On a new extensions of extended gamma and beta functions

Introduction

The Gamma and Beta functions are defined respectively, as follows (see) [8]: For each x, y ∈ (0, +∞), the Beta and Gamma functions have the following relation B(x, y) = Γ(x)Γ(y) Γ(x + y) (1.3) In 1903, Gosta Mittag-Leffler [4] introduced the functionE α (z) defined by

The generalized Mittag-Leffler function E α,β (z) [9] is defined by

(α, β ∈ ℂ, Re(α) > 0, Re(β) > 0). (1.5) E α,1 (z) = E α (z).

Recently, many authors have introduced certain extensions of extended Gamma and Beta functions (1.1) and (1.2) (see) [1][2][3][4][5][6][7] .

The following two extensions of Gamma function are introduced by Chaudhry and Zubair [1] and Pucheta [5] respectively:

Chaudhry et al. [2] presented the following extended Beta function:

( Re(p) > 0, Re(x) > 0, Re(y) > 0 ).

In 2014, Choi et al. [3] introduced the following extended Beta function:

( Re(p) > 0, Re(q) > 0, Re(x) > 0, Re(y) > 0 ).

The following extended Beta function is given by Rahman et al. [6] B p,q α (x, y)

) dt, (1.10) (Re(x) > 0, Re(y) > 0, α > 0, p, q ≥ 0).

In the present paper, we introduce a new extensions of Gamma and Beta functions in the following forms.

Re(x) > 0, p ≥ 0, Re(α) > 0, Re(β) > 0 and

) dt, (1.12) (Re(x) > 0, Re(y) > 0, Re(α) > 0, Re(β) > 0, p, q ≥ 0),

Remark 1.

(i) For p = 0, equation (1.11) reduces to the following extended Gamma function:

Remark 2.

(i) Forα = β = 1 and p = q, equation (1.12) reduces to extended Beta function (1.8).

(ii) Forα = β = 1, equation (1.12) reduces to extended Beta function (1.9). (iii) Forβ = 1, equation (1.12) reduces to extended Beta function (1.10).

(iv) Forα = β = 1 and p = q = 0, equation(1.12) reduces to the classical Beta function (1.2).

Properties and formulas

In this section, we present certain properties of extensions of Gamma and Beta functions including summation formulas, integral representations and Mellin transform Theorem 1. The following summation formulas holds true

Proof. From (1.12), we have

Using the generalized binomial theorem

Now, interchanging the order of summation and integration in the above equation and using (1.12), we obtain the required result (2.1). The proof of (2.2) is similar to that of (2.1) and we use here the generalized binomial theorem

Theorem 2. The following summation formula holds true

Proof. For prove of (2.3) we use the mathematical induction on (n ∈ ℕ 0 ) as follows:

Clearly, For n = 0 the equation (2.3) holds.

Forn = 1, we have B p,q (α,β) (x + 1, y) + B p,q (α,β) (x, y + 1)

) dt, = B p,q (α,β) (x, y). (2.4)

Therefore, the equation (2.3) holds for n = 1.

Continuing this process for all (n ∈ ℕ 0 ), we finally obtain the desired relation (2.3).

Remark 3.

(i) Ifα = β = 1 and p = q, then (2. (iv) If α = β = 1 and p = q = 0, then (2.4) reduces to a known result for the classical Beta function.

Theorem 3. For the product of two extended gamma function defined in (1.13) the following formula holds true

(Re(x + u) > 0, Re(y + v) > 0, Re(α) > 0, Re(β) > 0, Re(p), Re(q) > 0, Re(u), Re(v) > 0).

Proof.

Under stated conditions of the above integral, the order of integration can be inter change -ng. Therefore, we have

Finally, using the result (1.12), we obtain the desired result.

Remark 4.

The special case α = β = 1 of (2.5) reduces to a known result of Choi et al. [3, p.360 (2.1)].

Theorem 4. The following integral representations holds true:

Proof. For prove the formula (2.7), putting t = cos 2 θ in (1.12), we have

Similarly, the formulas(2.8) and (2.9) can be proved by taking the transformation t = u 1+u and t = u-a c-a in (1.12), respectively.

Theorem 5. The following Mellin transform holds true: ℳ {B p,q (α,β) (x, y); p ⟶ r, q ⟶ s} = B(x + r, y + s)Γ (α,β) (r)Γ (α,β) (s), (2. 10) Re(x + r) > 0, Re(y + s) > 0, Re(α) > 0, Re(β) > 0, Re(p), Re(q) > 0, Re(r), Re(s) > 0.

Proof. By taking the Mellin transform in both sides of (1.12), we have ℳ {B p,q (α,β) (x, y); p ⟶ r, q ⟶ s}

Under stated conditions of the above integral, the order of integration can be inter-changing. Therefore, we have

Using the substituting u =

Now, by using the definitions (1.2) and (1.13), we obtain the desired result. By taking the inverse Mellin transform on both sides of (2.15), we get the following Form-ula:

Corollary 1. The following formula of extended Beta function holds true B p,q (α,β) (x, y) = 1 (2πi) 2 ∫ ∫ B(x + r, y + s)Γ (α,β) (r)Γ (α,β) (s) γ 2 +i∞ γ 2 -i∞ γ 1 +i∞ γ 1 -i∞ × p -r q -s drds, (2.11) (Re(x) > 0, Re(y) > 0, Re(α) > 0, Re(β) > 0, p, q ≥ 0, γ 1 , γ 2 > 0).

References

1. Chaudhry MA, Zubair SM. Generalized incomplete Gamma functions with App-lications. J Comput. Appl. Math. 1994; 55:99-124.
2. Chaudhry MA, Qadir A, Rafique M, Zubair SM. Extension of Euler's Beta function. J Comput. Appl. Math. 1997; 78:19-32.
3. Choi J, Rathie AK, Parmar RK. Extension of extended Beta, Hypergeom-etric and confluent hypergeometric functions, Honam Mathematical J. 2014; 36(2):357-385.
4. Mittag-Leffler GM. Sur la nouvelle functionE α (z), C. R. Acad. Sci. Paris. 1903; 137:554-558.
5. Pucheta PI. A new extended Beta function, International Journal of Mathematics and its Applications. 2017; 5(3-C):255-260.
6. Rahman G, Kanwal G, Nisar KS, Ghaffar A. A new extension of Beta and hyper-geometric functions. 2018. doi:10.20944/preprints201801.0074.v1.
7. Shadab M, Jabee S, Choi J. An extension of Beta function and its application, Far East Journal of Mathematical Sciences. 2018; 103(1):235-251.
8. Srivastava HM, Manocha HL. A Treatise on Generating Functions, Halsted Press, New York, 1984.
9. Wiman A. Uber den Fundamentals at zin der Teorieder Funktionen E α (z), Acta Math. 1905; 29(1):191-207.