Extension of Extended Beta, Hypergeometric and Confluent Hypergeometric Functions

Honam Mathematical J. 36 (2014), No. 2, pp. 357–385 http://dx.doi.org/10.5831/HMJ.2014.36.2.357

EXTENSION OF EXTENDED BETA, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

Junesang Choi∗, Arjun K. Rathie, and Rakesh K. Parmar

Abstract. Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the con- ﬂuent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the above- mentioned extended functions and investigating various formulas for the further extended functions in a systematic manner. More- over, our extension of the Beta function is shown to be applied to Statistics and also our extensions ﬁnd some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

1. Introduction and Preliminaries

We begin by recalling the Gamma function Γ(z) Z ∞ (1.1) Γ(z) := e−t tz−1 dt (<(z) > 0), 0 which was appeared in two letters in 1729 and 1730 from Leonhard Euler (1707–1783) to Christian Goldbach (1690–1764) and developed by Euler who had been eager to extend factorials to values between the integers (see [11], [13, Chapter 1] and [14, Chapter 1]).

Received March 13, 2014. Accepted April 23, 2014. 2010 Mathematics Subject Classiﬁcation. Primary 33B20, 33C20; Secondary 33B15, 33C05. Key words and phrases. Gamma function, Beta function, hypergeometric function, extended Beta function, extended hypergeometric function, extended conﬂuent hypergeometric function, Mellin transform, summation formula. ∗Corresponding author 358 Junesang Choi, A. K. Rathie, R. K. Parmar

The Beta function B(α, β) is a function of two complex variables α and β deﬁned by  Z 1  tα−1(1 − t)β−1 dt (<(α) > 0; <(β) > 0),  (1.2) B(α, β) = 0 Γ(α) Γ(β)  (α, β 6∈ −),  Γ(α + β) Z0

− where Z0 denotes the set of nonpositive integers (see [11], [13, Chapter 1] and [14, Chapter 1]). The integrals in (1.2) and (1.1) are known as the Eulerian integrals of the ﬁrst and second kind, respectively. The classical Gauss’s hypergeometric function (GHF) is deﬁned by

∞ n a, b; X (a)n (b)n z (1.3) F z = = F (a, b; c; z), 2 1 c; (c) n! 2 1 n=0 n where (λ)n (for λ ∈ C) is the Pochhammer symbol deﬁned by (1.4) ( Γ(λ + n) 1 (n = 0) (λ)n := = Γ(λ) λ(λ + 1) ... (λ + n − 1) (n ∈ N := {1, 2, 3,...}).

Euler gave the following integral representation of GHF (see [11], [13, Chapter 1] and [14, Chapter 1]): Z 1 Γ(c) a−1 c−a−1 −b (1.5) 2F1 (a, b; c; z) = t (1 − t) (1 − zt) dt Γ(a) Γ(c − a) 0

(<(c) > <(a) > 0; | arg(1 − z)| ≤ π − (0 < < π)). The conﬂuent hypergeometric function (CHF) is deﬁned by (see [11], [13, Chapter 1] and [14, Chapter 1])

∞ n a; X (a)n z (1.6) F z = = F (a; b; z). 1 1 b; (b) n! 1 1 n=0 n An integral representation of CHF is given as follows (see [11]): (1.7) Z 1 1 a−1 b−a−1 zt 1F1(a; b; z) = t (1−t) e dt (<(b) > <(a) > 0). B(a, b − a) 0 Recently, extensions of some well known special functions have been considered by several authors (see [2, 3, 4, 5, 6, 7, 8]). In 1994, Chaudhry Extensions of Beta, hypergeometric, conﬂuent functions 359 and Zubair [5] introduced the following extension of the Gamma function: Z ∞ x−1 p (1.8) Γp(x) = t exp −t − dt (<(p) > 0). 0 t In 1997, Chaudhry et al. [2] presented the following extension of the Beta function: (1.9) Z 1 p B(x, y ; p) = tx−1 (1 − t)y−1 exp − dt (<(p) > 0), 0 t(1 − t) denoted by EBF, and proved that this extension has certain connections with Macdonald, error, and Whittaker functions. It is obviously seen that the cases p = 0 of (1.8) and (1.9) reduce to the Gamma and Beta functions, respectively. More recently, Chaudhry et al. [3] used B(x, y ; p) to extend the hypergeometric function and the conﬂuent hypergeometric function as follows: ∞ X B(b + n, c − b ; p) zn (1.10) F (a, b; c; z) = (a) p B(b, c − b) n n! n=0 (p = 0, |z| < 1 ; <(c) > <(b) > 0) and ∞ X B(b + n, c − b ; p) zn (1.11) Φ (b; c; z) = p B(b, c − b) n! n=0 (p = 0 ; <(c) > <(b) > 0) . They [3] presented the following integral representations: (1.12) Z 1 1 b−1 c−b−1 −a p Fp (a, b; c; z) = t (1−t) (1−zt) exp − dt B(b, c − b) 0 t(1 − t)

(p > 0; p = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) and (1.13) Z 1 1 b−1 c−b−1 p Φp (b; c; z) = t (1 − t) exp zt − dt B(b, c − b) 0 t(1 − t) (p > 0; p = 0 and <(c) > <(b) > 0) . It is clearly seen that the cases p = 0 of (1.10), (1.11), (1.12), and (1.13) reduce to the usual GHF and CHF. They called Fp (a, b; c; z) and Φp (b; c; z) extended Gauss hypergeometric function (EGHF) and 360 Junesang Choi, A. K. Rathie, R. K. Parmar extended confluent hypergeometric function (ECHF), respectively. They [3] investigated these functions in various ways to give their integral representations, beta distribution, certain properties of differentiation formulas, Mellin transform, transformation formulas, recurrence relations, summation and asymptotic formulas. Here we consider a further extension (denoted by EEBF) of the extended Beta function B(x, y ; p) in (1.9) by introducing an additional parameter q as follows: Z 1 p q B(x, y ; p, q) = tx−1 (1 − t)y−1 exp − − dt (1.14) 0 t 1 − t (<(p) > 0; <(q) > 0). It is easily seen that the cases p = q and p = 0 = q of (1.14) reduce to the extended Beta function (EBF) in (1.9) and the usual Beta function (1.2). First we aim at investigating various integral representations, certain properties, Mellin transforms, and beta distribution of EEBF. Next we generalize extended Gauss hypergeometric function (EGHF) (1.10) and extended confluent hypergeometric function (ECHF) (1.11) by making use of B(x, y ; p, q) and investigate various properties and certain interesting connections with some well known special functions such as Laguerre polynomials, Macdonald and whittaker functions for the further extended functions in the same ways as in [3].

