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A Laurent Series Expansion Formula and Its Applications

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A Laurent Series Expansion Formula and Its Applications Proe. Indian Acad. Sci. (Math. Sci.), Voi. 93, No. 1, November 1984, pp. 59--62. Printed in India. A Laurent series expansion formula and its applications R K RAINA Department of Mathematics,S.K.N. AgricultureCollege, Jobner 303 329, India Present address: Department of Mathematics,Sukhadia University,Udaipur 313 001, India MS received 1 May 1982 Abstract. Thispaper givesa certain Laurentseries expansion for a generalizedRodrigues type formula. The main result finds many applications which are enumerated briefly. Keywords. Laurent series; Lagrange'sexpansion formula;H-functions. 1. Introduction The literature is full of formulas giving series expansions relating the special functions of mathematical physics and one encounters several such results involving the higher classes of special functions of one and more variables. However, the closed form expressions for such series as the Laurent series ~ f(n), involving the special functions do not appear frequently in literature. Indeed, during the last decade significant contributions achieved in the field of fractional calculus saw these series expansions emerge. One finds some of these results in the papers of Osler ([3]-[5]). Further such expansions for higher classes of special functions were obtained by Raina and Co-workers ([8]-[1!]). This paper aims at deriving a Laurent series expansion for a certain type of function (equation (1)) defined below. Several applications can be motivated from our result because of the general nature of the function involved and these are briefly pointed out in the concluding section. For any real number a such that 0 < a ~< I, and for any arbitrary number r/(real or complex), define the generalized "Rodrigues Type" formula by 1 S~(F/; z) = O~"+~ { (cz + d)~ zaf(z)}, (1) F(an + tl + 1) where Re (,t) > - 1,f(z) is analytic in the circular domain [z[ < R, and is independent of n; c, d are constants (not both zero) and D~*§ denotes the derivative of arbitrary order. The derivative operator D~ (~ arbitrary) is the condensed form of the notation 0D~, which corresponds to the widely used notation ,D~ (see Ross [6] for its full definition). Under the various restrictions stated above, the operator D~*+" occurring in (1) is meaningful and consequently the function S~(r/; z) is well defined. 59 60 R K Raina 2. Main result We propose to establish the following expansion formula: Theorem. With the various constraints on the parameters and other restrictions, let S~(q; z) be defined by (1) Then (am+an+q+,~ .=-~o \ an+ti ] S~'+'(tl+ 7;z)t'+~ = a -t (1 -ct) -'~-~- 1S~, (7; (z+dt)/(l -ct)), (2) holds for all m = 0, + 1, + 2 .... ; provided that I tl = Iz/dl, where ~ is any arbitrary number. Proof. If I denotes the left-hand member of (2), then from the definition (1), we readily observe that I= ~ D~"+~176 'l an + Jl x F(an+ t/+ 1)F(am+~+ 1)" (3) Since Re(2) > - 1, the functions in (3) are analytic in the domain 0 < [z[ < R, the derivative operator ofarbitrary order involved in (3) would satisfy the law of exponents = D, O,, (4) so that I can be expressed as 1 Dm+~ ~ D*~'+"{(cz+d)~ I = F(am+r+ 1)-" .=_~ ta.+Jl x r(an + tl + 1)" (5) By appealing to the following generalized form of the Lagrange's expansion formula which is also a consequence of the more generalized form of Osier ([4] equation (1.2); see also Lavoie et al [2], equation (20.5)): tan + 'l .