where F (x) (called the Similarity Factor (SF)) denotes the function

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1 italian journal of pure and applied mathematic n ) 15 GENERALIZED EXPONENTIAL OPERATORS AND DIFFERENCE EQUATIONS Mohammad Aif 1 Anju Gupta Department of Mathematic Kalindi College Univerity of Delhi New Delhi India Abtract. The preent paper deal with the generalization of exponential operator ued by Dattoli and Levi for tranlation and diffuive operator which were utilized to etablih analytical olution of difference and integral equation. The generalization of their technique i expected to cover wide range of uch utilization. Keyword: Kampé de Feriét polynomial, exponential operator, group theory, Lie algebra of Lie group. PACS number: 0.0, 0.0.Sv. 1. Introduction In 000, Dattoli and Levi 1 dicued general method for the olution of difference equation, ariing in phyical and biological problem.their technique play crucial role in unifying the generalized familie of the difference equation. The preent paper deal with the generalization of exponential operator ued in 1 to operator of the type a λqx) d dx, where bae a a > 0, a 1) i a real number. In particular when a e, the operator reduce to the operator ued by Dattoli et al. 1. The action of the generalized exponential operator on a generic function fx) i defined a 1.1) d a λqx) d dx λ lna))qx) fx) e dx fx) ff 1 λ lna) + F x))). where F x) called the Similarity Factor SF)) denote the function 1 Correponding author. mohdaiff@gmail.com. Director, NCWEB, Univerity of Delhi.

2 16 mohammad aif, anju gupta F x) x q), and F 1 σ) i it invere. For qx) 1, the SF i given by x 1.) F x) x, therefore F 1 x) x, then operator 1.1) reduce to the ordinary tranlation or hift operator a follow: 1.3) a λ d dx fx) ff 1 λ lna) + x)) fλ lna) + x). Another example of application of operator 1.1), for qx) x, the SF i given by x 1.4) F x) lnx), o that F 1 x) e x, and hence operator 1.1) reduce to the dilatation operator 1.5) a λx d dx fx) ff 1 λ lna) + lnx))) fe λ lna)+lnx) ) fa λ x). The ordinary hift operator and their propertie play a central role within the context of the theory of difference equation 3. One can, therefore, upect that the above generalized exponential operator and the wealth of their propertie can be exploited to develop tool which allow the olution of different form of difference equation. 1a). Particular cae: The ubtitution of a e, into equation 1.1), 1.3) and 1.5) reduce to equation 1), ) and 3) of Dattoli et al. 1. A imple example of how the exponential operator can help u to olve difference equation may be illuminating. Let u conider the linear dilatation difference equation of the type 1.6) b 1 fa x) + b fax) + b 3 fx) 0, which, according to equation 1.5), equation 1.6) can be written in the following form 1.7) b 1 a x d dx + b a x d dx + b3 fx) 0. Suppoe fx) R lnx), we have a λx d dx R lnx) e λ lna)x d dx R lnx), where qx) x, o that F x) lnx) and F 1 x) e x or F 1 λ lna) + lnx)) e λ lna)+lnx) xa λ. Therefore, 1.8) a λx d dx R ln x R lnxaλ) R λ lna) R ln x.

3 generalized exponential operator and difference equation 17 Hence we can aociate with equation 1.7) the characteritic equation 1.9) b 1 R lna) + b R lna) + b 3 R lnx) 0, or b 1 R lna) + b R lna) + b 3 0, whoe root R lna) 1 and R lna) allow to write fx) in term of the following linear combination of independent olution: 1.10) fx) c 1 R lnx) 1 + c R lnx) α1 c α R lnx) α. The above example indicate that we can extend well-etablihed method of olution of difference equation to other type of equation reducible to ordinary difference equation, after a proper change of variable implicit in equation 1.1), 1.3). 1b). Particular cae: The replacement of a with e in equation 1.6), 1.7), 1.8) and 1.9) give raie to equation 5), 6), 7), and 8) of Dattoli et al. 1. To give a further example of the flexibility of the formalim aociated with exponential operator, let u conider the generalized Heat Equation of the following type 1.11) Qx, λ lna)) lna) qx) Qx, λ lna)), λ x Qx, 0) gx), which can formally be olved by rewriting equation 1.11) a Qx, λ lna)) lna) qx) Qx, λ lna)) 0, λ x which can formally be olved by conidering thi a ordinary linear differential equation of order one, whoe I.F. i determined a e lna)qx) x dλ e lna) qx) x λ a λ qx) x, we can, therefore, find it general olution a Qx, λ lna))a λ qx) x C, where C in any contant and uing the given initial condition, we get Qx, 0) gx) C, and, finally, we obtain the olution of the Heat equation 1.11) a 1.1) Qx, λ lna)) a λqx) x gx).

