Integration Involving Certain Products and I-Function Function of Two Variables

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(1)JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org. ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.382-389. Integration Involving Certain Products and I-Function Function of Two Variables S. S. Srivastava and Anshu Singh Department of Mathematics, Govt., P. G. College, Shahdol, M. P., INDIA. (Received on: August 13, 2014) ABSTRACT In this paper, we evaluate some integrals involving the products of I-function of two variables and other hypergeometric functions using E-operator. In section (3.3), we evaluate some integrals involving the products of I-function of two variables and other hypergeometric functions, while in section (3.4), some integrals involving the product of generalized hypergeometric function and I-function of of two variables have been derived by means of finite difference operator E. Keywords: Finite difference operator E, generalized hypergeometric function, double hypergeometric function.. 3.1 INTRODUCTION 3.1.1. Definition of I Function: m, n pi , q i : r. The I-function of one variable introduced by Saxena2, will be represented as follows:. [(aj, αj)1, n], [(aji, αji)n + 1, pi] [(bj, βj)1, m], [(bji, βji)m + 1, qi]. [(aj, αj)1, n], [(aji, αji)n + 1, pi] ] = (1/2πi) ∫ θ(s) xs ds I pm,, qn : r [x| [(b i i j, βj)1, m], [(bji, βji)m + 1, qi] L. m n Π Γ(bj – βjs) ΠΓ(1 – aj + αjs) j=1 j=1 θ (s) = , qi pi r Γ(1 – b + β s) Γ(a – α s) ji ji ji ji Π Π Σ i=1. j=m+1. j=n+1. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411). (1.3.1).

(2) 383. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). integral is convergent, when (B >0, A ≥ 0), where n. pi. m. qi. Β = Σ αj – Σ αji + Σ βj – Σ βji ,. A=. j=1. j=n+1. pi. qi. j=1. (1.3.2). j=m+1. Σ αji – Σ βji ,. j=1. j=1. s = (aj – 1 – v) | αj j = 1, 2, …, n; v = 0, 1, 2, …. lie to the left hand side. |arg x| < ½ Bπ, ∀ i ∈ (1, 2, …, r). pi (i = 1, 2, …, r); qi(i = 1, 2, …, r); m, n are integers satisfying 0 ≤ n ≤ pi, 0 ≤ m ≤ qi, (i = 1, 2, …, r); r is finite αj, βj, αji, βji are real and positive and aj, bj, aji, bji are complex numbers such that αj (bh + v) ≠ Bh (aj – 1 – k), for v, k = 0, 1, 2, ….. h = 1, 2, …, m; j = 1, 2, …, r; L is a contour runs from σ – i∞ to σ + i∞ (σ is real), in the complex s-plane such that the poles of 0, n. I[ ]= I. : m1, n1. :m2,. n2. x. [ |. pi, qi: r : pi´, qi´: r´ :pi´´, qi´: r´´ y. s = (bj + v) | βj j = 1, 2, …, m; v = 0, 1, 2, …. and right of L. The I–function of two variables introduced by Sharma & Mishra9, will be defined and represented as follows:. [(aj; αj, Aj)1, n], [(aji; αji, Aji) n + 1, pi] [(bji; βji, Bji)1, qi]. : [(cj; γj)1, n1], [(cji´; γji´) n1 + 1, pi´ ]; [(ej; Ej)1, n2], [(eji´´; Eji´´) n2 + 1, pi´´] : [(dj; δj)1, m1], [(dji´; δji´) m1 + 1, qi´ ]; [(fj; Fj)1, m2], [(fji´´; Fji´´) m2 + 1, qi´´] = ∫ ∫ φ1(ξ, η) θ2(ξ) θ3(η)xξ yη dξ dη, L1 L2. where φ1 (ξ, η) =. n Γ ( 1 − aj + αjξ + Ajη) Π j=1 r. pi. , qi. Σ [ Π Γ(aji - αjiξ − Αjiη) Π Γ (1 − bji + βjiξ + Bjiη) i=1. θ2 (ξ) =. j = n+ 1. j=1. n1 m1 Π Γ (dj − δjξ) Π Γ (1 − cj + γjξ) j=1 j=1 r´. qi´. Σ [Π i´ = 1. j=m +1 1. ,. pi´. Γ (1 − dji´ + δji´ξ) Π Γ (cji´ − γji´ξ) ] j=n +1 1. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411). (1.3.3).

