CLASSES OF TAYLOR MACLOURIN TYPE FORMULAE IN COMPLEX DOMAIN. 1. Introduction. In [1] the authors defined a system of functions ρ

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1 PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physcal ad Mathematcal Sceces 2, 2, p. 3 Mathematcs CLASSES OF TAYLOR MACLOURIN TYPE FORMULAE IN COMPLEX DOMAIN B. A. SAHAKYAN Char of Mathematcal Aalyss, YSU I the preset paper some systems of operators geerated by Rema Louvlle tegral ad dervatve of orders α (,), ad fuctos geerated by Mttag Leffler type fuctos are troduced. Some propertes of these systems are vestgated ad for a certa class of fuctos the geeralatos of Taylor Maclour type formulae are obtaed. Keywords: Rema Louvlle operators, Taylor Maclours type formulae.. Itroducto. I [] the authors defed a system of fuctos xξ ( ) e ξ ε ( x) dξ,, x (, ), 2π γ( ε; β) ( ξ λ ) as well as a systems of operators assocated wth the Rema Louvlle tegrodfferetal operators D γ / / / : L ( f) f( x), L f( x) ( D λ ) f( x), / / L f( x) D L f( x),, where, α /, λ λ,,,..., γ ( εβ ; ) s a cotour plae ξ ([2], p. 26). ( The fuctos ε ) ( x) admt represetatos of the form (see, []) ( ) ( ) ( S ) / ( x) C E ( x ;/ ) x ε λ,, where S( S ) s the multplcty, wth whch the umber λ appears o the terval { } ( ) λ of our sequece. Coeffcets { C } are determed from a ( ) Г( S ) C ( ξ λ ) S expaso R( ξ) ( ξ λ ), ad E (; μ), Г μ >, s the order etre fucto of Mttag Leffler type for ay value of parameter μ ([2], chap. VI, ). E-mal: maeat@rambler.ru

2 4 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p. 3. ( It s also ow [] that the fuctos ε ) ( x),, admt represetato of the form x t ( ) ( ) x x e xt dt e tt2 dt2 e t t e t dt / e( x; λ) E( λx ;/ ) x,, x (, ). ε () Ω () ( ; λ ) ( ; λ )... ( ; λ ) ( ; λ ), where Partcularly, [] the followg Taylor Maclours type formula was foud for some classes of fuctos: t x / ( ) ( ) ε ε f ( x) L f() ( x) ( xt) L f( t) dt, x (, l ). I the preset wor smlar problems are vestgated the complex doma. 2. Prelmares Iformato. Let f( x) L(, l) ( < l < ), α (, ). x α The fucto D f( x) ( xt) f( t) dt s called the Rema Louvlle Г( α) tegral of order α of fucto ( ) α, the fucto f x, ad for ( ] α d ( ) D f( x) D f( x) s called the Rema Louvlle dervatve of order α dx of fucto f ( x ). It s ow that lm D f( x) f( x) (almost everywhere) α all Lebeg pots x (, l) of the fucto f ( x ) ad, therefore, [ D f( x)] f( x) ad D ' f ( x ) f '( x ). α Let α [,), α,, x (, l). The operators D f( x) f( x), / d / D f( x) D f( x), D f( x) D D f( x), 2, are called Rema dx Louvlle operators of successve dfferetato of order / of fucto f ( x ) [2, 3]. π Itroduce some otatos. We deote Δ ; Arg <, < < 2. Ths doma for < < / 2 beg evdet by a mafold ad arraged o the Rema surface G of fucto L. H ( ) s the class of aalytc doma Δ fuctos f ( ). Let: (, l( ϕ)) { ;arg ϕ, < < }, π ϕ < π. Let α [,), α, ϕ ad f ( ) be a arbtrary fucto of a complex varable f( re ) L(, l( ϕ)), ϕ π, < r <. The fucto α D f( ) ( ξ ) f( ξ) dξ, (2.) Г( α)