4 18 mohammad aif, anju gupta Uing the identity e b 1 + e +b, π and replacing b by λ lna)qx) x, we have 1.13) e λ lna)qx) x a λqx) x 1 + π e + λ lna). Uing equation 1.1), finally yield the olution of equation 1.11) in the form of an integral tranform, which can be viewed a a generalized Gau tranform 1.14) a λqx) x gx) 1 + π or, in other word, we have e gf 1 λ lna) + F x))). 1.14) Qx, λ lna)) 1 + e gf 1 λ lna) + F x))). π It i evident that the formalim aociated with generalized exponential operator can be exploited in many flexible way in finding the general olution of a large number of problem. Thi paper i devoted to the dicuion of method which provide the olution of the clae of difference and generalized Heat equation and we hall ee that the technique we propoe offer reliable analytical tool and efficient numerical algorithm. 1c). Particular cae: To put a e, in the equation 1.11), 1.1), and 1.14) give raie the ame form of the equation 10), 11) and 13) repectively of Dattoli et al. 1.. Generalized difference equation Before dicuing the problem in it generality, let u conider the equation of the following type, a a further example, which reduce to 1; p )) when we conider a e, we have.1) b α fx coα lna)) + 1 x inα lna))) 0, which belong to the familie of generalized difference equation. Thi equation can be obtained by the action of the generalized exponential operator on the function fx).

5 generalized exponential operator and difference equation 19 b α finin 1 x) coα lna)) + coco 1 1 x ) inα lna))) 0, b α finin 1 x) coα lna)) + coin 1 x) inα lna))) 0, b α finin 1 x) + α lna)) 0, b α a α 1 x fx) 0. According to the dicuion of the previou ection, the ue of the exponential operator b α Â α fx) 0, where.) Â a 1 x d dx allow to cat.1) in the operator form where ΨÂ)fx) 0, N.3) ΨÂ) b α Â α. In thi cae, the SF aociated with.) i.4) F x) in 1 x). Independent olution of.1) can, therefore, be contructed in term of the function R in 1 x), which atifie the identity.5) Â α R in 1 x) R α lna) R in 1 x), the general olution of.1) can finally be written a.6) fx) c α R in 1 x) α. Similarly, if we conider the following example, we have N.1) b α fx coα lna)) 1 x inα lna))) 0,

6 0 mohammad aif, anju gupta which belong to the familie of generalized difference equation. According to the dicuion of the previou ection, the ue of the exponential operator.) Â a 1 x d dx allow to cat.1) in the operator form.3). In thi cae the SF aociated with.) i.4) F x) co 1 x). Independent olution of.1) can be therefore contructed in term of the function R co 1 x), which atifie the identity.5) Â α R co 1 x) R α lna) R co 1 x). The general olution of.1) can finally be written a.6) fx) c α R co 1 x) α, where R lna) α are the root of the characteritic equation.7) ΨR lna) ) 0. From the above dicuion it i now clear that, whenever one deal with equation of the type.8) b α ff 1 α lna) + F x))) 0, one can aociate it with the generalized exponential operator.9) Â a qx) d dx, which allow to cat.8) in the operator form.3) and we get the relevant olution in the form.10) fx) x c α R q) α. a). Particular cae: when we ubtitute a e in equation.1),.),.3),.5),.7),.8) and.9) then thee equation lead to equation 14), 15), 16), 18), 0), 1) and ) repectively due to Dattoli et al. 1.

7 generalized exponential operator and difference equation 1.11) A ueful example i given by the equation ) x b α f 0, 1 α lna)x by making ue of the hift operator a x d dx, which allow to cat.11) in the operator form.3), i.e., ) 1 b α f α lna) 1 0, x b α a αx d dx fx) 0, b α Â α fx) 0, where N Â a x d dx, ΨÂ)fx) 0 and ΨA) b α A)α. In thi cae, the SF aociated with.11) i F x) 1. It olution can thu x be written a.1) fx) α1 c α R 1 x α. The validity of the above olution i limited to the cae in which Rα lna) i not a multiple root of the characteritic equation; thi point will be dicued in the concluding ection. b). Particular cae: Replacing a with e in equation.11) reduce to Dattoli et al. 1; p.6564)). In the tune of Dattoli et al. 1; p. 6566)), let u introduce the following operational identitie: Â ±α x b q) b ±α lna) x b q),.13) Â ±α x b x q) φx)) b q) b lna) Â) ±α φx). valid for exponential operator of the form.9). We note that, according to the firt of.13), the non-homogeneou equation.14) ΨÂ)fx) Cb x q),

