with applications, see Prabhakar (Prabhakar, 1971). Further generalization of the Mittag-Leffler function E ρ υ,ω (z) was considered earlier by Shukla and Prajapati (Shukla & Prajapati, 2007) which is given as: which is the special case when q ∈ (0, 1) and min{ℜ(ω), ℜ(ρ)} > 0.
Integral formulae involving the Mittag-Leffler functions have been developed by many authors, see for example, (Prajapati & Shukla, 2012; Prajapati et al, 2013; Gehlot, 2021; Purohit et al, 2011. In this sequel, here, we aim to establish certain new generalized integral formulae involving the new generalization of the Mittag-Leffler function. The main result presented here is general enough to be specialized to give many interesting integral formulae which are derived as special cases.

Some special cases
If we replace η by ξ − 1 and put ϱ = σ = µ = θ = 0 then the integral I 2 transforms into the following integral involving the Legendre polynomial (Rainville, 1960) If σ = ϱ = 0, µ is replaced by µ − 1 and θ by θ − 1, then the integral I 3 transforms into the following integral involving the Legendre polynomial (Rainville, 1960) If ϱ = σ = 0, µ is replaced by µ − 1 and θ by θ − 1 then the integral I 3 transforms into the following integral involving the Legendre polynomial (Rainville, 1960) Integral with the Bessel Maitland function The special case of the Wright function (Erdelyi et al, 1953b), see also (Wright, 1935a,b) written in the form with complex z, a ∈ C and real A ∈ R. When A = η, a = ν + 1 and z is replaced by −z, then the function ϕ(η, ν and such a function is known as the Bessel Maitland function, or the generalized Bessel function, or the Wright generalized Bessel function, see (Mcbride, 1995).

Integrals with the Legendre functions
The Legendre functions are solution of Legendre's differential equation, see (Erdelyi et al, 1953a) ( where z, ν, ω are unrestricted. Under the subsitution w = (z 2 − 1) ω/2 ν in (5.1) becomes and with λ = 1/2−z/2 as the independent variable, this differential equation becomes This is the Gauss hypergeometric type equation with a = ω − ν, b = ν + ω + 1, c = ω + 1. Hence it follows that the function for | 1 − z |< 2 is a solution of (33). The function P ω ν (z) is known as the Legendre function of the first kind (Erdelyi et al, 1953a). It is one valued and regular on the z−plane, supposed cut along the real axis from 1 to −∞.

Now (38) becomes
which is the desired result.

Integrals with the Hermite polynomials
The Hermite polynomials H n (y), see (Rainville, 1960; Srivastava & Manocha, 1984 may be defined by means of the relation valid for all finite y and t. Since It follows from (43) that The examination of equation (44) shows that H n (y) is a polynomial of degree precisely n in y and that H n (y) = 2 n y n + π n−2 (y) in which π n−2 (y) is a polynomial of the degree (n − 2) in y.
Proof. Denoting the LHS of (9) by I 12 , we have now the integral in (47) can be solved by using the formula (Saxena, 2008) Again (47) becomes Proof. Denoting the LHS of (10) by I 13 , we have using the formula mentioned in (48), then the above expression (50), we get the desired result.

Integrals with the generalized hypergeometric functions
A generalized hypergeometric function (Rainville, 1960) may be defined by in which no denominator parameter σ j is allowed to be zero or a negative integer. If any numerator parameter ϱ i in (51) is zero or a negative integer, the series terminates.
Proof. Representing the LHS of (11) by I 15 , we have putting x = st and dx = tds, then we get The remaining theorems could be proved in a completely analogous fashion.
THEOREM 12. The following integral formula holds true, where f (k) is defined in (53) provided ϱ and σ are positive integers such that ϱ + σ ≥ 1.
THEOREM 13. The following integral formula holds true, where f (k) is defined in (53) provided ϱ and σ are positive integers such that ϱ + σ ≥ 1.

Conclusions
Certain new generalized integral formulae involving the Generalized Mittag-Leffler Type functions with many types of polynomials were established in this study. The results obtained here are general in nature and yield to many interesting formulae which are derived as particular cases.