(3) S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). m2 Π. j=1. θ3 (η) =. r ´´. n2. Γ (fj − Fjη) Π Γ (1 − ej + Ejη). j=1. ,. qi´´. pi´´. Σ [Π i´´ = 1. pi´´. Γ (eji´´ − Eji´´η) ]. Γ (1 − fji´´ + Fji´´η) Π. j=m +1 2. j=n +1 2. x and y are not equal to zero, and an empty product is interpreted as unity pi, pi´, pi´´, qi, qi´, qi´´, n, n1, n2, nj and mk are non negative integers such that pi ≥ n ≥ 0, pi´ ≥ n1 ≥ 0, pi´´ ≥ n2 ≥ 0, qi > 0, qi´ >0, qi´´ > 0, (i = 1, …, r; i´ = 1, …, r´; i´´ = 1, …, r´´; k = 1, 2) also all the A’s, α’s, B’s, β’s, γ’s, δ’s, E’s and F’s are assumed to be positive quantities for standardization purpose; the definition of Ifunction of two variables given above will however, have a meaning even if some of these quantities are zero. The contour L1 is in the ξ−plane and runs from – ω∞ to + ω∞, with loops, if necessary, to ensure that the. pi. 384. qi. poles of Γ(dj−δjξ) (j = 1, ..........., m1) lie to the right, and the poles of Γ (1 − cj + γjξ) (j = 1, ..., n1), Γ ( 1 − aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour. The contour L2 is in the η−plane and runs from – ω∞ to + ω∞, with loops, if necessary, to ensure that the poles of Γ (fj − Fjη) (j=1,....., n2) lie to the right, and the poles of Γ (1 − ej + Ejη) (j = 1, ..., m2), Γ ( 1 − aj + αjξ + Ajη) (j = 1, ..., n) to the left of the contour. Also R = Σ αji + Σ γji´ − Σ βji − Σ δji´ < 0,. qi´´. S = Σ Aji + Σ Eji´´ − Σ Bji − Σ Fji´´ < 0,. j=1. j=1. pi U=. qi. m1. qi´. n1. Σ αji − Σ βji + Σ δj − Σ. j=n+1. j=1. pi. V=− Σ. j =1. qi. m2. j=1. j =1. pi´. δji´ + Σ γj − Σ. j = m1 + 1. q. i´´. Aji − Σ Bji − Σ Fj − Σ. j=n+1. and. j=1. j =1. n2 F. j = m2 + ji´´ 1. γji´ > 0,. (1.3.4). j = n1 + 1. j =1. pi´´. + Σ Ej − Σ j =1. Eji´´ > 0,. j = n2 + 1. | arg x | < ½ Uπ, | arg y | < ½ Vπ. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411). (1.3.5).

(4) 385. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). 3.1.2 REVIEW OF LITRATURE. 3.3 MAIN INTEGRALS:. Srivastava Renu and Srivastava S. S. , Tiwari, I. P. and Sharma C. K.3,4 , Shrivastava, H. M.5, Burchnall J. L. and Chaundy T. Y.6, Appell, P. and Kampe de Feriet J7, Bromwich T. J. I. A.8, Erdelyi A.9 and several other authors have evaluated some integrals involving the product of Ifunction and other commonly used hypergeometric functions.. ****. 1,2. . z sin 2z F[ , ´ ;

(5) ;

(6) ´ ; . = ∑ . ,´;;´;. z F[,´;;´; − x z , −y z ] = ∑. !. J 2z. (3.2.1) and ,´;;´; z F[,´;;´; − x z , −y z ] = Γ1 + λ ∑ . ౤ J 2z ౤ ! . . భశಙషಓషమ౤ !. F[,,,´;;´; x , y ], , ´; ; ´;. (3.2.2) where A + A´ + C ≤ B + B´ + D, A + A´ + C´ ≤ B + B´ + D´, and for all values of λ with possible exception of zero and negative integers. (a) represents the sequence of A parameters a1, a2, …, aA and this convention will be retained throughout this chapter. The notation for double hypergeometric function is due to Burchnall and Chaundy6 in preference, for the sake of brevity, to an earlier one introduced by Kampe de Feriet7. The finite difference operator E8 has the following operations Eaf(a) = f(a + 1), E fa = E [E

(7) fa]. (3.2.3). ]dz. ,, , ´ ; ;´ ; F[ , ´ ;

(8) ;

(9) ´ ; x ,y ]. , ; , ; , . [. . . . . . . ……,…...:( / , ),…….:……,……. ], |. ……,……: ,,….….,. (

