An Extension of Beta Function, Its Statistical Distribution, And

Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 https://doi.org/10.1186/s13662-020-03142-6

R E S E A R C H Open Access An extension of beta function, its statistical distribution, and associated fractional operator Ankita Chandola1, Rupakshi Mishra Pandey1*,RituAgarwal2 and Sunil Dutt Purohit3

*Correspondence: [email protected] Abstract 1Amity Institute of Applied Sciences, Amity University, Uttar Recently, various forms of extended beta function have been proposed and Pradesh, India presented by many researchers. The principal goal of this paper is to present another Full list of author information is expansion of beta function using Appell series and Lauricella function and examine available at the end of the article various properties like integral representation and summation formula. Statistical distribution for the above extension of beta function has been defined, and the mean, variance, moment generating function and cumulative distribution function have been obtained. Using the newly defined extension of beta function, we build up the extension of hypergeometric and confluent hypergeometric functions and discuss their integral representations and differentiation formulas. Further, we define a new extension of Riemann–Liouville fractional operator using Appell series and Lauricella function and derive its various properties using the new extension of beta function. Keywords: Extended beta function; Appell series; Lauricella function; Extended hypergeometric function; Extended confluent hypergeometric function; Statistical distribution; Riemann–Liouville fractional operator

1 Introduction The classical beta function is given by [1, Eq. (16), p. 18] 1 ( )( ) 1–1 2–1 1 2 B(1, 2)= t (1 – t) dt = ,(1) 0 (1 + 2)

where (1), (2)>0, is the real part of the function. The Gauss hypergeometric and conﬂuent hypergeometric functions are deﬁned as [1, Eq. (6), p. 46; Eq. (1), p. 123]

∞ ( ) ( ) zn F ( , ; ; z)= 1 n 2 n ,(2) 2 1 1 2 3 ( ) n! n=0 3 n

where |z| <1,1, 2, 3 ∈ C; 3 =0,–1,–2,...and

∞ ( ) zn ( ; ; z)= 2 n ,(3) 1 1 2 3 ( ) n! n=0 3 n

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where |z| <1,2, 3 ∈ C and 3 =0,–1,–2,...and( γ )n is the Pochhammer symbol de- ﬁned by [1,Eq.(1),p.22;Eq.(3),p.23],forγ =0,–1,–2,..., ⎧ ⎨ n ∈ N k=1 (γ + k –1), n , (γ + n) (γ )n = ⎩ and (γ )n = . 1, n =0 (γ )

The integral representations of hypergeometric and conﬂuent hypergeometric function are [1, Theorem.16, p. 47; Eq. (9), p. 124] ( ) 1 3 2–1 3–2–1 –1 2F1(1, 2; 3; z)= t (1 – t) (1 – zt) dt,(4) (2)(3 – 2) 0

where (3)>(2)>0,| arg (1 – z)| < π,and ( ) 1 3 2–1 3–2–1 zt 11(2; 3; z)= t (1 – t) e dt,(5) (2)(3 – 2) 0

where (3)>(2)>0. Mubeen et al. [2, Eq. (2.1), p. 1552] deﬁned the extended beta function as follows:

α,σ α,σ B (1, 2; p, q)=Bp,q (1, 2) 1 –p –q 1–1 2–1 = t (1 – t) 1F1 α, σ ; 1F1 α, σ ; dt,(6) 0 t (1 – t)

where (1), (2)>0,(p), (q) ≥ 0, α ∈ C, σ =0,–1,–2,.... Goswami et al. [3, Eq. (12), p. 140] deﬁned an extension of beta function as follows: 1 –p q γ1,γ2 η1–1 η2–1 Bp,q (η1, η2)= t (1 – t) 1F1 γ1, γ2; – dt,(7) 0 t (1 – t)

where min{(p), (q)} >0,min{(η1), (η2)} >0,γ1 ∈ C,andγ2 =0,–1,–2,.... In the same paper the authors also defined hypergeometric and confluent hypergeometric functions using the newly defined extended beta function [3, Eqs. (13)–(14) p. 140]:

∞ γ1,γ2 n Bp,q (η + n, η – η ) z Fγ1,γ2 (η , η ; η ; z)= 2 3 2 (η ) ,(8) p,q 1 2 3 B(η , η – η ) 1 n n! n=0 2 3 2

where (p), (q) ≥ 0, (η3)>(η2)>0,γ1 ∈ C,andγ2 =0,–1,–2,...and

∞ γ1,γ2 n Bp,q (η + n, η – η ) z γ1,γ2 (η ; η ; z)= 2 3 2 ,(9) p,q 2 3 B(η , η – η ) n! n=0 2 3 2

where (p), (q) ≥ 0, (η3)>(η2)>0,γ1 ∈ C,andγ2 =0,–1,–2,.... The Appell series introduced by Paul Appell (1880) is the generalization of the Gauss

hypergeometric series 2F1 of one variable to two variables. Appell’s double hypergeometric functions are deﬁned as follows [4, Eqs. (1.4.1)–(1.4.4), p. 23]:

∞ m n (a ) (a ) (a ) u v F a , a , a ; a ; u, v = 1 m+n 2 m 2 n , (10) 1 1 2 2 3 (a ) m!n! m,n=0 3 m+n Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 3 of 17

∞ m n (a ) (a ) (a ) u v F a , a , a ; a , a ; u, v = 1 m+n 2 m 2 n , (11) 2 1 2 2 3 3 (a ) (a ) m!n! m,n=0 3 m 3 n ∞ m n (a ) (a ) (a ) (a ) u v F a , a , a , a ; a ; u, v = 1 m 1 n 2 m 2 n , (12) 3 1 1 2 2 3 (a ) m!n! m,n=0 3 m+n ∞ m n (a ) (a ) u v F a , a ; a , a ; u, v = 1 m+n 2 m+n . (13) 4 1 2 3 3 (a ) (a ) m!n! m,n=0 3 m 3 n

Convergence conditions for the Appell series are as follows:

• F2 converges for |u| + |v| <1; • F and F converge when |u| <1and |v| <1; 1 3 √ √ • F4 converges when | u| + | v| <1. Lauricella(1893) deﬁned the Lauricella functions as follows [4, Eqs. (2.1.1)–(2.1.4), p. 41]:

(n) FA a1, a1,...,an; a1,...,an; u1,...,un ∞ ··· ξ1 ··· mn (a1)ξ1+···+ξn (a1)ξ1 (an)ξn u1 un = ··· ··· , (14) (a1)ξ1 (an)ξn ξ1! ξn! ξ1,...,ξn=0 (n) ··· FB a1 an, a1,...,an; a1; u1,...,un ∞ ··· ··· ξ1 ··· ξn (a1)ξ1 (an)ξn (a1)ξ1 (an)ξn u1 un = ··· , (15) (a1)ξ1+···+ξn ξ1! ξn! ξ1,...,ξn=0 (n) FC a1, a1; a1,...,an; u1,...,un ∞ ξ1 ··· ξn (a1)ξ1+···+ξn (a1)ξ1+···+ξn u1 un = ··· ··· , (16) (a1)ξ1 (an)ξn ξ1! ξn! ξ1,...,ξn=0 (n) FD a1, a1,...,an; a1; u1,...,un ∞ ··· ξ1 ··· ξn (a1)ξ1+···+ξn (a1)ξ1 (an)ξn u1 un = ··· . (17) (a1)ξ1+···+ξn ξ1! ξn! ξ1,...,ξn=0

For the number of variables n = 2, Lauricella functions reduce to Appell series F2, F3, F4,

and F1 respectively, and for n = 1, these functions reduce to the Gauss hypergeometric

function 2F1(·). The Lauricella functions converge as per the following conditions: (n) | | ··· | | • FA converges when u1 + + un <1; • F(n) converges when |u | <1,...,|u | <1; B √1 n√ (n) | | ··· | | • FC converges when u1 + + un <1; (n) | | | | • FD converges when u1 <1,..., un <1. Fractional calculus has many applications in the areas of science and engineering like fluid flow, electrical networks, and many others. The classical Riemann–Liouville fractional integral of order β ∈ C with (β)>0ofa function g is given by [5] y β 1 β–1 Iy g (y)= g(t)(y – t) dt, y > 0. (18) (β) 0 Chandola et al. Advances in Difference Equations (2020)2020:684 Page 4 of 17

The classical Riemann–Liouville fractional derivative of order β ∈ C with (β)<0ofa function g is given by y β 1 –β–1 Dy g (y)= g(t)(y – t) dt, y >0, (β) < 0. (19) (–β) 0

2 Extension of beta function Many authors have studied various extensions and generalizations of beta function and hypergeometric functions (see, e.g., [6–10]). In this section, we have made eﬀorts to deﬁne the extension of beta function using the Appell series and the Lauricella function.

Deﬁnition 2.1 TheextensionsofbetafunctionusingAppellseries(10)–(13), respectively, are deﬁned as follows: 1 p q F1 1–1 2–1 (a) Bp,q( 1, 2)= t (1 – t) F1 a1, a2, a2; a3; r , r dt, (20) 0 t (1 – t)

≥ ≥ where (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0. 1 p q F2 1–1 2–1 (b) Bp,q( 1, 2)= t (1 – t) F2 a1, a2, a2; a3, a3; r , r dt, (21) 0 t (1 – t)

≥ ≥ where (a1), (a2), (a2), (a3), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0. 1 p q F3 1–1 2–1 (c) Bp,q( 1, 2)= t (1 – t) F3 a1, a1, a2, a2; a3; r , r dt, (22) 0 t (1 – t)

≥ ≥ where (a1), (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0. 1 p q BF4 t1–1 t 2–1F a a a a dt (d) p,q( 1, 2)= (1 – ) 4 1, 2; 3, 3; r , r , (23) 0 t (1 – t)

≥ ≥ where (a1), (a2), (a3), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0.