2. Integral Representations of B(x, y ; p, q)

Theorem 1. The following integral formula holds true.

Z ∞ Z ∞ (2.1) pr−1 qs−1 B(x, y ; p, q) dp dq = Γ(r) Γ(s) B(x+r, y +s) 0 0 (<(p) > 0, <(q) > 0, <(r) > 0, <(s) > 0, <(x + r) > 0, <(y + s) > 0) .

Proof. Multiplying each side of (1.14) by pr−1 qs−1 and integrating the resulting identity with respect to p and q (0 5 p, q < ∞), we obtain (2.2) Z ∞ Z ∞ pr−1 qs−1 B(x, y ; p, q) dp dq 0 0 Z ∞ Z ∞ Z 1 p q = pr−1 qs−1 tx−1 (1 − t)y−1 exp − − dt dp dq. 0 0 0 t 1 − t Extensions of Beta, hypergeometric, conﬂuent functions 361

The uniform convergence of the integral in (2.1) guarantees that the order of the triple integrals in (2.2) can be interchanged. We, therefore, have Z ∞ Z ∞ Z 1 pr−1 qs−1 B(x, y ; p, q) dp dq = tx−1 (1 − t)y−1 (2.3) 0 0 0 Z ∞ p Z ∞ q · pr−1 exp − dp · qs−1 exp − dq dt. 0 t 0 1 − t Moreover, the integrals in (2.3) can be simpliﬁed in terms of the Gamma function to prove Theorem 1: Z ∞ Z ∞ pr−1 qs−1 B(x, y ; p, q) dp dq 0 0 Z 1 = Γ(r) Γ(s) tx+r−1 (1 − t)y+s−1 dt = Γ(r) Γ(s) B(x + r, y + s). 0

Remark 1. The special case of (2.1) when r = 1 = s gives an interesting relation between the Beta function and EEBF: (2.4) Z ∞ Z ∞ B(x, y ; p, q) dp dq = B(x+1, y+1) (<(x) > −1, <(y) > −1). 0 0

Theorem 2. The following various integral representations for B(x, y ; p, q) hold true.

(2.5) Z π 2 2x−1 2y−1 h p q i B(x, y ; p, q) = 2 cos θ sin θ exp − 2 − 2 dθ; 0 cos θ sin θ

Z ∞ x−1 −p−q u p (2.6) B(x, y ; p, q) = e x+y exp −qu − du; 0 (1 + u) u

Z 1 B(x, y ; p, q) =21−x−y (1 + u)x−1 (1 − u)y−1 (2.7) −1 2(p + q) + 2(q − p) u · exp − du; 1 − u2 362 Junesang Choi, A. K. Rathie, R. K. Parmar

(2.8) Z c B(x, y ; p, q) =(c − a)1−x−y (u − a)x−1 (c − u)y−1 a c − a · exp − {(q − p) u + (pc − qa)} du (u − a)(c − u)

(<(p) > 0, <(q) > 0; p = 0 = q; <(x) > 0, <(y) > 0) .

Proof. Equations (2.5), (2.6), (2.7) and (2.8) can be obtained by 2 u 1+u u−a employing the transformations t = cos θ, t = 1+u , t = 2 and t = c−a in Equation (1.14), respectively.

Remark 2. The special case p = q of the integrals in Theorem 2 reduces to those corresponding results in [2]. It is also easily seen that the special case p = 0 = q of the integrals in Theorem 2 yields some known integral formulas for the Beta function.

3. Properties of B(x, y ; p, q)

Theorem 3. The following functional relation holds true. (3.1) B(x + 1, y ; p, q) + B(x, y + 1 ; p, q) = B(x, y ; p, q).

Proof. The left-hand side of (3.1) becomes B(x + 1, y ; p, q) + B(x, y + 1 ; p, q) Z 1 p q = tx (1 − t)y−1 + tx−1 (1 − t)y exp − − dt, 0 t 1 − t which, after a simple algebraic manipulation, proves the desired relation.

Remark 3. The special case p = q of (3.1) reduces to the corresponding result in [2]. It is also easily seen that the special case p = 0 = q of the identity in (3.1) yields a known relation for the Beta function.

Theorem 4. The following summation formula holds true. (3.2) ∞ X (y)n B(x, 1 − y ; p, q) = B(x + n, 1 ; p, q)(<(p) > 0, <(q) > 0). n! n=0 Extensions of Beta, hypergeometric, conﬂuent functions 363

Proof. Applying the generalized binomial theorem ∞ X tn (1 − t)−y = (y) (|t| < 1) n n! n=0 to the deﬁnition (1.14) of B(x, y ; p, q), we obtain ∞ Z 1 X tx+n−1 p q B(x, 1 − y ; p, q) = (y) exp − − dt. n n! t 1 − t 0 n=0 Now, interchanging the order of integration and summation in the last expression and using (1.14) proves the desired identity.

Theorem 5. The following inﬁnite summation formula holds true.

∞ X (3.3) B(x, y ; p, q) = B(x+n, y +1 ; p, q)(<(p) > 0, <(q) > 0). n=0

Proof. Replacing (1 − t)y−1 in (1.14) by its series representation: ∞ X (1 − t)y−1 = (1 − t)y tn, n=0 we obtain ∞ Z 1 X p q B(x, y ; p, q) = (1 − t)y tx+n−1 exp − − dt. t 1 − t 0 n=0 Now, interchanging the order of integration and summation in the last expression and using (1.14) proves the desired identity.

Theorem 6. For the product of two extended gamma function, we have the following product formula: ∞ Z 2(x+y)−1 2 p q Γp(x)Γq(y) = 2 r exp(−r )B x, y; , dr (3.4) r2 r2 0 (<(p) > 0, <(q) > 0; F or p = 0 = q, <(x) > 0, <(y) > 0).