8 mohammad aif, anju gupta where C i a contant and b lna) i not a root of the characteritic equation, admit the particular olution.15) fx) Cb x q) Ψb lna) ). In the lightly more complicated cae.16) ΨÂ)fx) Cb x q) φx), the econd of.13) yield x.17) fx) Cb q) 1 Ψb lna)â)φx). Further comment hall be dicued in the concluding ection. c). Particular cae: When a e, equation.14),.15),.16) and.17) convert into equation 7), 8), 9) and 30) of Dattoli et al Generalized hift operator and Jackon derivative In the previou ection, we have conidered linear equation involving dicrete power of the generalized exponential operator. Here, we hall dicu example in which the exponent are not necearily integer. The introductory example i 3.1) fa λ x) fx) λ lna) gx), where fx) i unknown, λ C, and gx) i an analytical function. The ue of the dilatation operator allow to cat equation.17) in the form of a the Jackon derivative 4, namely 3.) a λ d 1 fx) gx). λ lna) The operator on the left hand ide can formally be inverted and by writing the differentiation variable in term of the invere of the SF we find 3.3) fe ) λ lna) a λ d 1 ge ). The operator on the r.h.. of 3.3) can be expanded a

9 generalized exponential operator and difference equation 3 λ lna) a λ d 1 λ lna) λ lna) d + 1! λ lna) d ) + 1 3! λ lna) d )3 + 1 ) ) d λ lna) d + 1 λ lna) d! 3! + D 1 D ! λ lna) d + 1 3! 1 1! λ lna) d + 1 3! λ lna) d ) ) 1 + λ lna) d ) ) + 1 +! λ lna) d + 1 λ lna) d ) ) + + 3! D 1 D 1 D λ lna) d λ lna) d B 0 + B 1 1! λ lna) d + B! ) λ lna) d ) ) λ lna) d ) λ lna) d λ lna) d ) 1 λ lna) d ) ) + B 3 λ lna) d ) 3 + 3! or 3.4) λ lna) D 1 1 a λ d n0 ) n B n d λ lna))n, n! where 3.5) B 0 1, B 1 1, B 1 6, B 3 1,... are Bernoulli number ee 9; p.3009)) and D 1 i the invere of the derivative operator. Since gx) ha a Taylor expanion gx) b m x m ), we get from equation 3.3), 3.4) m0

10 4 mohammad aif, anju gupta fe ) D 1 D 1 D 1 D 1 n0 ) n ) B n d λ lna))n b m e m n! m0 B 0 + B 1 1! λ lna) d + B! b λ lna) d b 0 + λ lna) d λ lna) d ) + B 3 3! b m e m 1 λ lna) b m me m m1 + 1 λ lna)) 1 m0 b m m) e m 1 λ lna))3 6 m0 b m m em 1 λ lna) b m e m m1 + 1 λ lna)) 1 b λ lna) m0 b m m)e m 1 λ lna))3 6 m0 λ lna) d ) 3 + ) 1 λ lna) d ) m0 b m e m b m e m m0 b m m) 3 e m + m0 b m m) e m + m0 ) b 1 e λ lna)) λ lna)) λ lna) λ lna)) λ lna)) ) 1 λ lna) b 0 + b 1 e 1 λ lna)) + + +! 3! ) 1 λ lna) +b e 1 λ lna)) ! 3! b 0 + b 0 + b 1 e 1 + λ lna) λ lna)) +! + ) + b e λ lna) !! λ lna)b 1 e λ lna) λ lna)) λ lna)) !! 3! b 0 + λ lna)b 1e e λ lna) 1 + λ lna)b e e λ lna) 1 + λ lna)b 3e 3 e 3λ lna) 1 + ) b e + λ lna)) 3! + ) + λ lna)b e λ lna) λ lna)) λ lna)) !! 3! or fe ) m1 b m λ lna)e m e λm lna) 1 + b 0,