Remark 2.1 When q = 0, Appell’s double hypergeometric functions reduce to the hypergeometric function p q p F a , a , a ; a ; , = F a , a , a ; a , a ; ,0 1 1 2 2 3 tr (1 – t)r 2 1 2 2 3 3 tr p p = F a , a , a , a ; a ; ,0 = F a , a ; a , a ; ,0 3 1 1 2 2 3 tr 4 1 2 3 3 tr p = F a , a ; a ; , 2 1 1 2 3 tr

hence for q = 0 in Deﬁnition 2.1, we obtain the following result: The extension of beta function using hypergeometric series is deﬁned as follows: 1 p B t1–1 t 2–1 F a a a dt p( 1, 2)= (1 – ) 2 1 1, 2; 3; r , (24) 0 t

where (a1), (a2), (a3)>0,(p) ≥ 0, r ≥ 0, (1)>0,and(2)>0. Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 5 of 17

Subsequently, for p =1,r =0,anda1, a2, a3 =1,equations(20)–(23)reducetotheclas-

sical beta function B(1, 2), equation (1).

Deﬁnition 2.2 The extensions of beta function using Lauricella series (14)–(17)respectively, are deﬁned as follows:

(n) FA (a) Bp1,...,pn (1, 2) 1 p p 1–1 2–1 (n) 1 n = t (1 – t) FA a1, a1,...,an; a1,...,an; r ,..., r dt, (25) 0 t t

≥ ≥ where (a1), (a1),..., (an), (a1),..., (an)>0, (p1),..., (pn) 0, r 0, (1)>0, and (2)>0.

(n) FB (b) Bp1,...,pn (1, 2) 1 p p 1–1 2–1 (n) 1 n = t (1 – t) FB a1,...,an, a1,...,an; a1; r ,..., r dt, (26) 0 t t

≥ ≥ where (a1),..., (an), (a1),..., (an), (a1), > 0, (p1)..., (pn) 0, r 0, (1)>0, and (2)>0.

(n) FC (c) Bp1,...,pn (1, 2) 1 p p 1–1 2–1 (n) 1 n = t (1 – t) FC a1, a1; a1,...,an; r ,..., r dt, (27) 0 t t

≥ ≥ where (a1), (a1), (a1),..., (an)>0, (p1),..., (pn) 0, r 0, (1)>0,and (2)>0.

(n) FD (d) Bp1,...,pn (1, 2) 1 p p t1–1 t 2–1F(n) a a a a 1 n dt = (1 – ) D 1, 1,..., n; 1; r ,..., r , (28) 0 t t

≥ ≥ where (a1), (a1),..., (an), (a1)>0, (p1),..., (pn) 0, r 0, (1)>0,and (2)>0.

Remark 2.2 For n = 1, the Lauricella series reduces to the hypergeometric function p p F(n) a , a ,...,a ; a ,...,a ; 1 ;...; n A 1 1 n 1 n tr tr p p = F(n) a ,...,a , a ,...,a ; a ; 1 ;...; n B 1 n 1 n 1 tr tr p p p p = F(n) a , a ; a ,...,a ; 1 ;...; n = F(n) a , a ,...,a ; a ; 1 ;...; n C 1 1 1 n tr tr D 1 1 n 1 tr tr p = F a , a ; a ; 1 , 2 1 1 1 1 tr

and hence, for n = 1 in Deﬁnition 2.2,weobtainresult(24). Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 6 of 17

Subsequently, if p1 =1,r =0,anda1, a1, a1 =1,equations(25)–(28) reduce to the classical beta function B(1, 2), equation (1).

3 Important properties of the extended beta function In this section, recurrence relations and integral representations have been derived for the new extended beta function.

Theorem 3.1 The extension of beta function involving Appell series F1(·) satisﬁes the following:

F1 F1 F1 Bp,q(1, 2)=Bp,q(1 +1,2)+Bp,q(1, 2 + 1). (29)

Proof

F1 F1 RHS = Bp,q(1 +1,2)+Bp,q(1, 2 +1) 1 p q 1 2–1 = t (1 – t) F1 a1, a2, a2; a3; r , r dt 0 t (1 – t) 1 p q t1–1 t 2 F a a a a dt + (1 – ) 1 1, 2, 2; 3; r , r 0 t (1 – t) 1 p q 1–1 2–1 = t (1 – t) F1 a1, a2, a2; a3; r , r dt 0 t (1 – t)

F1 = Bp,q(1, 2)=LHS.

This proves the desired result (29).