Proof. Setting t = η2 and t = ξ2 in (1.8), we get ∞ Z p Γ (x) = 2 η2x−1 exp −η2 − dη p η2 0 364 Junesang Choi, A. K. Rathie, R. K. Parmar and ∞ Z q Γ (y) = 2 ξ2y−1 exp −ξ2 − dξ. q ξ2 0 We, therefore, have ∞ ∞ Z Z p q Γ (x)Γ (y) = 4 η2x−1ξ2y−1 exp −η2 − ξ2 exp − − dηdξ. p q η2 ξ2 0 0 Substituting η = r cos θ and ξ = r sin θ in the above equality, we get

π ∞ Z2 Z 2(x+y)−1 2 Γp(x)Γq(y) = 4 r exp(−r ) 0 0 p q · cos2x−1 θ sin2y−1 θ exp − − drdθ. r2 cos2 θ r2 sin2 θ Interchanging the order of the double integrals on the left-hand side, we have ∞ Z 2(x+y)−1 2 Γp(x)Γq(y) = 2 r exp(−r ) 0  π  2  Z p q  · 2 cos2x−1 θ sin2y−1 θ exp − − dθ dr. 2 2 2 2  r cos θ r sin θ   0  Using Eq.(2.5) of EEBF, we get the desired product formula.

Remark 4. Putting p = 0 = q in Eq. (3.4), we recover the classical relation between the gamma and beta functions given in Eq.(1.2).

4. Mellin Transforms of B(x, y ; p, q)

We begin by providing an identity for B(x, y ; p, q) which looks the ﬁnite Binomial expansion.

Theorem 7. The following identity holds true. n X n (4.1) B(α, −α−n ; p, q) = B(α+k, −α−k ; p, q)(n ∈ ) . k N0 k=0 Extensions of Beta, hypergeometric, conﬂuent functions 365

Proof. By fundamental relation we have B(x + 1, y ; p, q) + B(x, y + 1 ; p, q) = B(x, y ; p, q). Setting x = α and y = −α − n in the formula just obtained, we get B(α, −α − n ; p, q) = B(α, −α − n + 1 ; p, q) + B(α + 1, −α − n ; p, q). Starting with n = 1, we can write this formula recursively to obtain B(α, −α − 1 ; p, q) = B(α, −α ; p, q) + B(α + 1, −α − 1 ; p, q); B(α, −α − 2 ; p, q) = B(α, −α ; p, q) + 2 B(α + 1, −α − 1 ; p, q) + B(α + 2, −α − 2 ; p, q); B(α, −α − 3 ; p, q) = B(α, −α ; p, q) + 3 B(α + 1, −α − 1 ; p, q) + 3 B(α + 2, −α − 2 ; p, q) + B(α + 3, −α − 3 ; p, q) and so on. Now it is seen that the series arises exactly as the ﬁnite binomial expansion does. Therefore we can ﬁnally get the desired identity (4.1). It is also noted that the identity in (4.1) can be proved by the principle of mathematical induction on n. Theorem 8. The following Mellin transformation formula holds true. (4.2) B(x, y ; p, q)

Z γ1+i∞ Z γ2+i∞ 1 Γ(r) Γ(s) Γ(x + r) Γ(y + s) −r −s = 2 p q dr ds (2πi) γ1−i∞ γ2−i∞ Γ(x + y + r + s)

(<(p) > 0, <(q) > 0; γ1 > 0, γ2 > 0) .

Proof. Taking Mellin transform on each side of (1.14), we have M {B(x, y ; p, q); p → r, q → s} Z ∞ Z ∞ Z 1 p q = pr−1 qs−1 tx−1 (1 − t)y−1 exp − − dt dp dq. 0 0 0 t 1 − t Uniform convergence of the integral guarantees to interchange the order of integration: Z 1 M {B(x, y ; p, q); p → r, q → s} = tx−1 (1 − t)y−1 0 Z ∞ p Z ∞ q · pr−1 exp − dp · qs−1 exp − dq dt. 0 t 0 1 − t 366 Junesang Choi, A. K. Rathie, R. K. Parmar

By making use of the deﬁnitions of the Gamma and Beta functions (1.1) and (1.2), we obtain M {B(x, y ; p, q); p → r, q → s} Z 1 = tx−1 (1 − t)y−1 · tr Γ(r) · (1 − t)s Γ(s) dt 0 = Γ(r) Γ(s) B(x + r, y + s) Γ(r) Γ(s) Γ(x + r) Γ(y + s) = . Γ(x + y + r + s) Finally, taking the inverse Mellin transform on the ﬁrst and last sides of the last resulting identity proves (4.2). Theorem 9. The following recurrence type relation holds true. x B(x, y + 1 ; p, q) − y B(x + 1, y ; p, q) (4.3) = q B(x + 1, y − 1 ; p, q) − p B(x − 1, y + 1 ; p, q) (<(p) > 0, <(q) > 0).

Proof. We know that the further extension of the extended beta function (1.14) has the following Mellin transform representation: B(x, y ; p, q) = M {f(t ; y, p, q); x } where p q f(t ; y, p, q) := H(1 − t) (1 − t)y−1 exp − − t 1 − t and 0, t > 1, H(1 − t) := 1, t < 1. It is easy to see that the differentiation of f(t ; y, p, q) with respect to t gives ∂ h f(t ; y, p, q) = − δ(1 − t) (1 − t)y−1 − (y − 1) H(1 − t) (1 − t)y−2 ∂t p q i p q + H(1 − t) (1 − t)y−1 − exp − − , t2 (1 − t)2 t 1 − t d where dt H(1 − t) = −δ(1 − t) and δ denotes the dirac delta δ(1 − t) = δ(t − 1) = 0 for t 6= 0. By using a relationship between the Mellin transform of a function and its derivative: M[f(t); x] = F (x) ⇒ M[f 0(t); x] = −(x − 1) F (x − 1), Extensions of Beta, hypergeometric, confluent functions 367 after a little rearrangement, we find that (x − 1) B(x − 1, y ; p, q) − (y − 1) B(x, y − 1 ; p, q) = q B(x, y − 2 ; p, q) − p B(x − 2, y ; p, q). Replacing x and y by x + 1 and y + 1 in the last resulting identity, respectively, proves the desired identity (4.3).