= _~ D'~'+~{f(z)[g(z)]*"+~} F(an+q+ 1) = a- If(w) (I -- tO(W))- 1, (6) provided that I(w - z)/o(w)] = Iz/o(O)], where 0 < a ~< 1, ? is any arbitrary number, and w is given by w = z+to(w), (7) we thus obtain (1 -ct) -1 I= 07 +r {w "~(cw + d)*" +~f(w) }, (8) aF(am +~ + 1) Laurent series expansion 61 where w = (z + dt)/(1 - ct). (9) Noting that the function w given by (9) is analytic and univalent in the domain 0 < Iz ] < R, the derivative operator in (8) is also well defined for this transformation and thus (2) follows at once by using (1). 3. Applications At the outset, we observe that our expansion formula (2) in the special case a = 1, r/= y = 0 and 2 = 0 corresponds to the result of Srivastava ([13], equation (6)). Since many special functions of one or more variables can fairly easily be represented in the form (1) (see the list of such representations in the paper of Lavoie et al ([2], pp. 260-261), our expansion formula given in the theorem is widely applicable. We find it appropriate to obtain a particular expansion formula from (2) which would yield various special cases. To this end, let fl')l.Q-]' h > 0, (10) in (1), where the function described above is the well-known H-function of Fox ([1], p. 408) and is defined in terms of a Mellin-Barnes type contour integral. The expressions involved in the right side of (1) obviously suggest that we also require (for closed form expressions) in our analysis, the H-function of two variables which is an extension of the function appearing in (10). These functions being well known in literature, we skip details regarding their definition, notational format and the conditions of existence. For a full account of these details, one may refer to the paper of Srivastava and Panda ([12], pp. 265-267). The expansion formula (2) thus in conjunction with (10) on taking d = 1, and using suitably the known result of Raina and Koul ([7], equation (8)), with, of course, other slight changes yields the expansion tan+n /./0A :(M, N); [l, l] . = - oo F(an + q + 1)F( - an - am - tl -),) "" ~,~:(v,o;[~,1] x[ xl(-2;h, 1) : (a,,0t,),,e;(1 +an+am+q+y, 1)] y (-2+an+am+q+),;h, 1) (bi, fli)l.e;(0,1 ) = a- 1 (1 -yt) -~- x (1 + t) a-m-r rt0,1:tu, m[l,0 F( -am -y) "" I'I:(P'Q);tI'I] : (1 +a,.+,, 1)] • Ly((1 + t)/(1 - yt)) ( - 2 + am + y; h, 1): (bi, fli)~,Q; (0, 1) (11) valid for Iq=l; provided that h>0, Re[2+h(b~/fll)]>-1, i=1 ..... M, Q P ~'.fli-~i > 0 and [argx[ < [argy I < n, i 1 62 R K Raina M Q N P where A = E/~,- Z /~, + Z ~t, - E ~t, > 0. 1 M+I 1 N+I An interesting special case of (11) worth noting is the expansion for the H-function of Fox obtained recently by Raina ([9], equation (2.7)), which occurs (with one of its conditions corrected) when we formally let y ~ 0, set m -- ? = 0 and make other slight alterations. References [1] Fox C 1961 Trans. Am. Math. Soc. 98 395 [2] Lavoie J L, Osier T J and Tremblay R 1976 SIAM Rev. 18 240 [3] Osier T J 1970 SlAM J. Appl, Math. 18 658 [4] Osier T J 1971 SIAM J. Math. Anal. 2 37 [5] Osier T J 1972 SlAM J. Math. Anal. 3 1 [6] Ross B (ed.) 1975 Lectures on fractional calculus (New York: Springer Verlag) 457 [7] Raina R K and Koul C L 1977 Jh~nabha (Prof. Charles Fox memorial volume) 7 97 [8] Raina R K and Koul C L 1979 Proc. Am. Math. Soc. 73 188 [9] Raina R K 1979 Comment. Math. Univ. St. Pauli 28 115 [10] Raina R K 1979a Pure Appl. Math, Sci. 10 37 [11] Raina 1980 Univ. Nac. Tucumdn Rev. A28 (to appear) [12] Srivastava H M and Panda R 1976 J. Reine Anoew. Math. 283/284 265-274 [13] Srivastava H M 1975 Glasgow Math. J. 16 34 .
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