Theorem 3.2 The extension of beta function involving Appell series F1(·) satisﬁes the following:

∞ ( ) BF1 ( ,1– )= 2 n BF1 ( + n, 1). (30) p,q 1 2 n! p,q 1 n=0

Proof 1 p q BF1 t1–1 t –2 F a a a a dt LHS = p,q( 1,1– 2)= (1 – ) 1 1, 2, 2; 3; r , r . 0 t (1 – t) ∞ tn Using (1 – t)–2 = ( ) , |t| <1, n! 2 n n=0 ∞ 1 n t p q BF1 ( ,1– )= t1–1 ( ) F a , a , a ; a ; , dt. p,q 1 2 n! 2 n 1 1 2 2 3 tr (1 – t)r 0 n=0

Interchanging the order of integration and summation, we get ∞ 1 ( ) p q BF1 ( ,1– )= 2 n t1+n–1F a , a , a ; a ; , dt p,q 1 2 n! 1 1 2 2 3 tr (1 – t)r n=0 0 Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 7 of 17

∞ ( ) = 2 n BF1 ( + n,1)=RHS. n! p,q 1 n=0

This proves the desired result (30).

Theorem 3.3 The following result for the extension of beta function involving Appell series

F1(·) holds true: ∞ F1 F1 Bp,q(1, 2)= Bp,q(1 + n, 2 + 1). (31) n=0

Proof 1 p q BF1 t1–1 t 2–1F a a a a dt LHS = p,q( 1, 2)= (1 – ) 1 1, 2, 2; 3; r , r . 0 t (1 – t) ∞ Using (1 – t)2–1 =(1–t)2 tn, n=0 1 ∞ p q BF1 ( , )= t1–1(1 – t)2 tnF a , a , a ; a ; , dt. p,q 1 2 1 1 2 2 3 tr (1 – t)r 0 n=0

Interchanging the order of integration and summation, we obtain ∞ 1 p q BF1 ( , )= t1+n–1(1 – t)2 F a , a , a ; a ; , dt p,q 1 2 1 1 2 2 3 tr (1 – t)r n=0 0 ∞ F1 = Bp,q(1 + n, 2 +1)=RHS. n=0

This proves the desired result (31).

Theorem 3.4 The following result for the extension of beta function involving Appell series

F1(·) holds true:

n F1 n F1 Bp,q(d,–d – n)= CsBp,q(d + s,–d – s), (32) s=0

n n! where Cs = s!(n–s)! .

Proof From equation (29), we have

F1 F1 F1 Bp,q(1, 2)=Bp,q(1 +1,2)+Bp,q(1, 2 +1).

Let 1 = d and 2 =–d – n,then

F1 F1 F1 Bp,q(d,–d – n)=Bp,q(d +1,–d – n)+Bp,q(d,–d – n + 1). (33)

Substituting n =1,2,3,...inequation(33), we get

F1 F1 F1 Bp,q(d,–d –1)=Bp,q(d +1,–d –1)+Bp,q(d,–d), Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 8 of 17

F1 F1 F1 F1 Bp,q(d,–d –2)=Bp,q(d +2,–d –2)+2Bp,q(d +1,–d –1)+Bp,q(d,–d),

F1 F1 F1 Bp,q(d,–d –3)=Bp,q(d +3,–d –3)+3Bp,q(d +2,–d –2)

F1 F1 +3Bp,q(d +1,–d –1)+Bp,q(d,–d),

and so on. On generalizing, we get our desired result (32).

Theorem 3.5 (Integral representation) The following integral representations hold true:

π 2 F1 21–1 22–1 (i) Bp,q(1, 2)=2 cos θ sin θ 0 p q × F1 a1, a2, a ; a3; , dθ, (34) 2 cos2r θ sin2r θ ∞ u1–1 F1 (ii) Bp,q(1, 2)= (1 + u)1+2 0 r p(1 + u) × F a , a , a ; a ; , q(1 + u)r du, (35) 1 1 2 2 3 ur 1 F1 1–1–2 1–1 2–1 (iii) Bp,q(1, 2)=2 (1 + u) (1 – u) –1 r r 2 p 2 q × F a , a , a ; a ; , du, (36) 1 1 2 2 3 (1 + u)r (1 – u)r c F1 1–1–2 1–1 2–1 (iv) Bp,q(1, 2)=(c – z) (u – z) (c – u) z r r p(c – z) q(c – z) × F a , a , a ; a ; , du, (37) 1 1 2 2 3 (u – z)r (c – u)r π 4 F1 21–2 22 (v) Bp,q(1, 2)= tanh θ sech θ 0 p q × F1 a1, a2, a ; a3; , dθ, (38) 2 tanh2r θ sech2r θ

≥ ≥ where (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0.

2 u (1+u) (u–z) 2 Proof Substitute t = cos θ, t = (1+u) , t = 2 , t = (c–z) ,andt = tanh θ in equation (20)to get equations (34)–(38), respectively.

F2 F3 F4 Similarly, we can prove the above results for Bp,q(1, 2), Bp,q(1, 2), and Bp,q(1, 2).