3 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p where the tegrato s made alog the tercept coectg pots ad, α arg( ξ) ( α )arg, s called the Rema Louvlle tegral of order α of fucto f ( ), ad the fucto / d D f( ) D f( ) (2.2) d s called the Rema Louvlle dervatve of order / of fucto f ( ). The operators / / d D f( ) f( ), D f( ) D f( ), d (2.3) / D f( ) D D f( ), 2, are called Rema Louvlle operators of successve dfferetato of order / of fucto f ( ). The Mttag Leffler type fucto E (; μ), >, s a Г( μ ) etre fucto of order wth arbtrary value of parameter μ ([2], chap. VI, ). For ay μ >, α > the followg formula holds ([2], ch. III, formula (.6)): α / μ μ α / ( ξ ) E( λξ ; μ) ξ dξ E( λ ; μ α), Г( α) (2.4) ϕ ϕ re, ξ τe, < τ < r < l <, π ϕ < π. Lemma 2.. Let, λ, λ be arbtrary complex parameters. The formula holds (; ) (; ) ( ; ) ( ; ) e λ e e d e λ ξλ ξλ ϕ ξ, re, < r< l<, ϕ π, (2.5) λ λ / where e(; λ) E( λ ;/ ), but the tegrato s alog the segmet coectg ad pots. Note that for x (, ) ths Lemma was frst establshed more geeral form [4] (see also [2], ch. III, (.2)). 3. Ma Results. Let { λ } be a arbtrary creasg sequece of postve umbers. Cosder the sequece of operators o proper class of fuctos f ( ) for ay : / / / ϕ L f( ) f( ), L f( ) ( D λ ) f( ),, re, r < l <, ϕ π, [ ) / / (3.) L f( ) D L f( ), α ;, α,. (3.2)

4 6 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p. 3. ε ( ) The cosder the followg systems of fuctos { ε ( ) }, ( ) { Ω ( ) } ( ) e λ ( ) ( ; ), ( ) ( ) ( ) C e ε ( ) ε ( ; λ, λ,..., λ ) ( ; λ ), ( ) / where C ( λ λ), e(; λ) E( λ ;/ )., ( ) ( ) ( ) e Ω ( ) ( ; λ ), Ω ( ) Ω ( ; λ, λ,... λ ) t 2 2 t e ( t ; λ ) dt e ( t t ; λ ) dt... e ( t t ; λ ) e ( t ; λ ) dt,, ϕ ϕ ϕ : Δ, (3.3) (3.4) where re, t τe,..., t τe, < τ < τ <... < τ < r < l <. Note that for x (, ) operators (3.), (3.2) ad systems of fuctos (3.3), (3.4) were frst troduced []. More geeral systems of operators ad fuctos were troduced [5], ad for (,) smlar operators ad fuctos were troduced [6]. Lemma 3.. Let, α <. For ay the followg formula holds: { ( ) ( ε ( ; λ ), λ,..., λ, λ) ε ( ; λ, λ,..., λ, λ ) } λ λ (3.5) ( ) ε ( ; λ, λ,..., λ, λ ), < λ < λ,,,...,, Δ. ( ) Lemma 3.2. The systems of fuctos { ε ( ) } ad ( ) { Ω ( ) } assocated wth the gve sequece { λ } are detcal,.e. ad ( ) ( ) Ω ε ( ) ( ),, Δ. (3.6) Proof. Note that accordg to (3.3), (3.4) we have for ( ) ( ) / ( ) ( ) e ( ; ) E ( ;/ ) ε Ω λ λ. (3.7) Let. The ε () (; λ ) { (; λ ) (; λ )} (3.8) ( ) () C e e e λ λ ( ) ϕ ϕ ( ) e( t; ) e( t; ) dt, re, t e Ω λ λ τ. (3.9) Usg formula (2.9) we have ( ) Ω ( ) e( ; λ) e( ; λ) λ λ ( ) ( ) Ω { }, (3.).e. ε ( ) ( ). Now avalg of the methods of ductve reasog,.e. assumg that (3.6) holds for 2, we show that t s vald for. Accordg to (3.4), usg formula (2.9) ad Lemma 3. we have