Remark 5. The special case p = 0 = q of (4.3) yields a known relation (see [11, p. 31, Eq. 5(a)]).

5. Connection with Other Special Functions

Here we give certain relationships between B(x, y ; p, q) and some other special functions.

Theorem 10 (Laguerre polynomials). The following identity holds true. (5.1) ∞ −p−q X B(x, y ; p, q) = e B(x + m + 1, y + n + 1) Lm(p) Ln(q) m,n=0 (<(x) > −1, <(y) > −1),

(0) where the simple Laguerre polynomials Lm(p) = Lm (p)(m ∈ N0) are given by the generating function (see, e.g., [11, p. 202]):

∞ p t X (5.2) (1 − t)−1−α exp − = L(α)(p) tm (|t| < 1). 1 − t m m=0

Proof. Replacing t by 1 − t and setting α = 0 in (5.2), we have

( ∞ ) p X (5.3) exp − = e−p t L (p) (1 − t)m . t m m=0 Replacing t by 1 − t in (5.3) and writing q in place of p, we get

( ∞ ) q X (5.4) exp − = e−q (1 − t) L (q) tm . 1 − t m m=0 368 Junesang Choi, A. K. Rathie, R. K. Parmar

Multiplying (5.3) and (5.4) side by side, we ﬁnd that (5.5)  ∞  p q  X  exp − − = e−p−q L (p) L (q) tm+1 (1 − t)n+1 . t 1 − t m n m, n=0 

Multiplying both sides of (5.5) by tx−1 (1 − t)y−1 and integrating the resulting identity from 0 to 1, we obtain

∞ Z 1 −p−q X x+m y+n B(x, y ; p, q) = e Lm(p) Ln(q) t (1 − t) m,n=0 0 ∞ −p−q X = e B(x + m + 1, y + n + 1) Lm(p) Ln(q). m,n=0

This completes the proof.

By using Equation (4.2) and deﬁnition of Meijer’s G-function of two variables, we can express B(x, y ; p, q) in terms of Meijer’s G-function.

Theorem 11 (Meijer’s G-function). The following identity holds true. (5.6) B(x, y ; p, q) (x + y; 1, 1) : ; = G0,0:2,0;2,0 ; p, q , 1,0:0,2;0,2 : (0, 1), (x, 1) ; (0, 1), (y, 1) where G denotes Meijer’s G-function (see [15, p. 7, Eq. (1.2.3) and p. 88, Eq. (6.4.1)]; see also [10, pp. 843–849]).

Theorem 12 (Appell series). The following identity holds true. (5.7) B(x, y ; p, q) = B(x, y) F2 [1 − x − y, , ; 1 − x, 1 − y ; −p, −q] , where F2 [·] denotes one of the four Appell series Fj (j = 1, 2, 3, 4) (see, e.g., [16, pp. 22-23]).

p q p Proof. By Maclaurin series expansions of exp − t − 1−t = exp − t · q exp − 1−t in (1.14) and interchanging the integral and summations, Extensions of Beta, hypergeometric, conﬂuent functions 369 we have ∞ X (−p)m (−q)n Z 1 B(x, y ; p, q) = tx−m−1 (1 − t)y−n−1 dt m! n! m, n=0 0 ∞ X (−p)m (−q)n Γ(x − m) Γ(y − n) = , m! n! Γ(x + y − m − n) m, n=0 where the deﬁnition of the Beta function (1.2) is used. Now applying the known identity Γ(α − n) (−1)n (5.8) = (α 6= 0, ±1, ±2, ··· ) Γ(α) (1 − α)n to the Gamma functions in the last double series, we obtain ∞ m n X (1 − x − y)m+n (−p) (−q) B(x, y ; p, q) = B(x, y) . (1 − x) · (1 − y) m! n! m, n=0 m n

Now it is easy to see that, in view of the definition of F2 [·] (see, e.g., [16, p. 23, Eq. (3)]), the last double series is identified with the specified F2 [·] in (5.7). This completes the proof.

Theorem 13 (Laguerre polynomial and Whittaker function). The following identity holds true. ∞ x−1 − p −q X m B(x, y ; p, q) = Γ(y + 1) p 2 e 2 p 2 Lm(q) Wα,β(p) (5.9) m=0 (<(p) > 0, <(q) > 0, <(x) > −1, <(y) > −1), 1 1 1 where α := − y + 2 − 2 (x+m) and β := 2 (x+m), Lm(·) is Laguerre polynomial given in (5.1), and Wα,β(·) is Whittaker function (see, e.g., [10, pp. 1014–1018]). Proof. We begin by expressing p q p q exp − − = exp − exp − . t 1 − t t 1 − t Using (5.4), we obtain ( ∞ ) p q p X exp − − = exp − e−q (1 − t) L (q) tm t 1 − t t m m=0 (5.10) ∞ p X = exp − − q (1 − t) L (q) tm. t m m=0 370 Junesang Choi, A. K. Rathie, R. K. Parmar

Using (1.14), we have ∞ Z 1 p X B(x, y ; p, q) = tx−1 (1 − t)y−1 exp − − q (1 − t) L (q) tm dt t m 0 m=0 ∞ X Z 1 p = e−q L (q) tx+m−1 (1 − t)y exp − dt, m t m=0 0 where the interchange of order of integration and summation is guaran- teed by uniform convergence of the involved series. Applying a known integral formula (see [10, p. 362, Entry (3.471)(2)]): Z u ν−1 µ−1 − β x (u − x) e x dx 0 (5.11) ν−1 2µ+ν−1 β β 2 2 = β u exp − Γ(µ) W 1−2µ−ν , ν 2u 2 2 u (<(µ) > 0, <(β) > 0, u > 0) to the last integral, we get the desired representation (5.9). This completes the proof.