(10)

(11) , ),( / / ,):……,…….. (3.3.1) which is valid under the conditions A + A´ + C ≤ B + B´ + D, A + A´ + C´ ≤ B + B´ + D´, R(ρ + λ +. F[,,,´;;´; x, y] , ´; ; ´;. షమౣ. − x z , −y z ] I[. I , :; , :;, :´´. 3.2 FORMULA USED From Shrivastava5 (with z replaced by iz are required in the present work:. , ´ ; ;´ ;. R(ρ + λ +. . ) > − 1 (j = 1, …., k),. (

(12) ) . ) < 1 (j = 1, …., l) and. U > 0, > 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5). Proof of (3.3.1): To prove (3.3.1), take the expansion (3.2.1), multiply both side by f(z), integrate with respect to z between the limits 0 to ∞ and interchange the order of integration and summation we get . ,´;;´;. z F[,´;;´; − x z , −y z ]fzdz . = ∑ . . !. ,, , ´ ; ;´ ; F[ , ´ ;

(13) ;

(14) ´ ; x ,y ]. . J (2z) fzdz,. (3.3.3). for A + A´ + C ≤ B + B´ + D, A + A´ + C´ ≤ B + B´ + D´, R(λ + η + 1) > 0 and R(λ + ξ + 1) > 0, where f(z) = O(|z|η), for small z and f(z) = O(|z|ξ), for large z.. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411).

(15) 386. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). The change of integration and summation is justified [8] because (i) the series λ + 2nΓλ + n J 2z n! . F[

(16) , ,,´;;´; x , y ] ,´;;´;. is uniformly convergent in 0 ≤ z ≤ N, N being arbitrary; (ii) f(z) is a continuous function of z for all value of z ≥ z0 > 0; (iii) the integral on the left of (3.3.3) converges absolutely under the stated conditions. Now on taking . ]. in (3.3.3), replacing I-function of two variables on the right by its equivalent contour integral as given in (1.3.3), changing the order of integration which is justified due to the absolute convergence of the integrals, evaluating the inner integral with the help of [9] and interpreting it with (1.3.3), we get (3.3.1). **** . z cos 2z F[ , ´ ;

(17) ;

(18) ´ ; . [. (

(19) ). λ+. . ) > 0 (j = 1, …., k), R(ρ +. ) < 1 (j = 1, …., l) and U >. . Proof of 3.3.2: If we take. . f(z) = z

(20) cos 2z I[ ] proceed on the parallel lines as mentioned above and then in the light of the result9, we obtain (3.3.2). On considering the result (3.2.2), proceeding on the parallel lines as mentioned above and making use of the result [9, p.328(10); p.328(11)], we get the following different forms of the integral (3.3.1) and (3.3.2) as . z sin 2z F[ , ´ ;

(21) ;

(22) ´ ; , ´ ; ;´ ;. . షమౣ. − x z , −y z ] I[. ]dz. ∑ F[,, , ´ ; ;´ ; x , y ] , ´ ;

(23) ;

(24) ´ ; భశಙషಓ ౤ ! ,భ ;మ ,మ ;య ,య I , :;ᇲ ,ᇲ :ᇲ ;ᇲᇲ ,ᇲᇲ :´´ ౟ ౟ ౟ ౟ ౟ ౟. =. భ. , ´ ; ;´ ;. [. మౣ ……,…...:(మ,),…….:……,……. | భ ಓ ಙ ……,……: ,,….…., మ. షమౣ. − x z , −y z ] I[. ]dz. . F[

(25) , ,,´;;´; x , ,´;;´;

(26) ! , ; , ; , y ] I , :; , :;, :´´ = ∑ . . 0, > 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5).. . f(z) = z

(27) sin 2z I[. R(ρ + λ +. ……,…...:( / , ),…….:……,……. ], | ……,……: ,,….….,

(28)

(29) , , ( / / ,):……,…….. (3.3.2) which is valid under the conditions A + A´ + C ≤ B + B´ + D, A + A´ + C´ ≤ B + B´ + D´,. ],. (3.3.4). మ మ. (,),(//,):……,…….. which is valid under the same conditions as (3.3.1) and . z cos 2z F[ , ´ ;

(30) ;

(31) ´ ; , ´ ; ;´ ;. . =. షమౣ. − x z , −y z ] I[. . భశಙషಓ. ∑ . . ౤ !. F[,, , ´ ; ;´ ; x , y ] , ´ ;

(32) ;

(33) ´ ;. , ; , ; ,. I ,భ :;మ ᇲ ,మᇲ :ᇲయ;ᇲᇲయ,ᇲᇲ :´´ ౟. ౟. ౟. ౟. ]dz. ౟. ౟. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411).

