Generalized Beta-type integral operators
aa r X i v : . [ m a t h . C A ] J un GENERALIZED BETA-TYPE INTEGRAL OPERATORS
MUSHARRAF ALI, MOHD GHAYASUDDIN, AND R. B. PARIS ∗ Abstract.
This research note deals with the evaluation of some general-ized beta-type integral operators involving the multi-index Mittag-Lefflerfunction E ǫ i ) , ( ω i ) ( z ). Further, we derive a new family of beta-type inte-grals involving the product of a multi-index Mittag-Leffler function and agenerating function of two variables. Some concluding remarks regardingour present investigation are briefly discussed in the last section. Keywords:
Beta type integrals, Multi-index Mittag-Leffler function,Generating function, Wright hypergeometric function.
MSC(2010): Introduction
Several basic functions in the applied sciences are defined via improperintegrals, which are predominantly called special functions. Such functionsplay a remarkable role in various diverse fields of engineering and sciences.Therefore, many researchers have presented several extensions and associatedproperties of such type of functions.In recent years, a number of authors have established a list of integralformulas associated with different kinds of special functions (see, for details,[1]–[5], [7]–[12] and the references cited therein). In order to extend the above-mentioned literature, we present in this paper a new class of beta-type integraloperators associated with the multi-index Mittag-Leffler function and a gen-erating function of two variables.Throughout, let N , Z , R , R + and C be the sets of natural numbers, inte-gers, real numbers, positive real numbers and complex numbers, respectively.The generalized Wright hypergeometric function m Ψ n is defined by (see[19], see also [9] )(1.1) m Ψ n ( λ , G ) , · · · , ( λ m , G m );( µ , H ) , · · · , ( µ n , H n ); x = ∞ X k =0 m Q j =1 Γ( λ j + G j k ) n Q j =1 Γ( µ j + H j k ) x k k ! , where the coefficients G j ∈ R + ( j = 1 , . . . , m ) and H j ∈ R + ( j = 1 , . . . , n ) aresuch that(1.2) 1 + n X j =1 H j − m X j =1 G j ≥ . ∗ Corresponding author.
The Mittag-Leffler function E λ ( z ) was introduced by the Swedish math-ematician Gosta Mittag-Leffler [13] and is defined as follows:(1.3) E λ ( z ) = ∞ X k =0 z k Γ(1 + λk ) , where z ∈ C and λ ≥
0. The Mittag-Leffler function is a direct generalizationof the exponential function to which it reduces when λ = 1. Its importance hasbeen realized during the last two decades due to its involvement in problemsof physics, chemistry, biology, engineering and applied sciences. The Mittag-Leffler function occurs naturally as the solution of certain fractional-orderdifferential and integral equations.Wiman [20] introduced the following extension of E λ ( z ):(1.4) E λ,µ ( z ) = ∞ X k =0 z k Γ( µ + λk ) , which is known as the Wiman function. The properties of the Wiman function E λ,µ ( z ) and the Mittag-leffler function E λ ( z ) are very similar. These functionsplay a very important role in the solution of differential equations of fractionalorder.Subsequently, Kiryakova [6] introduced the following new class of multi-index Mittag-Leffler functions:(1.5) E ( ǫi ) , ( ω i ) ( z ) = ∞ X k =0 z k Γ( ω + ǫ k ) · · · Γ( ω l + ǫ l k ) , where l > ǫ , . . . , ǫ l > ω , . . . , ω l are arbitrary real num-bers. Some special cases of (1.5) for l = 2 are given as follows (see [6], [10]): • If ǫ = λ , ǫ = 0, and ω = µ , ω = 1 then (1.5) reduces to (1.4) (and to(1.3) when µ = 1). • If ǫ = ǫ = 1 and z is replaced by z / ω = 1 + ν, ω = 1 : E (1 , , (1+ ν, ( − z /
4) = (cid:18) z (cid:19) − ν J ν ( z ) , (1.7) ω = 3 + µ − ν , ω = 3 + µ + ν E (1 , , ( − ν + µ , ν + µ ) ( − z /
4) = 4 z µ +1 S µ,ν ( z ) , and(1.8) ω = 32 , ω = 32 + ν : E (1 , , ( , ν ) ( − z /
4) = 4 z µ +1 H ν ( z ) , where J ν ( z ), S µ,ν ( z ) and H ν ( z ) are respectively the Bessel function of the firstkind, the Struve function and the Lommel function. ENERALIZED BETA-TYPE INTEGRAL OPERATORS 3 Evaluation of Beta-type integrals
In this section, we present some theorems on the evaluation of the beta-type integral(2.1) J η ,η ,η a ,a ,ǫ i ,ω i [ h ( u ) , h ( u ); q ]= 1 B ( η , η ) Z a a ( u − a ) η − ( a − u ) η − [ h ( u )] η E ( ǫi ) , ( ω i ) [ qh ( u )] du, for some specific functions h ( u ), h ( u ) and q ∈ C , where B ( x, y ) = Γ( x )Γ( y ) / Γ( x + y ) is the Beta function. Theorem 2.1.