Theorem 14 (Macdonald function). The following identity holds true. (5.12) α p 2 √ B(α, −α ; p, q) = 2 e−p−q K (2 pq)(<(p) > 0, <(q) > 0), q α where Kα(·) is the Macdonald function (the modiﬁed Bessel function of the third kind)(see, e.g., [1, pp. 318–320, p. 675]). Also we have −p−q −α (5.13) B(α, −α ; p, q) = 2 e q Γqp(α), where Γqp(α) is the generalized Gamma function (see, e.g., [8, p. 10, Eq. (1.72)]). Proof. Setting x = α and y = −α in Equation (2.6), we get Z ∞ p B(α, −α ; p, q) = e−p−q uα−1 exp −qu − du 0 u which is a special case of the following known result (see [10, p. 363, Eq. (3.471)(9)]): ν Z ∞ 2 ν−1 β β p (5.14) x exp −γ x − dx = 2 Kν(2 βγ) 0 x γ (<(β) > 0, <(ν) > 0). Extensions of Beta, hypergeometric, conﬂuent functions 371

Again, using [8, p. 10, Eq. (1.72)], we get another form (5.13).

Using a result in Theorem 14, we obtain the following interesting special case of (3.4): ∞ √ Z p q 2 pq dr (5.15) Γ (x)Γ (x) = 4 qx exp −r2 − − K . p q r2 r2 x r2 r 0 Indeed, setting y = −x in (3.4), we have ∞ Z p q dr Γ (x)Γ (−x) = 2 exp(−r2) B x, −x; , . p q r2 r2 r 0 Using (5.12) in the form x √ p q − p − q p 2 2 pq B x, −x; , = 2 e r2 r2 K r2 r2 q x r2 and the reﬂection formula [8, p. 13, Eq. (1.88)] −x Γq(−x) = q Γq(x), we are immediately led to the desired result (5.15).

6. The Beta Distribution of B(x, y ; p, q)

It is expected that there will be many applications of B(x, y ; p, q) as in the extended beta function. One application that springs to mind is to statistics. For example, the conventional beta distribution can be extended, by using B(x, y ; p, q), to variables a and b with an inﬁnite range. It appears that such an extension is desirable for the project evaluation and review technique used in some special cases. We deﬁne the beta distribution of B(x, y ; p, q) by (6.1)  1 p q  ta−1 (1 − t)b−1 exp − − (0 < t < 1), f(t) = B(a, b ; p, q) t 1 − t  0, otherwise. If ν is any real number, then we have B(a + ν, b ; p, q) (6.2) E (Xν) = B(a, b ; p, q) (p > 0, q > 0, −∞ < a < ∞, −∞ < b < ∞) . 372 Junesang Choi, A. K. Rathie, R. K. Parmar

The particular case of (6.2) when ν = 1 B(a + 1, b ; p, q) (6.3) µ = E (X) = B(a, b ; p, q) represents the mean of the distribution, and σ2 = E X2 − {E (X)}2 (6.4) B(a, b ; p, q)B(a + 2, b ; p, q) − B2(a + 1, b ; p, q) = B2(a, b ; p, q) is the variance of the distribution. The moment generating function of the distribution is ∞ ∞ X tn 1 X tn (6.5) M(t) = E (Xn) = B(a + n, b ; p, q) . n! B(a, b ; p, q) n! n=0 n=0 The cumulative distribution of (6.1) can be expressed as B (a, b ; p, q) (6.6) F (x) = x , B(a, b ; p, q) where Z x a−1 b−1 p q (6.7) Bx(a, b ; p, q) = t (1 − t) exp − − dt 0 t 1 − t (p > 0, q > 0, −∞ < a < ∞, −∞ < b < ∞) is a new extension of the incomplete beta function.

7. Extension of Gauss and Conﬂuent Hypergeometric Func- tions

In this section we use B(x, y ; p, q) in (1.14) to extend Gauss and confluent hypergeometric functions defined by ∞ X B(b + n, c − b ; p, q) zn (7.1) F (a, b; c; z) = (a) p,q B(b, c − b) n n! n=0 (p = 0, q = 0, |z| < 1 ; <(c) > <(b) > 0) and ∞ X B(b + n, c − b ; p, q) zn (7.2) Φ (b; c; z) = p,q B(b, c − b) n! n=0 (p = 0, q = 0 ; <(c) > <(b) > 0) . Extensions of Beta, hypergeometric, confluent functions 373

We call Fp,q (a, b; c; z) and Φp,q (b; c; z) a further extension of the extended Gauss hypergeometric function in (1.10) (denoted by EEGHF) and a further extension of the extended conﬂuent hypergeometric function in (1.11) (denoted by EECHF), respectively.

8. Integral Representations of EEGHF and EECHF

Theorem 15. The following integral representations hold true. Z 1 1 b−1 c−b−1 Fp,q (a, b; c; z) = t (1 − t) B(b, c − b) 0 (8.1) n p q X (zt)n · exp − − (a) dt t 1 − t n n! n=0

(p > 0, q > 0; p = 0, q = 0 and |z| < 1; <(c) > <(b) > 0) .

1 Z 1 F (a, b; c; z) = tb−1 (1 − t)c−b−1 p,q B(b, c − b) (8.2) 0 p q · (1 − zt)−a exp − − dt t 1 − t

(p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) .

−p−q Z ∞ e b−1 a−c Fp,q (a, b; c; z) = u (1 + u) (8.3) B(b, c − b) 0 p · [1 + u(1 − z)]−a exp − − qu du u (p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) .

π 2b−1 2c−2b−1 2 Z 2 sin v cos v Fp,q (a, b; c; z) = 2 a (8.4) B(b, c − b) 0 1 − z sin v · exp −p sec2 v csc2 v + (p − q) sec2 v dv

(p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) . 374 Junesang Choi, A. K. Rathie, R. K. Parmar

(8.5) 2 Z ∞ sinh2b−1 v cosh2a−2c+1 v Fp,q (a, b; c; z) = 2 2 a B(b, c − b) 0 cosh v − z sinh v · exp −p cosh2 v coth2 v + (p − q) cosh2 v dv (p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) .

1 Φ (b; c; z) = p,q B(b, c − b) (8.6) Z 1 p q · tb−1 (1 − t)c−b−1 exp zt − − dt 0 t 1 − t (p > 0, q > 0; p = 0, q = 0 ; <(c) > <(b) > 0) .

ez Φ (b; c; z) = p,q B(b, c − b) (8.7) Z 1 q p · tc−b−1 (1 − t)b−1 exp −zt − − dt 0 t 1 − t (p > 0, q > 0; p = 0, q = 0 ; <(c) > <(b) > 0) .