4 Statistical distribution involving extended beta function In this section, application of the newly defined extension of beta function in statistics has been discussed. We define the extended beta distribution and derive the results for its mean, variance, moment generating function and cumulative distribution function. Chandola et al. Advances in Difference Equations (2020)2020:684 Page 9 of 17

Deﬁnition 4.1 Distribution of a new extended beta function involving Appell function

F1(·)isdeﬁnedby ⎧ ⎨ 1 1–1 2–1 p q F1 t (1 – t) F1(a1, a2, a2; a3; tr , (1–t)r ), 0 < t <1, B (1,2) f (t)=⎩ p,q (39) 0, otherwise,

≥ ≥ where (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (1)>0,and (2)>0.

If η is any real number, then the mean of the extended beta distribution deﬁned above is given by 1 BF1 ( + η, ) E Y n = Y nf (Y) dt = p,q 1 2 , (40) F1 0 Bp,q(1, 2)

where Y is any random variable. If η = 1, then the mean becomes

BF1 ( +1, ) E(Y)= p,q 1 2 . F1 Bp,q(1, 2)

Variance of the distribution is given by

2 BF1 ( , )BF1 ( +2, )–{BF1 ( +1, )} σ 2 = E Y 2 – E(Y)2 = p,q 1 2 p,q 1 2 p,q 1 2 . (41) F1 2 {Bp,q(1, 2)}

Moment generating function of the distribution is

∞ ∞ tn 1 tn n F1 M(t)= E Y = B (1 + n, 2) . (42) n! F1 p,q n! n=0 Bp,q(1, 2) n=0

Cumulative distribution function is given as

BF1 ( , ) F(y)= y,p,q 1 2 , (43) F1 Bp,q(1, 2)

with y p q F1 1–1 2–1 By,p,q( 1, 2)= t (1 – t) F1 a1, a2, a2; a3; r , r dt. 0 t (1 – t)

F2 F3 F4 Similarly, we can obtain the above results for Bp,q(1, 2), Bp,q(1, 2), and Bp,q(1, 2).

5 Extension of hypergeometric and confluent hypergeometric functions using extended beta function Here, we define a new extension of hypergeometric and confluent hypergeometric functions using the new extension of beta function. Chandola et al. Advances in Difference Equations (2020)2020:684 Page 10 of 17

Deﬁnition 5.1 The extension of hypergeometric function using the newly deﬁned beta

function involving Appell series F1(·)is

∞ F1 n Bp,q( + n, – ) w FF1 ( , ; ; w)= ( ) 2 3 2 , (44) p,q 1 2 3 1 n B( , – ) n! n=0 2 3 2

with (3)>(2)>0and(p), (q) ≥ 0and|w| <1.

Deﬁnition 5.2 The extension of conﬂuent hypergeometric function using the newly de-

ﬁned beta function involving Appell series F1(·)is

∞ F1 n Bp,q( + n, – ) w F1 ( ; ; w)= 2 3 2 , (45) p,q 2 3 B( , – ) n! n=0 2 3 2

with (3)>(2)>0and(p), (q) ≥ 0and|w| <1.

F2 F3 F4 Similarly, we can deﬁne Fp,q(1, 2; 3; w), Fp,q(1, 2; 3; w), Fp,q(1, 2; 3; w), F2 F3 F4 p,q(2; 3; w), p,q(2; 3; w), and p,q(2; 3; w).

Remark 5.1 When q =0andsubsequentlyifp =1,r =0,a1, a2, a3 =1,equations(44)–(45) reduce to the Gauss hypergeometric function (2) and conﬂuent hypergeometric function (3), respectively.

5.1 Integral representation Theorem 5.1 The extended hypergeometric function has the following integral representation: 1 1 FF1 ( , ; ; w)= t2–1(1 – t)3–2–1(1 – tw)–1 p,q 1 2 3 B( , – ) 2 3 2 0 p q × F a , a , a ; a ; , dt, (46) 1 1 2 2 3 tr (1 – t)r

≥ ≥ | | where (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (3)> (2)>0, w <1,and | arg (1 – t)| < π.

Proof By deﬁnition (44),

∞ F1 n Bp,q( + n, – ) w FF1 ( , ; ; w)= ( ) 2 3 2 . p,q 1 2 3 1 n B( , – ) n! n=0 2 3 2

By deﬁnition (20) of the extended beta function

∞ 1 1 FF1 ( , ; ; w)= t2+n–1(1 – t)3–2–1 ( ) p,q 1 2 3 B( , – ) 1 n 2 3 2 0 n=0 n p q w × F a , a , a ; a ; , dt . 1 1 2 2 3 tr (1 – t)r n! ∞ ( ) (tw)n Using 1 n =(1–t)–1 ,wegetthedesiredresult(46). n! n=0 Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 11 of 17

Theorem 5.2 The extended conﬂuent hypergeometric function has the following integral representation:

1 1 F1 ( ; ; w)= t2–1(1 – t)3–2–1 exp (wt) p,q 2 3 B( , – ) 2 3 2 0 p q × F a , a , a ; a ; , dt, (47) 1 1 2 2 3 tr (1 – t)r exp (w) 1 F1 ( ; ; w)= t2–1(1 – t)3–2–1 exp (–wt) p,q 2 3 B( , – ) 2 3 2 0 p q × F a , a , a ; a ; , dt, (48) 1 1 2 2 3 tr (1 – t)r

≥ ≥ | | where (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, (3)> (2)>0,and w <1.