The following integral formula holds true: J η ,η , , ,ǫ i ,ω i [(1 − z u ) − β (1 − z u ) − β , u (1 − u ); q ] = 1 B ( η , η ) ∞ X r,s =0 ( β ) r ( β ) s z r z s r ! s !(2.2) × Ψ l +1 ( η + r + s, , ( η , , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( η + η + r + s, q , where ℜ ( η ) > , ℜ ( η ) > and | z | < , | z | < .Proof. We have (see [4, p.962(7)])(2.3) Z u η − (1 − u ) η − (1 − z u ) − β (1 − z u ) − β du = B ( η , η ) × F ( η , β , β ; η + η ; z , z ) , where ℜ ( η ) > , ℜ ( η ) > | z | < , | z | < F is the Appell function[19]; see also [14, p. 413].On setting a = 0 , a = η = 1 , h ( u ) = u (1 − u ) and h ( u ) = (1 − z u ) − β (1 − z u ) − β in (2.1), expanding the multi-index Mittag-Leffler function E ( ǫ i ) , ( ω i ) by its defining series, applying the result (2.3) on the right side andafter some simplification, we obtain our required result (2.2). (cid:3) Theorem 2.2.
The following integral formula holds true: J η ,η , , ,ǫ i ,ω i [(1 − z u ) − β (1 − z (1 − u )) − β , u (1 − u ); q ] = 1 B ( η , η ) ∞ X r,s =0 ( β ) r ( β ) s z r z s r ! s !(2.4) × Ψ l +1 ( η + r, , ( η + s, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( η + η + r + s, q , where ℜ ( η ) > , ℜ ( η ) > and | z | < , | z | < .Proof. We have (see [18, p.279(17)])(2.5) Z u η − (1 − u ) η − (1 − z u ) − β (1 − z (1 − u )) − β du = B ( η , η ) × F ( η , η , β , β ; η + η ; z , z ) , MUSHARRAF ALI, MOHD GHAYASUDDIN, AND R. B. PARIS where ℜ ( η ) > , ℜ ( η ) > | z | < , | z | < F is the Appell function[19]; see also [14, p. 413].On putting a = 0 , a = η = 1 , h ( u ) = u (1 − u ) and h ( u ) = (1 − z u ) − β (1 − z (1 − u )) − β in (2.1), expanding the multi-index Mittag-Lefflerfunction E ( ǫ i ) , ( ω i ) by its defining series, applying the result (2.5) on the rightside and after some simplification, we obtain our required result (2.4). (cid:3) Theorem 2.3.
The following integral formula holds true: J η ,η ,η a ,a ,ǫ i ,ω i [ xu + y, ( u − a )( a − u ); q ] = 1 B ( η , η ) ∞ X r =0 ( − η ) r r ! (cid:26) − ( a − a ) xa x + y (cid:27) r (2.6) × Ψ l +1 ( η + r, , ( η , , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( η + η + r, q , where ℜ ( η ) > , ℜ ( η ) > , (cid:12)(cid:12)(cid:12) arg (cid:16) a x + ya x + y (cid:17)(cid:12)(cid:12)(cid:12) < π and a = a .Proof. We have (see [15, p.263])(2.7) Z a a ( u − a ) η − ( a − u ) η − ( xu + y ) η du = B ( η , η ) × F (cid:18) − η , η ; η + η ; − ( a − a ) xa x + y (cid:19) , where ℜ ( η ) > , ℜ ( η ) > (cid:12)(cid:12)(cid:12) arg (cid:16) a x + ya x + y (cid:17)(cid:12)(cid:12)(cid:12) < π , a = a and F is the Gausshypergeometric function [19]; see also [14, p. 384].On setting h ( u ) = xu + y and h ( u ) = ( u − a )( a − u ) in (2.1), expandingthe multi-index Mittag-Leffler function E ( ǫ i ) , ( ω i ) by its defining series, applyingthe result (2.7) on the right side and after some simplification, we obtain(2.6). (cid:3) Theorem 2.4.