Proof. The integral formula in (8.1) is easily obtained by replacing the B(b + n, c − b ; p, q) in (7.1) by the deﬁnition (1.14). Employing the generalized binomial expansion ∞ X (zt)n (1 − zt)−a = (a) n n! n=0 in (8.1), we ﬁnd the integral in (8.2). The integrals in (8.3), (8.4), and (8.5) can be obtained by setting u t = , t = sin2 v, t = tanh2 v 1 + u in (8.2), respectively. A similar argument will establish the integral (8.6) of EECHF in (7.2). The integral in (8.7) is obtained by replacing t by 1 − t in (8.6).

Remark 6. The special case p = q of the integrals in (8.2)-(8.7) leads to the corresponding integral representations given in Chaudhry et al. [3]. The special case p = 0 = q of the integrals in (8.2)-(8.7) is Extensions of Beta, hypergeometric, conﬂuent functions 375 easily seen to reduce to those integral representations of the GHF and CHF (see [12, pp. 20–23]).

9. Diﬀerentiation Formulas of EEGHF and EECHF

Diﬀerentiation formulas for EEGHF and EECHF can be found by dif- ferentiating (7.1) and (7.2) with respect to z together with the following formulas: c B(b, c − b) = B(b + 1, c − b) and (λ) = λ (λ + 1) . b n+1 n

Theorem 16. The following diﬀerentiation formulas hold true. d a b (9.1) F (a, b; c; z) = F (a + 1, b + 1; c + 1; z). dz p,q c p,q

n d (a)n (b)n (9.2) n Fp,q (a, b; c; z) = Fp,q (a + n, b + n; c + n; z) dz (c)n

(n ∈ N0 := N ∪ {0}) .

n d (b)n (9.3) n Φp,q (b; c; z) = Φp,q (b + n; c + n; z)(n ∈ N0) . dz (c)n

Proof. Diﬀerentiating (7.1) with respect to z, we have

∞ d X B(b + n, c − b ; p, q) zn−1 (9.4) F (a, b; c; z) = (a) , dz p,q B(b, c − b) n (n − 1)! n=1 which, upon replacing n by n + 1 and using (8.2), proves (9.1). A repeated application of this process gives the general form (9.2). A similar argument for EECHF proves (9.3).

Remark 7. The special case p = q of (9.2) and (9.3) leads to the corresponding results given in Chaudhry et al. [3]. The special case p = 0 = q of (9.2) and (9.3) is easily seen to yield those corresponding formulas of the GHF and CHF (see [12]). 376 Junesang Choi, A. K. Rathie, R. K. Parmar

10. Mellin Transforms of EEGHF and EECHF

Theorem 17. The following Mellin-Barnes integral representations hold true.

(10.1) 1 Z γ1+i∞ Z γ2+i∞ Fp,q (a, b; c; z) = 2 Γ(r)Γ(s) (2πi) B(b, c − b) γ1−i∞ γ2−i∞ · B(b + r, c + s − b) F (a, b + r; c + r + s; z) p−r q−s dr ds

(γ1 > 0, γ2 > 0) .

(10.2) 1 Z γ1+i∞ Z γ2+i∞ Φp,q (b; c; z) = 2 Γ(r)Γ(s) (2πi) B(b, c − b) γ1−i∞ γ2−i∞ · B(b + r, c + s − b) Φ(b + r; c + r + s; z) p−r q−s dr ds

(γ1 > 0, γ2 > 0) .

Proof. In order to ﬁnd the Mellin transform of EEGHF, we multiply both sides of (8.2) by pr−1 qs−1 and integrate the resulting identity with respect to p from 0 to ∞ and q from 0 to ∞ to yield (10.3) Z ∞ Z ∞ r−1 s−1 M {Fp,q (a, b; c; z): r, s} = p q Fp,q (a, b; c; z) dp dq 0 0 1 Z 1 = tb−1 (1 − t)c−b−1 (1 − zt)−a B(b, c − b) 0 Z ∞ p Z ∞ q · pr−1 exp − dp · qs−1 exp − dq dt. 0 t 0 1 − t By making use of (1.1) and (1.3) in (10.3), we ﬁnd

M {Fp,q (a, b; c; z): r, s} Γ(r) Γ(s) Z 1 = tb+r−1 (1 − t)c+s−b−1 (1 − zt)−a dt (10.4) B(b, c − b) 0 Γ(r) Γ(s) B(b + r, c + s − b) = F (a, b + r ; c + r + s ; z). B(b, c − b) Now taking the inverse Mellin transforms of both sides of (10.4) proves (10.1). A similar argument will establish (10.2). Extensions of Beta, hypergeometric, conﬂuent functions 377

Theorem 18. We have the following diﬀerence formula for EEGHF :

(b − 1)B(b − 1, c − b + 1)Fp,q(a, b − 1; c; z)

= (c − b − 1)B(b, c − b − 1)Fp,q(a, b; c − 1; z)

(10.5) − azB(b, c − b)Fp,q(a + 1, b; c; z)

− pB(b − 2, c − b)Fp,q(a, b − 2; c − 2; z)

+ qB(b, c − b − 2)Fp,q(a, b; c − 2; z).

Proof. We know that B(b, c − b)Fp,q(a, b; c; z) is the Mellin transform of the function p q f (t : z; p, q) = H(1 − t)(1 − t)c−b−1(1 − zt)−a exp − − . a,b,c t 1 − t

Hence B(b, c − b)Fp,q(a, b; c; z) has the Mellin transform representation

B(b, c − b)Fp,q(a, b; c; z) = M{fa,b,c(t : z; p, q): b} .

Diﬀerentiating fa,b,c(t : z; p, q) with respect to t, we get ∂ h (f (t : z; p, q)) = − δ(1 − t)(1 − t)c−b−1(1 − zt)−a ∂t a,b,c + (c − b − 1)H(1 − t)(1 − t)c−b−2(1 − zt)−a − azH(1 − t)(1 − t)c−b−1(1 − zt)−a−1 p −q i p q − H(1 − t)(1 − t)c−b−1(1 − zt)−a + exp − − . t2 (1 − t)2 t 1 − t Since n 0 o M f (t): b = −(b − 1)M{f(t): b − 1} , we get

(b − 1)B(b − 1, c − b + 1)Fp,q(a, b − 1; c; z)

= (c − b − 1)B(b, c − b − 1)Fp,q(a, b; c − 1; z) − azB(b, c − b)Fp,q(a + 1, b; c; z)

− pB(b − 2, c − b)Fp,q(a, b − 2; c − 2; z) + qB(b, c − b − 2)Fp,q(a, b; c − 2; z).

