Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 687 https://internationalpubls.com Multiple Integrals Involving Pragathi-Satyanarayana’s I-Function, Generalized Gamma and Generalized Hypergeometric Functions B. Satyanarayana1,*D. K. Pavan Kumar2,Y. Pragathi Kumar3 and N.Srimannarayana4 1Department of Mathematics, Acharya Nagarjuna University, Guntur, India. *2Department of Mathematics, Seshadri Rao Gudlavalleru Engineering College, Gudlavalleru, India. 3Department of General Studies, University of the People (Remote), Pasadena, USA. 4Department of Mathematics, KL University, Vaddeswaram, India. Article History: Received: 14-04-2024 Revised: 22-05-2024 Accepted: 04-06-2024 Abstract: This paper evaluates the general multiple integrals involving Pragathi- Satyanarayana’s I-function, generalized gamma function and generalized hypergeometric function. The result is perceived as innovative and possesses the ability to generate the previous findings. Furthermore, a collection of corollaries will be revealed at the end. Keywords: Generalized Gamma function, Pragathi-Satyanarayana’s I-function, Mellin-Barnes integral contour and generalized hyper geometric function. 1. Introduction Saxena [6] defined the I-function as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1, 1, 1 1, , , , , , 1, 1, 1 1 1` , , , 11 2, , , 1 i i i i i i i i n m j j ji ji j j j jn n p j jm n m n s p q r p q r r q pL j j ji jim m q ji ji ji ji j m j n i a A a A a A s b B s I z I z z ds b B b B b B s a A s  + = = + = + = + =     − +  −  = =        − +   −        (1.1) This function is a generalization of the H-function [4]. It converges if: 0, | arg( ) | , 1,..., 2 i iz i r      = (1.2) 0, | arg( ) | 2 i iz      and ( )Re 1 0i +  (1.3) Where 1 1 1 1 1 max i ip qn m i j j ji ji i r j j j n j m A G A G   = = = + = +    = + − +        (1.4) and 1 1 1 1 , 1,2,..., . 2 i iq pm n i i i j j ji ji j j j m j n p q g a g a i r = = = + = +   − = − + − + =        (1.5) Earlier to Saxena, Arjun K Rathie [3] defined a generalized H-function[4], this function was defined as- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 688 https://internationalpubls.com ( ) ( ) ( ) ( ) 1,, , , , 1, , ; 1 2, ; j j j pm n m n s p q p q Lj j j q a A I z I z s z ds b B      = =         (1.6) Where ( ) ( ) ( ) ( ) ( ) 1 1, , 1 1 1 1 j j j j m n B A j j j j j jm n p q p q A B j j j j j n j m b s a s s a s b s     = = = + = +  −  − +  =   −   − + (1.7) The quantities jA and jB are positive reals. The integral (1.6) converges when 1 | arg( ) | 2 z   , ( 0 ) and 1 1 1 1 q pm n j j j j j j j j j j m j j n B B A A    = = + = = +  = − + −    . (1.8) Now we have a new function defined by Pragathi and Satyanarayana [7]. This function is a unification of the above mentioned I – functions. It’s called Pragathi-Satyanarayan’ I-function. We have this integral representation as: ( ) ( ) ( ) ( ) ( ) 1, 1,, , , , , , 1, 1, , ; ; , ; , ; ; , ; i i i i i i j j j ji ji jin n pm n m n p q r p q r j j j ji ji jim m q a A a A z z b B b B       + +    = =       ( ) ( ) ( ) ( ) 1 1 1 1 1 11 2 1 j j i i ji ji n m A B j j j j j j s r q p B AL ji ji ji ji j m j n i a s b s z ds b s a s      = = = + = + =  − +  −     − +   −      (1.9) The general convergence criterion of the above integral is defined in [7]. 2. Required Multiple integrals In this section, we remark two generalized multiple integral formulae. In lemma 1, we use the formula about the generalized Gamma-function. In lemma 2, we have a unified multiple integrals involving the generalized hypergeometric function [1, 5]. Lemma 1. Prudnikov et al. ([2], Ch 3.3.3, 2 page 588) equality is given by 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx         + +          1 1 1 . 