Simultaneous Quadruple Series Equations Involving Lagueree Polynomials

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1 Global Journal of Pure and pplied Mathematic. ISSN Volume 13, Number 7 (217), pp Reearch India Publication Simultaneou Quadruple Serie Equation Involving Lagueree Polynomial Indu Shukla Department of Phyical cience, M.G.C.G.V., Chitrakoot, Satna, M.P.(India) btract Lownde [2] [3] have obtained the olution of ome dual erie equation involving laguerre polynomial and then olved triple erie equation involving Laguerre polynomial. Singh, Rokne and Dhaliwal [4] obtained cloed from olution of triple erie equation involving Laguerre polynomial and Srivatava [6] have alo obtained the olution of certain dual erie equation involving Lagueree polynomial. In the preent paper, an exact olution ha been obtained for the imultaneou quadruple erie equation involving Laguerre polynomial by Noble` [5] modified multiplying factor technique. Key word: Integral equation, Serie equation, Serie equation, Laguerre polynomial. Subject Claification: 45XX, 45F1, 1552, 33C45, 42C5. 1. INTRODUCTION We conider the following Quadruple erie equation b [(x + d) k ] = 1i (x) ; < x < a (1.1) [(x + d) k ] = 2i (x) ; a < x < b (1.2) [(x + d) k ] = 3i (x) ; b < x < c (1.3)

2 3774 Indu Shukla c Γ(+β+ni+p) [(x + d) k ] = 4i (x) ; c < x < (1.4) Where + β + 1 > β > 1 m, + 1 > + β >, m i a poitive integer and < h <, b < and h and b are finite contant. [(x + d) k ] i a laguerre polynomial, p i a non- negative integer. are unknown coefficient to be determine and 1i (x), 2i (x), 3i (x) and 4i (x) are precribed function for i = 1,2,.. n Γ+n+1 L n (x) = 1 (x) ; < x < a (1.5) n Γ+n+1 L n (x) = 2 (x) ; a < x < b (1.6) n Γ+n+1 L n (x) = 3 (x) ; b < x < c (1.7) n Γ+n+β L n (x) = 4 (x) ; c < x < (1.8) The Quadruple erie equation (1.5), (1.6), (1.7) and (1.8) are a pecial cae of imultaneou Quadruple erie equation (1.1), (1.2),(1.3) and (1.4) when p =, d =, k =1, a = b = c = 1, i replaced by n and n i i replaced by n for j = 1,2,. and i= 1,2,. and =. L n (x) = n ( n+ ) ( x) k k= n k k, n =, 1, 2 (1.9) i the laguerre polynomial of order and degree n in x. 2. PRELIMINRY RESULTS (i) The orthogonal property of the laguerre polynomial i given by Erdelyi ( ) e x x L m (x)l n where δ m,n i the kronecker delta. (x)dx = Γ(+n+1) δ n! m,n, > 1 (2.1) (ii) Formula (27), pp.19 of Erdelyi ( ) in the form; d m dx m {x+m L (+m) n (x)} = Γ(+m+n+1) Γ(+n+1) x L n (x) (2.2) (iii) The following form of the known reult Erdelyi ( )

3 Simultaneou Quadruple Serie Equation Involving Lagueree Polynomial 3775 ( x) β 1 x L n (x)dx = Γ(n++1)Γ(β) Γ(n++β+1) +β L n +β () (2.3) when β >, > 1 and the econd integral (x ) β 1 e x L n where + 1 > β >. (x)dx = Γ(β). e L n β () (2.4) 3. SOLUTION OF QUDRUPLE SERIES EQUTIONS We aume that x + d = X 1 k, 1i (X 1 k d) = 1i (X), 2i (X 1 k d) = 2i (X), 3i (X 1 k d) = 3i (X), 4i (X 1 k d) = 4i (X), d k = e, (a + d) k = f, (b + d) k = g, (c + d) k = h. (3.1) Then the imultaneou quadruple erie equation (1.1), (1.2),(1.3) and (1.4) can be written in the following form: c We aume that Γ(+β+ni+p) (X) = 1i (X) ; e < X < f (3.2) (X) = 2i (X) ; f < X < g (3.3) (X) = 3i (X) ; g < X < h (3.4) (X) = 4i (X) ; h < X < (3.5) (X) = 1i (X) ; < X < e (3.6) Combining the erie equation (3.6) and (3.2), we can write the imultaneou quadruple erie equation (3.6) and (3.2) in the form (X) = ф 1i (X) ; < X < f (3.7) (X) = ф 2i (X) ; f < X < g (3.8) (X) = ф 3i (X) ; g < X < h (3.9)