(34) 387. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014) భ. [. మౣ ……,…...:(మ,),…….:……,……. | ಓ ಙ ……,……: ,,….….,. (3.3.5). ],. మ మ. The conditions of validity for (3.3.5) are the same as for (3.3.2). 3.4 INTEGRALS BY MEANS OF FINITE DIFFERENCE OPERATOR E: In this section we evaluate some integrals by means of finite difference operator E: /. "# $%" $. sin θ cos ! θ I[. . . . ౨ ∏౫ ౠసభ(ౠ ,) ,భ ;మ ,మ ;య ,య I , :;ᇲ ,ᇲ :ᇲ ;ᇲᇲ ,ᇲᇲ :´´ ∏౬ ! ,"! ౟ ౟ ౟ ౟ ౟ ౟ ౠసభ ౠ. (3.4.2) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in. (1.2.40) and (1.2.41). . /. "# $%" $. sin θ cos ! θ I[. . ]. . uFv [eu; fv; c sin2µθ cos2νθ] dθ. = ∑ . ౨ ∏౫ ౠసభ(ౠ ,) ,భ ;మ ,మ ;య ,య I ᇲ ᇲ ᇲᇲ ᇲᇲ ౬ ∏ౠసభ !ౠ ,"! ౟ ,౟ :;౟ ,౟ :ᇲ ;౟ ,౟ :´´. ……,…...:/μ,# ,…….:……,……. [ |…...,……:/$%,& ,…….,$(μ%),#& :……,……. ],. (3.4.3) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.2.40) and (1.2.41).. ]. . uFv [eu; fv; c sin2µθ cos2νθ] dθ. = ∑ . . ……,…...:/$%,& ,……,$(μ%),#& :……,……. [ |…...,……:/μ,# ,…….:……,……. ],. (,),(//,):……,…….. . = ∑ . ౨ ∏౫ ౠసభ(ౠ ,) ,భ ;మ ,మ ;య ,య I ᇲᇲ ౬ ∏ౠసభ !ౠ ,"! ౟ ,౟ :;ᇲ౟ ,ᇲ౟ :ᇲ ;ᇲᇲ ౟ ,౟ :´´. ……,…...: /

(35)

(36) μ,', /

(37) !

(38) &,(,…….:……,……. [ |…...,……:….….,

(39)

(40) !

(41) μ&,'(:……,……. ],. (3.4.1) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in H-function of two variables which is convergent if. . /. "# $%" $. sin θ cos ! θ I[. . ]. . uFv [eu; fv; c sin2µθ cos2νθ] dθ. = ∑ . ౨ ∏౫ ౠసభ(ౠ ,) ,భ ;మ ,మ ;య ,య I ᇲᇲ ౬ ∏ౠసభ !ౠ ,"! ౟ ,౟ :;ᇲ౟ ,ᇲ౟ :ᇲ ;ᇲᇲ ౟ ,౟ :´´. ……,…...:…….., !μ&,'(:……,……. [ |…...,……: / μ,', / !&,(,……..:……,……. ],. U = − Σ αj − Σ βj − Σ δj − Σ δj Σ γj − Σ γj > 0, (1.2.40). +. (3.4.4) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.2.40) and (1.2.41).. V = − Σ Aj − Σ Bj − Σ Fj − Σ Fj Σ Ej − Σ Ej > 0, (1.2.41). +. Proof of (3.4.1):. and. | arg x | < ½ Uπ, | arg y | < ½ Vπ.. As we know by the definite integral, . . . . /. "# $%" $ sin θ cos ! θ I[ ]. . uFv [eu; fv; c sin2µθ cos2νθ] dθ. /. "# $%" $. sin θ cos ! θ I[. ] dθ. , ; , ; ,. = (1/2) I , :; , :; , :´´ . . . . . . ……,…...: /

(42) ,', /

(43) !,(,…..:……,…….. [ |…...,……:……..,

(44)

(45) !,'(:……,…….. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411). ],(2.3.18).