1 1 . 1 ,. . ., . . . 1 k n n n k k n n k k k x dx dx       − = =         =     +      1= P kn k k k u a b (2.1) Provided , , ( 1,..., ) 0.k k ka k n  =  Lemma 2. Prudnikov et al. ([2], Ch3.3.5, 14 page 595) equality is given by ( ) 1 1 1 1 0 0 1 1 .... 1 1 ...k kk n n k k k n k k x x x z dx dx    − − − = =   − − =       Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 689 https://internationalpubls.com 1 1 1 1 1 1 1 ,..., , ,..., , ,..., .. ,..., ,..., n n n n n n n n F z              + − −             (2.2) Provided k kRe(υ),Re(α ),Re(β )>0, (k=1, …, n) and ( )| arg 1 |z −  . 3. Main Multiple integrals With reference to above lemmas, we have the generalized integrals as – Theorem 1. Prove that 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 . . .k i i n n m n k k p q r i n i x z x dx dx  − = =     =    1 n k k k a  = . ( ) ( ) ( ) ( ) 1, 1,, , 1, 1, 1, , , ; ; , ; , ; ; , ; , i i i i n j j j ji ji jin n pm n p n q r j j j ji ji ji nm m q A a A a A z b B b B B      ++ + + +           n (3.1) Provided k k k kα , β ,μ ,υ >0, 0)Re(min 1 +  j j j mj k k k k b B     , (k=1, …, n) and 0,  ( )| arg | , 2 z    where 1 1 1 1 1 max i iq pm n j j j j ji ji ji ji i r j j j m j n B A B A      = = = + = +    = + − +        (or) | arg( ) | 2 z  =  with 0 if one of the two conditions cited in [7] be satisfied and 1 1 1 1 1 1 1 1 ; ;1 ,. . . , 1 ; ;1 ; ; , . . . , ; 1 nn n k n n n k n n k n A B             =      = − − = −            (3.2) Proof: To prove the theorem 1, using (1.9), express the Pragathi-Satyanarayana’s I-function in Mellin- Barnes contour integral and swap the order of integration, which is reasonable because of the absolute convergence of the integral involved in the process. Collecting the powers of ( )1,...,kx k n= , this gives I . I = 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 . . .k i i n n m n k k p q r i n i x z x dx dx  − = =     =    1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          1 1 kn k kx  − = ( ) ( ) ( ) ( ) 1 1 1 1 1 11 2 1 j j i i ji ji n m A B j j j j j j r q p B AL ji ji ji ji j m j n i a s b s b s a s      = = = + = + =  − +  −     − +   −      1 1 1 . . . s n i n i z x ds dx dx  =       Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 690 https://internationalpubls.com We obtain I = 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 . . .k i i n n m n k k p q r i n i x z x dx dx  − = =     =    1 n k k k a  = . . ( ) ( ) ( ) ( ) 1 1 1 1 1 11 2 1 j j i i ji ji n m A B j j j j j j s r q p B AL ji ji ji ji j m j n i a s b s z b s a s      = = = + = + =  − +  −     − +   −      . 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          1 1 . . .k k sn k nx dx dx ds  + = Applying the lemma, this gives: I = 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 1 . . .k i i n n m n n k k k p q r i n k i k a x z x dx dx    − = = =     =     ( ) ( ) ( ) ( ) 1 1 1 1 1 11 2 1 j j i i ji ji n m A B j j j j j j s r q p B AL ji ji ji ji j m j n i a s b s z b s a s      = = = + = + =  − +  −     − +   −      1 1 1 . . 