11 generalized exponential operator and difference equation 5 or 3.6) fe ) m1 Going back to the original variable, we get 3.7) fx) m1 b m λ lna)e m a λm 1 + b 0. b m λ lna)x m a λm 1 + b 0 lnx). The erie on the right hand ide of equation 3.6) provide the olution of our problem. 1) m x m+1 Taking another example, gx) inx), we find m + 1)! fe ) D 1 D 1 D 1 D B n λ lna)) n n0 n! B 0 + B 1 1! λ lna) d + B! m0 ) n d 1) m e m+1) m + 1)! m0 λ lna) d ) + B 3 λ lna) d ) 3 + 3! 1 1 λ lna) d + 1 λ lna) d ) 1 1 λ lna) d ) m0 1) m e m+1) m + 1)! λ lna) d ) m0 m0 1 1) m e m+1) m + 1)! m0 m0 1) m e m+1) m + 1)! λ lna) d ) 1) m e m+1) m + 1)! m0 ) λ lna) d m0 1) m m + 1)m + 1)! em+1) 1 λ lna) 1) m m + 1)! em+1) + 1 λ lna)) 1 m0 1) m m + 1) m + 1)! 1 6 λ lna))3 e m+1) m0 m0 1) m e m+1) m + 1)! 1) m e m+1) m + 1)! 1) m m + 1) e m+1) m + 1)!

12 6 mohammad aif, anju gupta 1) m e m+1) λ lna)m + 1) 1 m + 1)m + 1)! m0 + λ lna)) m + 1) λ lna))3 m + 1) 3 + m0 1 1) m e m+1) m + 1)m + 1)! 6 λ lna)m + 1) λ lna)) m + 1) +! 3! 1 or or fe ) λ lna) 1) m a λm+1) 1 m + 1)! em+1) 3.8) fx) λ lna) 1) m a λm+1) 1 m + 1)! xm+1. It i eentially the erie defining gx), provided that b m i replaced by If, e.g., we take gx) cox), we find b m λ ln a a λm 1 3.8) fx) λ lna) 1) m a λm) 1 m)! xm, and for gx) e xq, we get 3.9) fx) m1 λ lna) x qm a λqm 1 m! + lnx). We can, therefore, conclude that the primitive of a Jackon derivative can be contructed according to the above-quoted recipe. Thi method can alo be generalized and the concept of Jackon derivative extended to other form of exponential operator. In thi cae we conider equation of the type 3.10) fx) coλ lna)) + 1 x inλ lna)) λ lna) gx), with the aitance of equation.), we write equation 3.10) a follow: 3.11) a λ 1 x fx) fx) λ lna) gx) or a λ 1 x 1) fx) gx), λ lna)

13 generalized exponential operator and difference equation 7 by auming gx) i an odd function there taking x in, we have d d dx dx co d dx, 3.1) a λ d 1) fin ) gin ), λ lna) fin ) λ lna) a λ d 1 gin ), or let u find out the expanion of the firt factor of the r.h.. of equation 3.1) with the help of equation 3.4), we have 3.13) fin ) D 1 n0 ) n B n d λ lna))n gin ), n! ince gx) in an odd and analytic function, then gin ), can be expanded by Taylor expanion uch a gin ) b m+1 in ) m+1, we have from 3.13), fin ) D 1 D 1 D 1 D 1 n0 n0 m0 ) n B n d λ lna))n b m+1 in ) m+1 n! n0 m0 ) n B n d e i e i m+1 λ lna))n b m+1 n! i m0 ) n B n d b m+1 i) m+1 λ lna))n e i ) m+1 1 e i m+1 n! m+1 m0 B 0 + B ) 1λ lna)) d + B ) λ lna)) d + B ) 3λ lna)) 3 d 3 1!! 3! b m+1 i) m+1 e i ) m+1 m+1 m0 m+1 0 m+1 ) 1) m+1 e m+1 )i. Subtituting the value of Bernoulli number from equation 3.5), we have ) ) ) ) D 1 λ lna)) d λ lna)) d λ lna))3 d b m+1 i) m+1 m1 m1 m+1 m+1 0 b m+1 i) m+1 D 1 m+1 m+1 m+1 0 ) 1) e im )+1 ) ) m+1 1) e im )+1 λ lna)) d