(46) 388. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5). On multiplying both side of (2.3.18) by. ∏ (* ). ∏ +, -. and then applying the operator. μ ಕ. /. "# $%" $. sin θ cos ! θ I[. , ; , ; , . e'ಙ 'ಚ 'ಓ sin θ cos $ θ I[. )*మ౞ +,)మౡ +. . ∏.- Γ e- + λ

(47) c ∏/- Γ f- + λ

(48). . . . . . ౫ ಓ ……,……:/,,/,,…..:……,……. ∏ౠసభ !("ౠ #)$ [ |……,……:……..,,:……,……. ] ∏౬ }. #' !%& ౠ ౠసభ. (3.4.5). . /. Expanding both sides of (3.4.5) and applying (3.2.3), we have { sin(μ) θ cos ($%) θ I[ . .. )*మ౞ +,)మౡ +. . ಓశ౨ ∏౫ ౠసభ ౠ ". ∏౬ ౠసభ !ౠ "!. =∑. {. ]. dθ}. ಓశ౨ ∏౫ ౠసభ ౠ " ౬ ∏ౠసభ !ౠ "!. "# $%" $. ]dθ. , ; , ; , . . . . . ……,…...: /

(49) ,',……:……,…….. /. "# $%" $. sin θ cos ! θ I[. ] dθ. , ; , ; ,. = (1/2) I , :; , : ; , :´´. . . , ; ,మ ;య ,య I ,భ :;మ ᇲ , ᇲ ᇲ ᇲᇲ ᇲᇲ ౟ ౟ ౟ ౟ : ;౟ ,౟ :´´. Further, using α, n =. ], (2.3.19). [ |…...,……: / !,(,……..,

(50)

(51) !,'

(52) (:……,……. ], (2.3.20) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5). . . ……,……:/μ),,/(),,….:…,….. . = (1/2) I , :; , :;, :´´. . . [ |……,……:……..,μ(),:……,……. ]. . sin θ cos ! θ I[ . ( . . provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5).. , ; , ; , . . [ |…...,……: / ,',……..:……,…….. dθ}. . . ……,…...: /

(53) !,(,….., !,'

(54) (:……,…….. = e. . . { I , :; , : ; , :´´. . ] dθ. = (1/2) I , :; , :;, :´´. ( . μ . . . e'ಙ 'ಚ 'ಓ , we get μ ಕ. replacing (e0 + λ) by e0 and (f0 + λ) by f0 , we get (3.4.1). The results (3.4.2) to (3.4.4) can be derived on the same lines as mentioned above with the help of the results. . . . . . ……,…...:…….., !,'(:……,…….. ಓ ∏౫ ౠసభ !("ౠ #)$. ∏౬ౠసభ !%&ౠ #'. (/ ) (/). }.. and. changing the order of summation and integration on left hand side and then. [ |…...,……: / ,', / !,(,…..:……,……. ], (2.3.21) provided that ρ > 0, σ > 0, U > 0, >. 0, |argξ| < Uπ, where U and V are given in (1.3.4) and (1.3.5). respectively.. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411).

(55) 389. S. S. Srivastava, et al., J. Comp. & Math. Sci. Vol.5 (4), 382-389 (2014). REFERENCES 1. Srivastava, Renu and Srivastava, S. S.: Integration of Certain Products Involving I-function and Double Hypergeometric Function, Vikram Mathematical Journal, Vol. 17, p. 74-82 (1997). 2. Srivastava, Renu: Some finite double integral formulae involving I-function, Vijnana Parishad Anusandhan Patrika, Vol. 46, No.02, p.127-137 April (2002). 3. Tiwari, I. P. and Sharma, C. K.: Application of E-operator in oscilcation of water in a lake, Ganita, Vol.45, N0.182, p. 137-140 (1994). 4. Tiwari, I. P. and Sharma, C. K.: Evaluation of a definite integral by using E-operator and its application in heat conduction, Journal of M.A.C.T. Vol.. 26, p. 1-6 (1993). 5. Shrivastava, H. M.: Generalized Neumann expansions involving hypergeometric functions of one and two variables with Applications, South Asian publishers, New Delhi, (1982). 6. Burchnall, J. L. and Chaundy, T. Y.: Expansion of Appell's double hypergeometric functions, Proc. Camb. Phil, Soc., 63, 425-429 (1967). 7. Appell, P. and Kampe de Feriet, J: Functions Hypergeometriques et hypershperiques, polynomes d' Hermite, Gauthier Villers, Paris, (1926). 8. Bromwich, T. J. I. A.: An Introduction to the Theory of Infinite Series (1965). 9. Erdelyi, A.: A Table of Integral Transform, Vol. I, McGraw-Hill, New York, (1954).. Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411).

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