The following integral formula holds true: J η ,η , − ( η + η ) a ,a ,ǫ i ,ω i (cid:20) ( a − a ) + ξ ( u − a ) + σ ( a − u ) , ( u − a )( a − u )[( a − a ) + ξ ( u − a ) + σ ( a − u )] ; q (cid:21) (2.8)= ( ξ + 1) − η ( σ + 1) − η B ( η , η )( a − a ) Ψ l +1 ( η , , ( η , , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( η + η , q ( ξ + 1)( σ + 1) , where ℜ ( η ) > , ℜ ( η ) > , ( a − a ) + ξ ( u − a ) + σ ( a − u ) = 0 and a = a .Proof. We have (see [15, p.261(3.1)])(2.9) Z a a ( u − a ) η − ( a − u ) η − du [( a − a ) + ξ ( u − a ) + σ ( a − u )] η + η = B ( η , η )( ξ + 1) − η ( σ + 1) − η ( a − a ) , ENERALIZED BETA-TYPE INTEGRAL OPERATORS 5 where ℜ ( η ) > , ℜ ( η ) >
0, ( a − a ) + ξ ( u − a ) + σ ( a − u ) = 0 and a = a .On taking h ( u ) = ( a − a ) + ξ ( u − a ) + σ ( a − u ), h ( u ) = ( u − a )( a − u ) /h ( u ) and η = − ( η + η ) in (2.1), expanding the multi-index Mittag-Leffler function E ( ǫ i ) , ( ω i ) by its defining series, applying the result (2.9) on theright side and after some simplification, we obtain our desired result (2.8). (cid:3) Beta-type integrals involving a generating function of twovariables
This section deals with some beta-type integrals involving a generatingfunction of two variables. The generating function of two variables G ( u, t ) isdefined as follows [19]:(3.1) G ( u, t ) = ∞ X r =0 a r g r ( u ) t r , where each member of the generated set { g r ( u ) } ∞ r =0 is independent of t , andthe coefficient set { a r } ∞ r =0 may contain the parameters of the set { g r ( u ) } ∞ r =0 but is independent of u and t . Theorem 3.1.
Let the generating function G ( u, t ) defined by (3.1) and besuch that G { u, ty µ (1 − y ) ν } is uniformly convergent for y ∈ (0 , , µ, ν ≥ and µ + ν > . Then we have (3.2) Z y m − (1 − y ) n − m − G { u, ty µ (1 − y ) ν } E ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r =0 a r g r ( u ) t r Ψ l +1 ( m + µr, , ( n − m + νr, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( n + µr + νr, q , where n > m > and q ∈ C .Proof. On using (3.1) in the left side of (3.2), expanding the multi-indexMittag-Leffler function E ( ǫ i ) , ( ω i ) as its defining series, applying the definitionof beta function and after some simplification, we arrive at (3.2). (cid:3) Corollary 3.2.
On taking n = 2 m and µ = ν in (3.2) , we have (3.3) Z y m − (1 − y ) m − G [ u, t { y (1 − y ) } ν ] E ( ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r =0 a r g r ( u ) t r Ψ l +1 ( m + νr, , ( m + νr, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , (2 m + 2 νr, q . Next, by making use of Theorem 3.1, we derive a few more interestingintegrals in the following examples:
Example 3.1.
Let us consider the generating function of the hypergeometricfunction [19, p. 44(8)] G ( u, t ) = (1 − ut ) − c = ∞ X r =0 ( c ) r ( ut ) r r ! = F ( c ; − ; ut ) ( | ut | < . MUSHARRAF ALI, MOHD GHAYASUDDIN, AND R. B. PARIS
By making use of this generating function (with u = 1) and Theorem 3.1, weobtain(3.4) Z y m − (1 − y ) n − m − [1 − ty µ (1 − y ) ν ] − c E ( ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r =0 c r t r r ! Ψ l +1 ( m + µr, , ( n − m + νr, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( n + µr + νr, q , where n > m > µ, ν ≥ µ + ν > Example 3.2.
Let us consider the generating function [16, p.409(2)](3.5) G ( u, t ) = ∞ X r =0 ( c ) r ( d ) r F ( c ; d + r ; u ) t r r ! = Φ [ c, c ; d ; u, t ] ( | u | , | t |h ∞ ) , where Φ is Humbert’s confluent hypergeometric function [19].Use of this generating function and Theorem 3.1, yields(3.6) Z y m − (1 − y ) n − m − Φ [ c, c ; d ; u, ty µ (1 − y ) ν ] E ( ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r =0 ( c ) r t r ( d ) r r ! F [ c ; d + r ; u ] Ψ l +1 ( m + µr, , ( n − m + νr, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( n + µr + νr, q , where n > m > µ, ν ≥ µ + ν > | u | < ∞ . Example 3.3.