Proof By deﬁnition (45)

∞ F1 n Bp,q( + n, – ) w F1 ( ; ; w)= 2 3 2 . p,q 2 3 B( , – ) n! n=0 2 3 2

By deﬁnition (20) of the extended beta function

∞ 1 1 F1 ( ; ; w)= t2+n–1(1 – t)3–2–1 p,q 2 3 B( , – ) 2 3 2 n=0 0 n p q w × F a , a , a ; a ; , dt . 1 1 2 2 3 tr (1 – t)r n! ∞ (tw)n Using = exp (wt), we get the desired result (47). n! n=0

Replacing t by (1 – t)inequation(47), we get result (48).

Theorem 5.3 The following integral representations for the extended hypergeometric function hold true: ∞ 1 – (i) FF1 ( , ; ; w)= a2–1(1 + a)1–3 1+a(1 – w) 1 p,q 1 2 3 B( , – ) 2 3 2 0 r p(1 + a) × F a , a , a ; a ; , q(1 + a)r da, (49) 1 1 2 2 3 ar π 2 2 F1 22–1 23–22–1 (ii) Fp,q(1, 2; 3; w)= cos θ sin θ B(2, 3 – 2) 0 – × 1–cos2 θw 1 p q × F1 a1, a2, a ; a3; , dθ, (50) 2 cos2r θ sin2r θ Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 12 of 17

21+1–3 1 F1 3–2–1 –1 (iii) Fp,q(1, 2; 3; w)= (1 – a) 2–w(1 + a) B(2, 3 – 2) –1 × (1 + a)2–1 r r 2 p 2 q × F a , a , a ; a ; , da, (51) 1 1 2 2 3 (1 + a)r (1 – a)r (c – u)1+1–3 c F1 2–1 –1 (iv) Fp,q(1, 2; 3; w)= (a – u) (c – u)–w(a – u) B(2, 3 – 2) u × (c – a)3–2–1 r r p(c – u) q(c – u) × F a , a , a ; a ; , da, (52) 1 1 2 2 3 (a – u)r (c – a)r π 1 4 F1 22–2 23–22 (v) Fp,q(1, 2; 3; w)= tanh θ sech θ B(2, 3 – 2) 0 – × 1–tanh2 θw 1 p q × F1 a1, a2, a ; a3; , dθ. (53) 2 tanh2r θ sech2r θ

a 2 1+a a–u 2 Proof Substitute t = 1+a , t = cos θ, t = 2 , t = c–u ,andt = tanh θ in equation (46)toget equations (49)–(53).

Theorem 5.4 The following integral representations for the extended conﬂuent hypergeometric function hold true:

1 ∞ wa F1 2–1 –3 (i) p,q(2; 3; w)= a (1 + a) exp B(2, 3 – 2) 1+a 0 r p(1 + a) × F a , a , a ; a ; , q(1 + a)r da, (54) 1 1 2 2 3 ar π 2 2 (ii) F1 ( ; ; w)= cos22–1 θ sin23–22–1 θ exp w cos2 θ p,q 2 3 B( , – ) 2 3 2 0 p q × F1 a1, a2, a ; a3; , dθ, (55) 2 cos2r θ sin2r θ 21–3 1 w(1 + a) F1 2–1 3–2–1 (iii) p,q(2; 3; w)= (1 + a) (1 – a) exp B(2, 3 – 2) 2 –1 r r 2 p 2 q × F a , a , a ; a ; , da, (56) 1 1 2 2 3 (1 + a)r (1 – a)r (c – u)1–3 c w(a – u) F1 2–1 3–2–1 (iv) p,q(2; 3; w)= (a – u) (c – a) exp B(2, 3 – 2) (c – u) u r r p(c – u) q(c – u) × F a , a , a ; a ; , da, (57) 1 1 2 2 3 (a – u)r (c – a)r π 1 4 (v) F1 ( ; ; w)= tanh22–2 θ sech23–22 θ exp w tanh2 θ p,q 2 3 B( , – ) 2 3 2 0 p q × F1 a1, a2, a ; a3; , dθ. (58) 2 tanh2r θ sech2r θ Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 13 of 17

a 2 1+a a–u 2 Proof Substitute t = 1+a , t = cos θ, t = 2 , t = c–u ,andt = tanh θ in equation (47)toget equations (54)–(58).