Theorem 19. We have the following diﬀerence formula for EECHF :

(b − 1)B(b − 1, c − b + 1)Φp,q(b − 1; c; z)

= (c − b − 1)B(b, c − b − 1)Φp,q(b; c − 1; z)

(10.6) − zB(b, c − b)Φp,q(b; c; z)

− pB(b − 2, c − b)Φp,q(b − 2; c − 2; z)

+ qB(b, c − b − 2)Φp,q(b; c − 2; z). 378 Junesang Choi, A. K. Rathie, R. K. Parmar

Proof. We know that B(b, c−b)Φp,q(b; c; z) is the Mellin transform of the function p q f (t : z; p, q) = H(1 − t)(1 − t)c−b−1ezt exp − − . b,c t 1 − t

Hence B(b, c − b)Φp,q(b; c; z) has the Mellin transform representation

B(b, c − b)Φp,q(b; c; z) = M{fb,c(t : z; p, q): b} .

Taking derivative of fb,c(t : y; p, q) with respect to t, we get ∂ (f (t : z; p, q)) ∂t b,c h = − δ(1 − t)(1 − t)c−b−1ezt + (c − b − 1)H(1 − t)(1 − t)c−b−2ezt − zH(1 − t)(1 − t)c−b−1ezt p −q i p q − H(1 − t)(1 − t)c−b−1ezt + exp − − . t2 (1 − t)2 t 1 − t Since n 0 o M f (t): b = −(b − 1)M{f(t): b − 1} , we get

(b − 1)B(b − 1, c − b + 1)Φp,q(b − 1; c; z)

= (c − b − 1)B(b, c − b − 1)Φp,qp, q(b; c − 1; z) − zB(b, c − b)Φp,q(b; c; z)

− pB(b − 2, c − b)Φp,q(b − 2; c − 2; z) + qB(b, c − b − 2)Φp,q(b; c − 2; z).

11. Transformation Formulas for EEGHF and EECHF

Theorem 20. The following transformation formulas hold true. z (11.1) F (a, b; c; z) = (1 − z)−a F a, c − b; c; − p,q q,p 1 − z (p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) .

1 (11.2) F a, b; c; 1 − = za F (a, c − b; c; 1 − z) p,q z q,p (p > 0, q > 0; p = 0, q = 0 and | arg z| < π; <(c) > <(b) > 0) .

z (11.3) F a, b; c; = (1 + z)a F (a, c − b; c; −z) p,q 1 + z q,p Extensions of Beta, hypergeometric, conﬂuent functions 379

(p > 0, q > 0; p = 0, q = 0 and | arg(1 + z)| < π; <(c) > <(b) > 0) .

z (11.4) Φp,q (b; c; z) = e Φq,p (c − b; c; −z).

Proof. By writing z −a [1 − z(1 − t)]−a = (1 − z)−a 1 + t 1 − z and replacing t by 1 − t in (8.2), we get −a Z 1 (1 − z) c−b−1 b−1 Fp,q (a, b; c; z) = t (1 − t) B(b, c − b) 0 z −a q p · 1 + t exp − − dt 1 − z t 1 − t (p > 0, q > 0; p = 0, q = 0 and | arg(1 − z)| < π; <(c) > <(b) > 0) , 1 z which proves (11.1). Then, replacing z by 1 − z and 1+z in (11.1) yields (11.2) and (11.3), respectively. Also the formula (11.4) is seen to follow from (8.6) and (8.7).

Remark 8. The special case p = q of (11.1) and (11.4) leads to the corresponding results given in Chaudhry et al. [3]. The special case p = 0 = q of (11.4) is easily seen to yield Kummer’s ﬁrst formula (see [11, p. 125, Theorem 42]).

12. Diﬀerential and Diﬀerence Relations for EEGHF and EECHF

Theorem 21. The following relations hold true. b z (12.1) 4 F (a, b; c; z) = F (a + 1, b + 1; c + 1; z); a p,q c p,q

d a (12.2) F (a, b; c; z) = 4 F (a, b; c; z); dz p,q z a p,q

(12.3) b 4b Φp,q (b; c + 1; z) + c 4c Φp,q (b; c; z) = 0;

d b (12.4) Φ (b; c; z) = Φ (b; c + 1; z) − 4 Φ (b; c; z); dz p,q c p,q c p,q 380 Junesang Choi, A. K. Rathie, R. K. Parmar where 4α denotes the diﬀerence operator deﬁned by

4α f(α, . . .) = f(α + 1,...) − f(α, . . .).

c c (12.5) Φ (b + 1; c + 1; z) = Φ(b; c; z) + 1 − Φ(b; c + 1; z). b b

Proof. It is seen from (8.2) and the difference operator 4a that (12.6) 4a Fp,q (a, b; c; z) = Fp,q (a + 1, b; c; z) − Fp,q (a, b; c; z) z Z 1 p q = tb (1 − t)c−b−1 (1 − zt)−a−1 exp − − dt. B(b, c − b) 0 t 1 − t Replacing the parameters a, b, and c by a + 1, b + 1, and c + 1 in (8.2), respectively, we get (12.7) Z 1 1 b c−b−1 Fp,q (a + 1, b + 1; c + 1; z) = t (1 − t) B(b + 1, c − b) 0 p q · (1 − zt)−a−1 exp − − dt. t 1 − t Now using (9.4) and (12.7) in (12.6) is seen to yield (12.1). Next, using the differentiation formula (9.1) proves (12.2). Replacing b and c by b + 1 and c + 1 in (8.7), respectively, and using (9.4), after a little simplification, we obtain (12.3). Using the differentiation formula (9.3) with n = 1, and considering (12.3) is seen to prove (12.4).

Remark 9. The special case p = q of (12.2) leads to the corresponding result given in Chaudhry et al. [3]. The special case p = 0 = q of (12.3) is easily seen to reduce to (12.5).

