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 (MittagLeffler, 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 . For various properties of this function with applications, see Prabhakar (Prabhakar, 1971). Further generalization of the Mittag-Leffler function 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.

In continuation of this study, Salim and Faraj (Salim & Faraj, 2012; Nadir et al, 2014) introduced a new generalization of the Mittag-Leffler function which was given as:

Numerous generalizations and cases of the Mittag-Leffler function have been studied and investigated, see for details (Singh & Rawat, 2013; Wright, 1935b; Faraj et al, 2013; Dorrego & Cerutti, 2012;Srivastava & Tomovski, 2009; Saxena et al, 2011; Khan & Ahmed, 2012).

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.

Integrals with the Jacobi polynomials

The Jacobi polynomials (Rainville, 1960; Srivastava & Manocha, 1984) may be defined by

When ϱ = σ = 0, the polynomial in (6) becomes the Legendre polynomial (Rainville, 1960). From (6), it follows that is a polynomial of the degree n and that

Here, we obtain the following integrals.

THEOREM 1. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and ℜ(ξ) > −1, ϱ > −1, σ > −1 then the following integral formula holds true

Proof. Naming the left-hand side (LHS) of (8) as I1 and using the definition (5), we have

interchanging the order of integration and summation which is permissible under the conditions of the theorem, the above expression becomes

Apply the following formula (Saxena, 2008) on (9)

provided that ϱ > −1 and σ > −1, and we get the desired result.

THEOREM 2. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and ℜ(ξ) > −1, ϱ > −1, σ > −1 then the following integral formula holds true

Proof. Denoting the LHS of (11) by I₂ and using definition (5), we get

Now, using (6) in (12), we get

Again using (6) in (13), we attain

but by the formula found in (Rainville, 1960; Srivastava & Manocha, 1984)

using it in (14a, 14b), we get the required result.

THEOREM 3. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and ϱ > −1, σ > −1, then

Proof. Denoting the LHS of (16) by I₃,

Now, using (6) in (17) we have

further, using (15) in (18) we attain the desired result.

THEOREM 4. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and ϱ > −1, σ > −1, then

Proof. Denoting the LHS of (19a, 19b) by I₄,

Now, using (6) in (20) we attain

further, using (15) in (21) we attain the required result.

THEOREM 5. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and ϱ > −1, σ > −1, then

Proof. Denoting the LHS of (22) by I₅,

now using (6) in (23) we attain

further, using (15) in (24) we attain the required result.

Some special cases

If we replace η by ξ − 1 and put ϱ = σ = µ = θ = 0 then the integral I2 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 I3 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 I3 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 + 1; −z) is defined by

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).

THEOREM 6. If p, q > 0 z, υ, ω, ρ, δ, ∈ C ,ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0, ϱ − ϱτ > −1, ϱ > 0, 0 < τ < 1 and ℜ(µ + 1) > 0, then the following integral formula holds true.

Proof. Naming the LHS of (30) as I₉, we obtain

Now we know the formula, see (Saxena, 2008)

provided ℜ(µ) > −1, 0 < τ < 1.

Using (32) in (31), we attain

hence proved.

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 = (z² − 1)^ω/2ν 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

s a solution of (33).

The function 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 −∞.

THEOREM 7. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and θ > 0 and η is a positive integer then

Proof. Denoting the LHS of (37) by I₁₀,

Now the integral in (38) can be solved by using the formula (Erdelyi et al, 1953a)

provided ℜ(θ) > 0, η = 1, 2, 3, . . . .

Now (38) becomes

which is the desired result.

THEOREM 8. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and θ > 0 and η is a positive integer then

Proof. Denoting the LHS of (40a, 40b) by I₁₁,

now the integral in (41) can be solved by using the formula (Erdelyi et al, 1953a)

provided ℜ(θ) > 0, η = 1, 2, 3, ....

Again (41) becomes

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

in which π_n−2(y) is a polynomial of the degree (n − 2) in y.

THEOREM 9. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and h > 0 ℜ(µ) = 0, 1, 2 . . . then

Proof. Denoting the LHS of (9) by I₁₂, we have

now the integral in (47) can be solved by using the formula (Saxena, 2008)

Again (47) becomes

THEOREM 10. If p, q > 0 z, υ, ω, ρ, δ, ∈ C, ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and h > 0 ℜ(µ) = 0, 1, 2... then

Proof. Denoting the LHS of (10a, 10b) by I₁₃, 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.

THEOREM 11. The following integral formula holds true,

where

provided

(1) ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and p, q > 0,

(2) ℜ(ϱ) ≥ 0, ℜ(v) ≥ 0 (both are not zero simultaneously),

(3) ϱ and σ are positive integers such that ϱ + σ ≥ 1.

Proof. Representing the LHS of (11) by I₁₅, 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

(1) ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and p, q > 0,

(2) ℜ(ϱ) ≥ 0, ℜ(v) ≥ 0 (both are not zero simultaneously),

(3) 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

(1) ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and p, q > 0,

(2) ℜ(ϱ) ≥ 0, ℜ(v) ≥ 0 (both are not zero simultaneously),

(3) ϱ and σ are positive integers such that ϱ + σ ≥ 1.

THEOREM 14. The following integral formula holds true,

provided

(1) ℜ(υ) > 0, ℜ(ω) > 0, ℜ(ρ) > 0, ℜ(δ) > 0 and p, q > 0,

(2) ℜ(ϱ) ≥ 0, ℜ(v) ≥ 0 (both are not zero simultaneously),

(3) 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.

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Author notes

nicola.fabiano@gmail.com

Additional information

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

EDITORIAL NOTE: The fourth author of this article, Nicola Fabiano, is a current member of the Editorial Board of the Military Technical Courier. Therefore, the Editorial Team has ensured that the double blind reviewing process was even more transparent and more rigorous. The Team made additional effort to maintain the integrity of the review and to minimize any bias by having another associate editor handle the review procedure independently of the editor – author in a completely transparent process. The Editorial Team has taken special care that the referee did not recognize the author’s identity, thus avoiding the conflict of interest.

КОММЕНТАРИЙ РЕДКОЛЛЕГИИ: Четвертый автор данной статьи Никола Фабиано является действующим членом редколлегии журнала «Военно-технический вестник». Поэтому редколлегия провела более открытое и более строгое двойное слепое рецензирование. Редколлегия приложила дополнительные усилия для того чтобы сохранить целостность рецензирования и свести к минимуму предвзятость, вследствие чего второй редактор-сотрудник управлял процессом рецензирования независимо от редактора-автора, таким образом процесс рецензирования был абсолютно прозрачным. Редколлегия во избежание конфликта интересов позаботилась о том, чтобы рецензент не узнал кто является автором статьи.

РЕДАКЦИЈСКИ КОМЕНТАР: Четврти аутор овог чланка Никола Фабиано је актуелни члан Уређивачког одбора Војнотехничког гласника. Због тога је уредништво спровело транспарентнији и ригорознији двоструко слепи процес рецензије. Уложило је додатни напор да одржи интегритет рецензије и необјективност сведе на најмању могућу меру тако што је други уредник сарадник водио процедуру рецензије независно од уредника аутора, при чему је тај процес био апсолутно транспарентан. Уредништво је посебно водило рачуна да рецензент не препозна ко је написао рад и да не дође до конфликта интереса.

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