5.2 Differentiation formula Theorem 5.5 The following diﬀerentiation formulas for the extended hypergeometric and conﬂuent hypergeometric function hold true:

dn ( ) ( ) FF1 w 1 n 2 n FF1 n n n w n p,q( 1, 2; 3; ) = p,q( 1 + , 2 + ; 3 + ; ) (59) dw (3)n

and

dn ( ) F1 w 2 n F1 n n w n p,q( 2; 3; ) = p,q( 2 + ; 3 + ; ). (60) dw (3)n

Proof Diﬀerentiating equation (44)withrespecttow,weget

∞ F1 n–1 d Bp,q( + n, – ) w FF1 ( , ; ; w) = ( ) 2 3 2 . dw p,q 1 2 3 1 n B( , – ) (n –1)! n=1 2 3 2

Replacing n by n +1

∞ F1 n d Bp,q( + n +1, – ) w FF1 ( , ; ; w) = ( ) 2 3 2 . (61) dw p,q 1 2 3 1 n+1 B( , – ) n! n=0 2 3 2

k Using B(h, k – h)= h B(h +1,k – h)inequation(61), we get

∞ F1 n d Bp,q( + n +1, – ) w FF1 ( , ; ; w) = ( ) 2 2 3 2 (62) dw p,q 1 2 3 1 n+1 B( +1, – ) n! n=0 3 2 3 2 ∞ F1 n d Bp,q( + n +1, – ) w ⇒ FF1 ( , ; ; w) = ( ) 1 2 2 3 2 dw p,q 1 2 3 1 n B( +1, – ) n! n=0 3 2 3 2 1 2 F1 = Fp,q(1 +1,2 +1;3 +1;w). 3

Again diﬀerentiating equation (62)withrespecttow,weget

d2 ( +1) ( +1) FF1 w 1 1 2 2 FF1 w 2 p,q( 1, 2; 3; ) = p,q( 1 +2, 2 +2; 3 +2; ). dw 3(3 +1)

Continuing like this, n times, we get the desired result (59). We can prove result (60)inasimilarway.

F2 F3 Similarly, we can prove the above results for Fp,q(1, 2; 3; w), Fp,q(1, 2; 3; w), F4 F2 F3 F4 Fp,q(1, 2; 3; w), p,q(2; 3; w), p,q(2; 3; w), and p,q(2; 3; w).

6 Extension of Riemann–Liouville fractional operators In this section, we extend the Riemann–Liouville fractional operators using the Appell series and derive its properties using the new extension of beta function. Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 14 of 17

Deﬁnition 6.1 The extended Riemann–Liouville fractional integral is deﬁned as follows: y r r 1 py qy Iβ g y p q g t y t β–1F a a a a dt y ( ): , = ( )( – ) 1 1, 2, 2; 3; r , r , (63) (β) 0 t (y – t)

≥ ≥ where (β)>0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0.

Deﬁnition 6.2 The extended Riemann–Liouville fractional derivative is deﬁned as follows: y r r β 1 –β–1 py qy Dy g(y):p, q = g(t)(y – t) F1 a1, a2, a2; a3; r , r dt, (64) (–β) 0 t (y – t)

≥ ≥ where (β)<0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0.

Similarly, we can deﬁne the extended Riemann–Liouville fractional operators using Ap-

pell series F2(·), F3(·), and F4(·).

≥ ≥ Theorem 6.1 If (β)>0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, then

1 Iβ g(y)=yη : p, q = BF1 (η +1,β)yη+β . (65) y (β) p,q

Proof From Deﬁnition 6.1 y r r β η 1 η β–1 py qy Iy g(y)=y : p, q = t (y – t) F1 a1, a2, a2; a3; r , r dt. (66) (β) 0 t (y – t)

Let t = yu,thenequation(66)becomes 1 β η 1 η+β η β–1 p q Iy g(y)=y : p, q = y u (1 – u) F1 a1, a2, a2; a3; r , r du. (67) (β) 0 u (1 – u)

Using equation (20)in(67), we get our desired result (65).

Theorem 6.2 β ≥ ≥ If ( )>0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, and g(t)= ∞ m | | m=0 bmt , t <1,then

∞ 1 (i) Iβ g(y):p, q = b BF1 (m +1,β)ym+β . (68) y (β) m p,q m=0 ∞ yλ+β–1 (ii) Iβ tλ–1g(y):p, q = b BF1 (m + λ, β)ym. (69) y (β) m p,q m=0

Proof (i) From Deﬁnition 6.1

∞ y r r 1 py qy Iβ g(y):p, q = b tm(y – t)β–1F a , a , a ; a ; , dt. y (β) m 1 1 2 2 3 tr (y – t)r 0 m=0 Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 15 of 17

Interchanging the order of integration and summation, we get

∞ y r r 1 py qy Iβ g(y):p, q = b tm(y – t)β–1F a , a , a ; a ; , dt. (70) y (β) m 1 1 2 2 3 tr (y – t)r m=0 0

Using equation (65)in(70), we get our desired result (68). (ii) From Deﬁnition 6.1,wehave

∞ y r r 1 py qy Iβ tλ–1g(y):p, q = b tm+λ–1(y – t)β–1F a , a , a ; a ; , dt. y (β) m 1 1 2 2 3 tr (y – t)r 0 m=0

Interchanging the order of integration and summation, we get β Iy g(y):p, q ∞ y r r 1 py qy = b tm+λ–1(y – t)β–1F a , a , a ; a ; , dt. (71) (β) m 1 1 2 2 3 tr (y – t)r m=0 0

Using equation (65)in(71), we get our desired result (69).