1 1 ,. . ., 1 n n n n k k k n k k k k ss a ds s          = = ++           + +      By using the definition of the Pragathi-Satyanarayana’s I-function (1.9) and the generalized Gamma- function, after a few simplifications, we get - 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 1 . . .k i i n n m n n k k k p q r i n k i k a x z x dx dx    − = = =     =     . ( ) ( ) ( ) ( ) 1, 1,, , 1, 1, 1, , , ; ; , ; , ; ; , ; , i i i i n j j j ji ji jin n pm n p n q r j j j ji ji ji nm m q A a A a A z b B b B B      ++ + + +           n Theorem 2. Prove that ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k x x x z x x dx dx       − − − − = = =     − − −            ( ) ( ) ( ) ( ) ( ) 1, 1,, 2 2 , , , 0 1, 1, , , ; ; , ; ! , ; ; , ; , i i i i i n j j j ji ji jin n pn m n nn p n q n r n j j j ji ji ji nm m q A a A a A z z n b B b B B        + + + + = +    =        (3.3) Provided ( ) ( ) ( ) ( ) ( )Re ,Re ,Re ,Re ,Re 0k k k k      ( )1,..., ,k n= ( ) 1 | arg 1 | , 0 min Re j k k j j m j b z B       −   +      and ( ) 1 0 min Re , 1,..., j k k j j m j b B k n       + =     . Also, 0,| arg( ) | , 2 z      where Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 691 https://internationalpubls.com 1 1 1 1 1 max i iq pm n j j j j ji ji ji ji i r j j j m j n B A B A      = = = + = +    = + − +        (or) | arg( ) | 2 z  =  , 0 if one of the two conditions cited by (ii) be satisfied and ( ) ( )Re Rek k  where ( ) ( ) ( ) ( )1 1 1 1 1 11 ; ;1 ,..., 1 ; ;1 , 1 ; ;1 ,..., 1 ; ;1n n n n n n nA n n            = − − − − − + − − + − (3.4) ( ) ( )1 11 ; ;1 ,..., 1 ; ;1n n nB n n    = − − − − (3.5) Proof: To prove the theorem 2, using (1.9), express the Pragathi-Satyanarayana’s I-function in Mellin-Barnes contour integral and swap the order of integration, which is reasonable because of the absolute convergence of the integral involved in the process. Collecting the powers of kx and ( )1 1,...,kx k n− = , this gives I as - ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) 1 1 1 0 0 1 1 .... 1 1 .k kk n n k k k k k x x x z    − − − = =   = − −       ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1 11 1 ... 2 1 j j k kk i i ji ji n m A B nj j j j s sj j ss k k nr q p kB AL ji ji ji ji j m j n i a s b s z x x dsdx dx b s a s        −= = = = + = + =  − +  −  −     − +   −      We can transform the above formula and leads to ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) 1 1 1 1 1 11 . 2 1 j j i i ji ji n m A B j j j j j j s r q p B AL ji ji ji ji j m j n i a s b s z b s a s      = = = + = + =  − +  − =     − +   −      ( ) 1 1 1 1 0 0 1 1 .... 1 1 ...k k k kk k n n s ss k k k n k k x x x z ds dx dx       − − + − −+ = =   − −       Evaluating the inner ( )1,..., nx x − integrals using the lemma, this gives by applying the generalized Gamma- function, we obtain ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) 1 1 1 1 1 11 . 2 1 j j i i ji ji n m A B j j j j j j s r q p B AL ji ji ji ji j m j n i a s b s z b s a s      = = = + = + =  − +  − =     − +   −      Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 692 https://internationalpubls.com 1 1 1 1 1 1 1 1 ,..., , ,..., ,..., n n k k n n n n s s s s s s s s                 + + − + − − + −        + +  . . . 1 1 1 1 1 , ,..., ,..., n n n n n n s s F z ds s s          +  + +    + +  Using the expression of the generalized hypergeometric function [5] in terms of series 0 , n  =  under the hypothesis, by interchanging this series and the integrals, we obtain: ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) ( ) 1 1 0 1 1 1 11 . . ! 2 1 j j i i ji ji n m A B j j j j j jn sn r q p B An L ji ji ji ji j m j n i a s b s z z n b s a s       = = = = + = + =  − +  − =      − +   −       ( ) ( ) 1 1 1 1 1 1 1 1 1 ,..., , ,..., ,..., n n k k n n n k k n k k k n n n s s s s s s s ds s s s                      =  + + − + − − + −  +     +  + +  . . We can change the expression by using the generalized Gamma-function: ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) ( ) 1 1 0 1 1 1 11 . . ! 