14 8 mohammad aif, anju gupta m+1 0 m+1 0 m1 λ lna)) ) m+1 1) e im )+1 λ lna)) + 1 λ lna)) + 1 ) m+1 1) e im )+1 b m+1 i) m+1 D 1 m+1 m+1 0 m+1 0 m+1 0 ) d ) m+1 1) e im )+1 ) m+1 1) i{m ) + 1}e im )+1 m+1 b m+1 i) m+1 m1 λ lna)) λ lna)) + 1 m+1 m+1 0 m+1 0 m+1 m+1 ) 1) i{m ) + 1} e im )+1 0 m+1 b m+1 i) m+1 m1 { 1 m+1 b m+1 i) m+1 m1 m+1 ) m+1 1) e im )+1 im ) + 1 ) 1) e im )+1 m+1 ) 1) i{m ) + 1}e im )+1 0 ) m+1 1) e im )+1 im ) + 1 λ lna)) λ lna)) i{m ) + 1} + i{m ) + 1} 1 } λ lna))3 i{m ) + 1} b m+1 i) m+1 m+1 ) m+1 m+1 m1 0 1) e im )+1 im )+1 { 1+ 1 i{m )+1}λ lna)+ 1 i{m )+1}λ! 3! lna) + } m+1 0 ) m+1 1) λ lna)e i ) m )+1 i{m )+1}λ lna)+ 1 i{m )+1}λ! lna) + 1 i{m )+1}λ 3! lna)3 + b m+1 i) m+1 m+1 ) m+1 1) λ lna)co + i in ) m )+1 m+1 e im )+1λ lna) 1 m1 0

15 generalized exponential operator and difference equation 9 or fin ) b m+1 λ lna) i) m+1 m+1 m1 m+1 m+1 1) 1 in + i in ) m )+1 e im )+1λ lna). 1 0 Finally, 3.15) fx) b m+1 λ lna) i) m+1 m+1 m1 m+1 m+1 1) 1 x + ix) m )+1 a im )+1λ 1 0 It i intereting to note that, in thi cae too, the criterion to evaluate the primitive of the Jackon derivative, aociated with the operator.), can eaily be inferred. Let u note that the procedure we have dicued can alo be extended to the cae involving the generalized Gau tranform. In fact the olution of or in other word, we have let u uppoe x e, then a λx d Now, from equation 3.17), we have dx ) 1 fx) gx) 3.16) λ lna) λ lna) gx) fx), 3.17) a λx d dx ) 1 d d dx dx d e dx x d dx 3.18) λ lna) a λ d ) 1 ge ) fe ). Further, after following the tep a we followed in getting the reult 3.4), we obtain the expanion of firt factor of l.h.. a 3.19) λ lna) a λ d ) 1 D n0 ) n B n d λ lna))n. n!

16 30 mohammad aif, anju gupta Now by the virtue of the analyticity of gx), we expand gx), by Taylor erie, i.e., ) n fe ) D B n d λ lna))n b m e mx, n! n0 m0 proceeding of the tep a proceeded in finding the reult 3.7), we obtain fe ) m1 b m or in other word, if we take b 0 0, then 3.0) fx) λ lna) e λ lna)m 1 em + b 0 m1 b m λ lna) a λm 1 xm. Further comment on the reult of thi ection will be dicued in the forthcoming concluding ection. 3a). Particular cae: If we ubtitute a e, in equation 3.1), 3.), 3.3), 3.4), 3.7), 3.8), 3.9), 3.10), 3.15), 3.16) and 3.0), then we obtain the equation 31), 3), 33), 34), 36), 37), 38), 39), 40), 41) and 4) on page number due to Dattoli et al Remark In the previou ection we have conidered linear difference equation, a trivial) non-linear example, imilar to Riccati equation, i given blow 4.1) fax) fx) + b lnx) fax)fx) 0, which can be olved uing the auxiliary function gx) 1 and thu getting fx) 4.) 1 fx) 1 fax) + blnx) 0, gax) gx) b lnx). or From operator 1.5), we have 4.3) a x d dx gx) gx) b lnx),

17 generalized exponential operator and difference equation 31 or, in other word, we write the above equation 4.3) a blnx) gx) a x d dx 1 1 a x d dx 1 b lnx) 1 + a x d dx + a x d dx + a 3x d dx + b lnx) b lnx) + a x d dx b lnx) + a x d dx b lnx) + a 3x d dx b lnx) + b lnx) + b lna) b lnx) + b lna) b lnx) + b 3 lna) b lnx) b lna) + b lna) + b 3 lna) + b lnx) 1 blnx) 1 blna) blnx) b lna) 1 thu finding a a particular olution 4.4) fx) 1 gx) blna) 1 b lnx). Moreover, in general, equation of the type 4.5) b α ff 1 α lna) + F x))) ɛfx) n, tandard perturbation method can be ued. At the lowet order in ɛf f 0 +ɛf 1 ), we find 4.6) b α f 0 F 1 α lna) + F x))) 0 and 4.7) b α f 1 F 1 α lna) + F x))) R n x q), where R lna) i one of the root of the characteritic equation aociated with 4.5). The firt-order contribution f 1 can therefore be evaluated by uing equation.15), which hould be modified a follow: 4.8) fx) C x q) )b x Ψ b lna) ) Ψ b lna) ) d dr ΨRlna) ) if b lna) i a imple root of the characteritic equation. q) 1,, Rb