≥ ≥ Theorem 6.3 If (μ – η)>0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, then

(η) Iμ–η yη–1(1 – y)–β : p, q = yμ–1FF1 (β, η; μ; y). (72) y (μ) p,q

Proof From Deﬁnition 6.1 μ–η η–1 –β Iy y (1 – y) : p, q y r r 1 py qy tη–1 t –β y t μ–η–1F a a a a dt = (1 – ) ( – ) 1 1, 2, 2; 3; r , r . (73) (μ – η) 0 t (y – t)

Let t = yu,thenequation(73)becomes μ–η η–1 –β Iy y (1 – y) : p, q μ–1 1 y η–1 –β μ–η–1 p q = u (1 – yu) (1 – u) F1 a1, a2, a2; a3; r , r du. (74) (μ – η) 0 u (1 – u)

Using equation (46)in(74), we get

yμ–1 Iμ–η yη–1(1 – y)–β : p, q = B(η, μ – η)FF1 (β, η; μ; y). y (μ – η) p,q

On simpliﬁcation, we get our desired result (72).

≥ ≥ Theorem 6.4 If (β)<0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, then

1 Dβ g(y)=yη : p, q = BF1 (η +1,–β)yη–β . (75) y (–β) p,q Chandola et al. Advances in Diﬀerence Equations (2020)2020:684 Page 16 of 17

Proof Using Deﬁnition 6.2 and following the same method as in Theorem 6.1,wegetour desired result (75).

β ≥ ≥ Theorem 6.5 If ( )<0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, and g(t)= ∞ m | | m=0 bmt , t <1,then

∞ 1 (i) Dβ g(y):p, q = b BF1 (m +1,–β)ym–β . (76) y (–β) m p,q m=0 ∞ yλ–β–1 (ii) Dβ tλ–1g(y):p, q = b BF1 (m + λ,–β)ym. (77) y (–β) m p,q m=0

Proof Using Deﬁnition 6.2 and following the same method as in Theorem 6.2,wegetour desired results (76)and(77).

≥ ≥ Theorem 6.6 If (μ – η)<0, (a1), (a2), (a2), (a3)>0, (p), (q) 0, r 0, then

(μ) Dμ–η yμ–1(1 – y)–β : p, q = yη–1FF1 (β, μ; η; y). (78) y (η) p,q

Proof Using Deﬁnition 6.2 and following the same method as in Theorem 6.3,wegetour desired result (78).

Similarly, we can derive the above results for the extended Riemann–Liouville fractional

operators involving Appell series F2(·), F3(·), and F4(·). Also, in a similar way, we can prove all the above properties (from Sects. 3–6)forthe extension of beta function using Lauricella functions (Deﬁnition 2.2).

7Conclusion In this paper, we have discussed some extensions of beta function, Gauss hypergeometric, and confluent hypergeometric function. A new extension of the classical beta function using Appell series and Lauricella function has been obtained, which reduces to the classical beta function for specific values of the parameters. The integral representations and properties of the newly defined beta function have been evaluated. Further, the statistical distribution using the newly defined beta function has been defined and the mean, variance, moment generating function and cumulative distribution function have been discussed here. Thereafter, the extended beta function has been used to define a new extension of hypergeometric and confluent hypergeometric functions and to discuss their properties like integral representations and differentiation formulas. Moreover, extension of the Riemann–Liouville fractional operators using Appell series and Lauricella function has been defined and its various properties have been discussed using the new extension of beta function. All the results obtained here are reducible to a variety of known results involving classical beta function, Gauss hypergeometric, confluent hypergeometric functions, and many others. In the future, the operators of fractional derivatives, fractional integration, and integral transforms can be applied to the extended beta, hypergeometric, and confluent hypergeometric functions, and several image formulas can be established (see, e.g., [11–13]). Chandola et al. Advances in Difference Equations (2020)2020:684 Page 17 of 17

Acknowledgements The authors are thankful to the referees for their valuable suggestion in the improvement of the present work.

Funding Not applicable.

Availability of data and materials Not applicable.

Competing interests The authors declare that they have no competing interests.

Authors’ contributions The authors contributed equally and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Author details 1Amity Institute of Applied Sciences, Amity University, Uttar Pradesh, India. 2Malaviya National Institute of Technology, Jaipur, India. 3Rajasthan Technical University, Kota, India.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 16 July 2020 Accepted: 24 November 2020

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