2 1 j j i i ji ji n m A B j j j j j jn sn r q p B An L ji ji ji ji j m j n i a s b s z z n b s a s       = = = = + = + =  − +  − =      − +   −       . ( ) ( ) ( ) ( ) ( ) 1 1 1 n n n k kk k n k k k k k k k k k k k n ss s s ds s s             = = =  + +    − + −  + + then we can use the known relation ( ) ( ) ( ) n a a a n =  + where ( )Re 0a  , we get: ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) ( ) 1 1 0 1 1 1 11 . . ! 2 1 j j i i ji ji n m A B j j j j j jn sn r q p B An L ji ji ji ji j m j n i a s b s z z n b s a s       = = = = + = + =  − +  − =      − +   −       . ( ) ( ) ( ) 1 1 n n k k k k k k k k k k s n s s ds s n        = =  + +   − + −  + + Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 693 https://internationalpubls.com Interpreting the above Mellin-Barnes integral contour with the help of the definition (1.9), we have the desired relation. Now, we observe the special cases. IV. Special Cases Corollary 1. From Theorem 1, The Pragathi-Satyanarayana’s I-function reduces to I-function defined by Rathie [3], in this situation, we have r = 1 then: 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , 1 1 . . .k n n m n k k p q i n i x I z x dx dx  − = =     =    ( ) ( ) 1, , . 1 , 1 . 1, , , ; . , ; , n j j j p n m nk k p n q k j j j nq A a A a I z b B B    + = + +              n (4.1) Under the conditions and notations verified by the theorem 1 of the above section. nA and nB are defined by the equation (3.2). Corollary 2. From Theorem 1,The Pragathi-Satyanarayana’s I-function reduces to I-function defined by Saxena [6] when 1j j ji jiA B A B= = = = and we have the result: 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , , 1 1 . . .k i i n n m n k k p q r i n i x I z x dx dx  − = =     =    ( ) ( ) ( ) ( ) 1, 1, , 2 . 1 , 1, . 1, 1, , , ; , . , ; , , i i i i n j j ji jin n p n m nk k p n q r k j j ji ji nm m q A a a a I z b b B      + + = + + +              n (4.2) Where 1 1 1 1 1 1 1 1 ; ,. . . , 1 ; ; ; , . . . , nn n k n n n k n n k n A B             =       = − − = −            Corollary 3. Taking r = 1, (4.2) reduces to H-function [4] and we have the result: 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , 1 1 . . .k n n m n k k p q i n i x H z x dx dx  − = =     =    ( ) ( ) 1, , . 1 , 1 . 1, , , . , , n j j p n m nk k p n q k j j nq A a a H z b B    + = + +              n (4.3) under the same conditions verified by the corollary 2. Corollary 4. We suppose that ( ) ( ) 1, 1, 1j jp q A  = = = , then (4.3) leads to Meijer G-function [1], this gives: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 694 https://internationalpubls.com 1 1 1 1 0 0 ..... 1 . . . . n n n n x x xx a a       + +          11 , 1 , 1 1 . . .k n n m n k k p q i n i x G z x dx dx  − = =     =    ( ) ( ) 1, , . 1 , 1 . 1, , . , n j p n m nk k p n q k j nq A a a G z b B  + = + +              n (4.4) Corollary 5. From Theorem 2, The Pragathi-Satyanarayana’s I-function reduces to I-function defined by Rathie [3], in this situation, we have r = 1 then: ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z I x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) 1, , 2 . 2 , . 