4 3776 Indu Shukla c Γ(+β+ni+p) (X) = ф 4i (X) ; h < X < (3.1) where, ф 1i (X) = { 1i (X) ; < X < e (X) ; e < X < f 1i (3.11) ф 2i (X) = 2i (X), ф 3i (X) = 3i (X), ф 4i (X) = 4i (X) (3.12) Multiplying equation (3.7) by x ( x) β+m 2 and integrating with repect to X over (, ) and firt fractional integral formula (2.3) we get, = β m+1 L β++m 1 Γ(ni+β+p++m) ni+p () Γ(β+m 1) X ( X) β+m 2 ф 1i (X)dX (3.13) Now multiplying both ide of equation (3.13) by β++m 1 and differentiating both ide m time with repect to and uing the derivative formula (2.2) we get, b L β+ 1 Γ(ni+β+p+) ni+p () = e β +1 d m X ( Γ(β+m 1) d m X) β+m 2 ф 1i (X)dX (3.14) where e are the element of the matrix [b ][a ]. 1 and < < y, > 1, β + m > 1, i = 1,2,3... Equation (3.14) can be written a, Γ(ni+β+p+) β+ 1 () = e β+1 ф 1 () (3.15) Γ(β+m 1) Multiplying both ide of equation (3.1) by e X (X ) β δ and integrate with repect to X over (, ) and uing the econd fractional integral formula (2.4) we get, which can be written a, L +β 1 Γ(ni++p+β) ni+p () = L +β 1 Γ(ni++p+β) ni+p () = Where < x < and + 1 > β + δ >. e Γ( β +1) e X (X ) β ф 4i (X)dX (3.16) e Γ( β +1) ф 4() (3.17)

5 Simultaneou Quadruple Serie Equation Involving Lagueree Polynomial 3777 Left hand ide of equation (3.15), (3.17), (3.8) and (3.9) are identical hence on uing the orthogonal relation (2.1) = b a e (ni+p)! d [ Γ(β+m 1) a e L β+ 1 ni+p ()ф 1 ()d + e β+ 1 L β+ 1 ni+p () ф 2 ()d + e β+ 1 L β+ 1 ni+p ()ф 3 ()d g (ni+p)! c b + β+ 1 L β+ 1 ni+p ()ф 4 ()d] (3.18) Γ( β +1) c Where, n =,1,2,. and d are the element of the matrix [b ] 1. ф 1 () = dm X ( X) β+m 2 ф d m 1i (X)dX, ф 2 () = ф 2i (X), ф 3 () = ф 3i (X), ф 4 () = e X (X ) β ф 4i (X)dX (3.19) Provided that + β + 1 > β > 1 m, + 1 > + β >, m being a poitive integer with the help of (3.11), (3.19) can be written in the form e ф 1 () = dm [ d m X ( X) β+m 2 1i (X) dx + X ( X) β+m 2 1i (X) dx], e REFERENCES e < (3.2) [1] Erdelyi, : ( ) Higher Trancendental function, Mc Graw Hill, Newyork vol. I, II, III, pp , [2] Lownde, J.S.: (1968) Some dual erie equation involving Laguerre polynomial, Pacific J. Math.25, pp [3] Lownde, J.S.: (1969) Triple erie equation involving Laguerre polynomial, Pacific J. Math.29 (1), pp [4] Singh, B.M. Rokne, J. and Dhaliwal, R.S.,:(21) On cloed form olution of triple Serie equation involving Laguerre polynomial Ukrainian Mathematic Jour., 62(2), pp [5] Noble, B.,:(1963) Some dual erie equation involving Jacobi polynomial, proc. Camb. phil. oc., 59, pp [6] Srivatava, H.M.,:(1969) note on certain dual erie equation involving Laguerre polynomial pacific J. Math, 3, pp

6 3778 Indu Shukla [7] Panda, R.:(1977) Certain Dual erie equation involving Laguerre polynomial, Indag Math, 39, pp [8] Mathur, P.K. and Singh,. :( 212) Certain imultaneou five tuple erie equation Involving Laguerre polynomial ultra-cientit, 24, pp [9] Narain, K.:(213) Certain imultaneou triple erie equation involving Laguerre Polynomial Mathematical theory & modelling, 3, pp [1] Mudaliar, R.K. and Narain, K.:(216) Certain Dual erie equation involving Generalized Laguerre polynomial Int. Jour. of computational and applied Mathematic, 11, pp

Certain Dual Series Equations Involving Generalized Laguerre Polynomials

International Journal of Computational and Applied Mathematics. ISSN 1819-4966 Volume 11, Number 1 (2016), pp. 55-59 Research India Publications http://www.ripublication.com Certain Dual Series Equations