18 3 mohammad aif, anju gupta 4a). Particular cae: The replacement of a with e in the equation 4.1), 4.4) and 4.8) reduce to the equation 43), 45) and 49) of Dattoli et al. 1. Let u now go back to the problem of treating exponential operator of the type 4.9) Â m,λ a λqx) d dx )m. We have een that, for m and λ > 0, they can be viewed a generalized Gau tranform. Before dicuing the problem more deeply, we recall the following important relation : a λ d dx )m x n H n m) x, λ lna)), 4.10) n H n m) m λ lna)) r x n mr x, λ lna)) n!. r!n mr)! which hold for negative or poitive λ and H n m) x, λ lna)) are Kampé de Feriét polynomial, and atify the identity ) m 4.11) λ Hm) n x, λ lna)) lna) H n m) x, λ lna)). x According to equation 4.8) we alo find 4.1) r0 a λ d dx )m gx) a λ d dx )m Therefore, it i eay to realize that 4.13) A m,λ x n0 n0 n0 b n x n b n H n m) x, λ lna)). φ n H n m) F x), λ lna)), where we have aumed that the function F 1 can be expanded in power erie 4.14) F 1 ζ) φ n ζ n. It i clear that equation 4.1) can be further handled to extend the action of operator 4.8) to any function gx). It i worth conidering the poibility of extending the definition of operator 4.8) to the cae of not necearily integer m. In the cae of m 1, equation 4.9) hould be replaced by a λ d dx ) 1 x n H 1 ) n x, λ lna)), 4.15) n H 1 ) λ lna)) r x n r n x, λ lna)) n! r!γn r + 1). n0 r0

19 generalized exponential operator and difference equation 33 It i evident that in thi cae H 1 ) n x, λ lna)) i a relation analogou to 4.10) hold, namely 4.16) 1 λ H ) n or involving emi derivative ) 1 λ H ) n ) x, λ lna)) lna)) x x, λ lna)) lna) H 1 ) n x, λ lna)). ) 1 H 1 ) n x, λ lna)). x Thi definition can be extended to any m 1 p p, integer). 4b). Particular cae: Equation 4.9), 4.10), 4.11), 4.1), 4.13), 4.15) and 4.16) lead to Dattoli et al. 1 equation 50), 51), 5), 53), 54), 56) and 57). Concluding remark. It i hope that for the value other than e ome more ue of the generalized exponential operator can be obtained. Reference 1 Dattoli, G., Levi, D., Exponential Oprator and Generalized Difference Equation, Riv. Nuovo Cimento, B, ), Dattoli, G., Ottaviani, P.L., Torre, A., Vazquez, L., Evolution operator equation: Integration with algebraic and finite difference method. Application to phyical problem in claical and quantum field theory, Riv. Nuovo Cimento, ), Jordan, Ch., Calculu of Finite Difference, Rotting and Romwalter, Sopron, Hungary, Jackon, F.H., Q.J. Math. Oxford Ser., 1951). 5 Khan, M.A., Aif, M., Generalized Operational Method, Fractional Operator and Special Polynomial, Math. Sci. Re. J., 15 7) 011), Khan, M.A., Aif, M., Shift Operator On the Bae aa > 0, 1, ) and Peudo-Polynomial of Fractional Order, Int. J. of Math. Analyi, 5 3) 011),

20 34 mohammad aif, anju gupta 7 Khan, M.A., Aif, M., Shift Operator On the Bae aa > 0, 1, ) and Monomial Type Function, Int. Tran. in Math. Sci. and Comp., ) 009), Olver, P.J., Application of Lie Group to Differential Equation, Springer- Verlag, New York, Rainville, E.D., Special Function, Macmillan, New York, Accepted:

Bogoliubov Transformation in Classical Mechanics

Bogoliubov Transformation in Classical Mechanics Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How