0 1, , , . ! , , n j j p n m nn p n q n n j j nq A a A z I z n g G B   + + + =      =         n (4.5) Under the conditions and notations verified by the theorem 2 of the above section. nA and nB are defined by the equations (3.4) and (3.5). Corollary 6. From Theorem 2, The Pragathi-Satyanarayana’s I-function reduces to I-function defined by Saxena [6] when 1j j ji jiA B A B= = = = and we have the result: ( ) ( ) 1 1 1 , , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k i i n n n m n k k k p q r k k n k k k I x x x z I x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) ( ) ( ) 1, 1, , 2 . 2 , , . 0 1, 1, , , ; , . ! , ; , , i i i i n j j ji jin n p n m nn p n q n r n j j ji ji nm m q A a a z I z n b b B      +   + + + = +      =         n (4.6) Where, ( ) ( ) ( ) ( )1 1 1 1 1 11 ; ,..., 1 ; , 1 ; ,..., 1 ;n n n n n n nA n n             = − − − − − + − − + − and ( ) ( )1 11 ; ,..., 1 ;n n nB n n     = − − − − . Corollary 7. Taking r = 1, (4.6) reduces to H-function [4] and we have the result: ( ) ( ) 1 1 1 , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k n n n m n k k k p q k k n k k k I x x x z H x x dx dx       − − − − = = =     = − − −            Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 695 https://internationalpubls.com ( ) ( ) ( ) 1, , 2 . 2 , . 0 1, , , . ! , , n j j p n m nn p n q n n j j nq A a z H z n b B      + + + =      =         n (4.7) under the same conditions verified by the corollary 6. Corollary 8. Taking ( ) ( ) 1, 1, 1j jp q A  = = = in (4.7), then Meijer G-function [1] replace the H-function and we have: ( ) ( ) 1 1 1 , , 1 0 0 1 1 1 .... 1 1 1 ...k k k kk k n n n m n k k k p q k k n k k k I x x x z G x x dx dx       − − − − = = =     = − − −            ( ) ( ) ( ) 1, , 2 . 2 , . 0 1, , . ! , n j p n m nn p n q n n j nq A a z G z n b B   + + + =      =         n (4.8) 5. Conclusion In the study of Pragathi-Satyanarayana’s I-function [7] by specializing several parameters as well as variables, obtained like [4], lead to a large number of results concerning remarkably wide variety of useful special functions (or product of such special functions) expressible in terms of I-function defined by Saxena [6], defined by Rathie [3], H-function [4], Meijer’s G-function [1] and hypergeometric function of one variable [1, 5]. The nature of the multiple integrals involving Pragathi-Satyanarayana’s I-function, generalized Gamma function and generalized Hypergeometric function was studied. The theorems developed in this study are quite broad in nature and may be helpful in a number of interesting examples that emerge in literature relating to pure and applied mathematics as well as mathematical physics. References [1] A. A. Kilbas, R. K. Saxena, M. Saigo and J. J. Trujillo, “The generalized hypergeometric function as the Meijer G-function”, Journal Analysis, Vol. 36 no. 1, 1 -14, 2016. [2] A.P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, “Integrals and series: Special functions, Gordon & Breach Science Publishers”, Vol. 3, 589-595, 1986. [3] Arjun K Rathie, “A new generalization of generalized hypergeometric functions”, Le Matematiche, Vol. 52 no. 2,297-310, 1997. [4] H. M. Srivastava, K.C. Gupta and S.P. Goyal, “The H-function of one and two variables with applications”, South Asian Publishers, New Delhi (1982). [5] H. M. Srivastava, M. J. Luo and R. K. Raina, “Extended generalized hypergeometric functions and their applications”, Bulletin of mathematical analysis and applications, Vol. 5 no. 4, 65-77, 2013. [6] V. Jat, V. P. Saxena and P. L. Sanodia, “On certain special cases of existence conditions of I-function”, Jnanabha, Vol. 48 no. 1, 72- 78, (2018). [7] Y. Pragathi Kumar and B. Satyanarayana, “A study of Psi-function, Journal of Informatics and mathematical sciences”, Vol. 12 no. 2, 159-171, 2020.