Let us consider the generating function [17, p.276(1)](3.7) G ( u, t ) = (1 − ut + t ) − α = ∞ X r =0 C ( α ) r ( u ) t r , where the C ( α ) r ( u ) are the Gegenbauer, or ultraspherical, polynomials [14,p. 444].By making use of this generating function (with u = 1), Theorem 3.1 and factthat C ( α ) r (1) = (2 α ) r r ! (see [14, Eq.(18.5.9)] ), we find(3.8) Z y m − (1 − y ) n − m − [1 − ty µ (1 − y ) ν ] − α E ( ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r =0 (2 α ) r t r r ! Ψ l +1 ( m + µr, , ( n − m + νr, , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( n + µr + νr, q , where n > m > µ, ν ≥ µ + ν > Remark.
On setting l = 2 with ǫ = λ , ǫ = 0, and ω = ω = 1 in all theintegrals derived in Sections 2 and 3, we easily recover the results of Jabee etal. [5]. ENERALIZED BETA-TYPE INTEGRAL OPERATORS 7 Concluding remarks
In the present research note, we have evaluated a list of generalizedbeta-type integral operators involving the multi-index Mittag-Leffler function E ǫ i ) , ( ω i ) . We have also pointed out some particular cases of our main results.In this section, we briefly consider the following generalizations of Theorem2.1 and Theorem 3.1: Theorem 4.1.
The following integral formula holds true: J η ,η , , ,ǫ i ,ω i " n Y i =1 (1 − z i u ) − β i , u (1 − u ); q = 1 B ( η , η ) ∞ X r , ··· ,r n =0 ( β ) r · · · ( β n ) r n z r · · · z nr n r ! · · · r n !(4.1) × Ψ l +1 ( η + r + · · · + r n , , ( η , , (1 , ω , ǫ ) , · · · ( ω l , ǫ l ) , ( η + η + r + · · · + r n , q , where ℜ ( η ) > , ℜ ( η ) > and max {| z | , · · · , | z n |} < .Proof. We have (see [4, p.965(20)])(4.2) Z u η − (1 − u ) η − n Y i =1 (1 − z i u ) − β i du = B ( η , η ) × F ( n ) D ( η , β , · · · β n ; η + η ; z , · · · z n ) , where ℜ ( η ) > , ℜ ( η ) >
0, max {| z | , · · · | z n |} < F ( n ) D is Lauricella’shypergeometric function of n variables [19, p.60(4)].On setting a = 0 , a = η = 1 , h ( u ) = u (1 − u ) and h ( u ) = n Q i =1 (1 − z i u ) − β i in (2.1), expanding the multi-index Mittag-Leffler function E ( ǫ i ) , ( ω i ) by its defining series, applying the result (4.2) on the right side and after somesimplification, we obtain our required result (4.1). (cid:3) Clearly, for n = 2 Theorem 4.1 immediately reduces to Theorem 2.1. Theorem 4.2.
Let us consider the conditions for G ( u, t ) defined by (3.1) tobe same as in Theorem 3.1. Then we have (4.3) Z y m − (1 − y ) n − m − k Y i =1 (1 − z i y ) − β i G { u, ty µ (1 − y ) ν }× E ( ǫ i ) , ( ω i ) [ qy (1 − y )] dy = ∞ X r,r , ··· ,r k =0 a r g r ( u ) t r ( β ) r · · · ( β k ) r k z r · · · z kr k r ! · · · r k ! × Ψ l +1 ( m + µr + r + · · · + r k , , ( n − m + νr, , ( ω , ǫ ) , · · · ( ω l , ǫ l ) , (1 , n + µr + νr + r + · · · + r k , q MUSHARRAF ALI, MOHD GHAYASUDDIN, AND R. B. PARIS
Proof.
The proof of this theorem is similar to that of Theorem 3.1. Thereforewe omit its proof. (cid:3)
Also, it is noticed that, by using the special cases of the multi-indexMittag-Leffler function E ( ǫ i ) , ( ω i ) (given in ( ?? ), (1.6), (1.7) and (1.8)) in ourmain results derived in this paper, we can establish some new families of beta-type integrals involving the Wiman function, Bessel function, Struve functionand Lommel function. References [1] Ahmed, S and Khan, M.A.: Euler type integral involving generalized Mittag-Lefflerfunction,
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ENERALIZED BETA-TYPE INTEGRAL OPERATORS 9 [20] Wiman, A.: Uber den Fundamentalsatz in der Theorie der Funktionen E α ( x ), ActaMath. , 191–207 (1905).(Musharraf Ali) Department of Mathematics, G.F. College, Shahjahanpur-242001, India
E-mail address : [email protected] (Mohd Ghayasuddin) Department of Mathematics, Faculty of Science, Inte-gral University, Lucknow-226026, India
E-mail address : [email protected] (R. B. Paris) Division of Computing and Mathematics, Abertay University,Dundee DD1 1HG, UK
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