13. Summation Formula and Connection with Other Special Functions for EEGHF and EECHF

Gauss found the following useful summation formula (see [12, p. 2]): (13.1) Γ(c) Γ(c − a − b) B(b, c − a − b) F (a, b; c; 1) = = (<(c−a−b) > 0). 2 1 Γ(c − a) Γ(c − a) B(b, c − b) We can also provide a summation formula for EEGHF like (13.1) as in the following theorem: Extensions of Beta, hypergeometric, conﬂuent functions 381

Theorem 22. The following summation formula hold true. B(b, c − a − b; p, q) (13.2) F (a, b; c; 1) = p,q B(b, c − b)

(p > 0, q > 0; p = 0, q = 0 and <(c − a − b) > 0) .

Proof. Setting z = 1 in (8.2) and using (1.14) is seen to yield the summation formula (13.2).

Remark 10. The special case p = q of (13.2) leads to the corresponding result given in Chaudhry et al. [3]. The special case p = 0 = q of (13.2) is easily seen to reduce to the Gauss summation formula (13.1). By letting c = a in (13.2) and using (5.11), we get a connection with the Macdonald function: b √ p 2 K (2 pq) (13.3) F (a, b; a; 1) = 2 e−p−q b . p,q q B(b, c − b)

Theorem 23. For the EEGHF we have the following representation in terms of simple Laguerre and hypergeometric functions: (13.4) ∞ X (b)m+1(c − b)n+1 F (a, b; c; z) = exp(−p − q) p (c) m,n=0 n+m+2

· Lm(p)Ln(q) 2F1(a, b + m + 1; c + n + m + 2; z) (p > 0, q > 0; p = 0, q = 0 and | arg(1 − z) |< π; <(c) > <(b) > 0).

Proof. Applying Eq.(5.5) to Eq.(8.2), we get

1 exp(−p − q) Z F (a, b; c; z) = tb−1(1 − t)c−b−1(1 − zt)−a p,q B(b, c − b) 0 ∞ X m+1 n+1 · Lm(p)Ln(q)t (1 − t) dt. m,n=0 Interchanging the order of integration and summation, which is valid since the series involved is uniformly convergent and the integral involved is absolutely convergent for p > 0, q > 0; p = 0, q = 0 and | arg(1 − z) |< π; <(c) > <(b) > 0, 382 Junesang Choi, A. K. Rathie, R. K. Parmar we have

∞ 1 1 X Z F (a, b; c; z) = L (p)L (q) tm+b(1 − t)n+c−b(1 − zt)−adt. p,q B(b, c − b) m n m,n=0 0

Now using Eq. (1.5) of the Euler’s integral representation of the hypergeometric functions, we have

∞ X B(b + m + 1, c + n + 1 − b) F (a, b; c; z) = p,q B(b, c − b) m,n=0

· Lm(p)Ln(q)2F1(a, b + m + 1; c + n + m + 2; z). Finally, an application of Eq. (1.4) proves the desired representation.

Theorem 24. For the EECHF, we have the following representation in terms of simple Laguerre and conﬂuent hypergeometric functions:

∞ X (b)m+1(c − b)n+1 Φp,q(b; c; z) = exp(−p − q) (c)n+m+2 (13.5) m,n=0 · Lm(p)Ln(q)1F1(b + m + 1; c + n + m + 2; z) (p ≥ 0, q ≥ 0; <(c) > <(b) > 0).

Proof. A similar argument for EECHF in Eq. (8.6) is easily seen to prove our desired result.

Theorem 25. For the EEGHF, we have the following representation in terms of the simple Laguerre polynomials and Whittaker functions: (c − b)Γ(c) √ −p F (a, b; c; z) = ( p)b−1 exp − q p,q Γ(b) 2 ∞ √ n m/2 (13.6) X pz p · (a)nLm(q)W b−1−n−m−2c , n+m+b (p) 2 2 n! m,n=0 (p ≥ 0, q ≥ 0; <(c) > <(b) > 0, | z |< 1).

Proof. Using Eq. (5.11) and the generalized binomial expansion in Eq. (8.2), we have Extensions of Beta, hypergeometric, conﬂuent functions 383

1 exp(−q) Z F (a, b; c; z) = tb−1(1 − t)c−b p,q B(b, c − b) 0 ∞ ∞ X (a)n −p X · zn exp L (q)tmdt. n! t m n=0 m=0 Interchanging the order of summations and integration, we have ∞ exp(−q) X (a)n F (a, b; c; z) = znL (q) p,q B(b, c − b) n! m m,n=0 1 Z −p · tn+m+b−1(1 − t)c−b exp dt. t 0 Finally, using the integral representation given in Eq. (5.11), we get the desired representation.

Theorem 26. For the EECHF, we have the following representation in terms of the simple Laguerre polynomials and Whittaker functions: (c − b)Γ(c) √ −p Φ (b; c; z) = ( p)b−1 exp − q p,q Γ(b) 2 ∞ √ n m/2 (13.7) X pz p · Lm(q)W b−1−n−m−2c , n+m+b (p) 2 2 n! m,n=0 (p ≥ 0, q ≥ 0; <(c) > <(b) > 0) Proof. In a similar manner, applying Eq. (5.11) to Eq. (8.6), we get the desired representation.

References [1] Y. A. Brychkov, Handbook of Special Functions (Derivatives, Integrals, Series and Other Formulas), Taylor & Francis Group, LLC; Chapman & Hall/CRC, 2008. [2] M. A. Chaudhry, A. Qadir, M. Raﬂque, and S. M. Zubair, Extension of Euler’s Beta function, J. Comput. Appl. Math. 78 (1997), 19-32. [3] M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and conﬂuent hypergeometric functions, Appl. Math. Comput. 159 (2004), 589-602. 384 Junesang Choi, A. K. Rathie, R. K. Parmar

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Junesang Choi Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea. E-mail: [email protected]

A. K. Rathie Department of Mathematics, School of Mathematical & Physical Sci- ences, Central University of Kerala, Riverside Transit Campus, Padennakad P.O. Nileshwar, Kasaragod-671 328, India. Extensions of Beta, hypergeometric, conﬂuent functions 385

E-mail: [email protected]

R. K. Parmar Department of Mathematics, Government College of Engineering and Technology, Bikaner-334001, Rajasthan State, India. E-mail: [email protected]