5 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p t ( ) ( ) e( t; ) dt e( tt2; ) dt2... t e( t; λ) e( t; λ ) e t t λ dt λ λ t t Ω λ λ... ( ; ) e( t; λ) dt e( t t2; λ) dt2... e( t t; λ ) e( t; λ) dt λ λ t t e( t; λ) dt e( tt2; λ) dt2... e( t t; λ ) e( t; λ ) dt ( ) ( ) ( ε ( ; λ, λ,..., λ, λ) ε ( ; λ, λ,..., λ, λ ) ) λ λ ( ) ε ( ; λ, λ,..., λ, λ, λ ). Lemma 3.2 s proved. ϕ Lemma 3.3. Let, f( ) H( ), f( re ) L(; l( ϕ)), Δ. The for ay the formula holds t e ( t ; λ ) dt e ( t t ; λ ) dt... e ( t t ; λ ) dt 2 2 t ( ) e ( t t ; λ) f( t ) dt ε ( τ) f( τ) dτ, t τ e ϕ,, t τ e ϕ, τ τ... τ t (3.) where re ϕ, < < < < < r < l <. Lemma 3.3 s proved the same way as the proof of Lemma 4. for x (, ) []. Lemma 3.4. Let, f( ) H( ), f( re ) L(, l( ϕ)). The the fucto ( ) ε, y (;{ λ }) ( t) ftdt () s the soluto of the followg Cauchy type problem: ϕ re ϕ e ϕ, t τ, (3.2) L y( ) f( ), (3.3) / L y( ),,,...,. (3.4) The proof of Lemma 3.4 s ot gve here, because smlar problems for x (, ) have bee dscussed [] (see Lemma 3.2). Lemma For ay the followg relatos hold: / ( ) / ( ) D α { L [ ε ( )]} L [ ε ( )],,, Δ, (3.5) α / ( ) / ε λ D { L [ ( )]} E ( ;), Δ. (3.6) 2. For ay / ( ) lm D α { L [ ε ( )]},, Δ. (3.7)

6 8 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p. 3. L Proof.. Let, the accordg to (3.), (3.5) we have ( ) ( ) ( ) / ε λ λ λ. (3.8) L [ ( )] L C e ( ; ) C ( D ) e ( ; ) / But sce ( D λ) e ( ; λ) (see [7]), the from (3.8) we get Sce ( ) ( ) ( ). (3.9) L [ ε ( )] C ( λ λ ) e ( ; λ ) ( λ λ) for,,...,, t follows from (3.9) that [ ε ( )]. It s easy to see uder the assumpto of 2 that / ( ) / / ( ) ε λ λ ε L ( ) ( D ) ( D ) ( ) / ( ) λ ε ( D ) L ( ). / ( ) / ( ) Now ote that L ε ( ) D α L ε ( ),. Further we have / ( ) ( ) / ( ) ε λ λ λ λ λ L () C ( D ) e (; ) C ( ) e (; ) ( ) λ λ λ C ( ) e ( ; ), ( ) but sce C ( λ λ ) ( λ λ ) ( λ λ ), follows that (3.2) the from (3.2) t / ( ) / ε ( ) ( ; λ) ( λ ;/ ) L e E. (3.2) Usg formula (2.4) we obta from (3.2) / ( ) / ( ) / D α L ε ( ) L ε ( ) E ( λ ;). 2. Let ow,. The we have: { } / ( ) ( ) / D L ε ( ) D C ( D λ) e ( ; λ) ( ) ( ) D C ( λ λ) e(; λ) C ( λ λ) D { e(; λ)} (3.22) ( ) / C λ λ E λ ( ) ( ;). Note that / λ E ( ;), ad

7 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p ( ) ( ) ( ) ( ) ( ),,, C λ λ λ λ λ λ λ λ t follows from (3.22) / ( ) lm D { L [ ε ( )]} ( λ λ),, Lemma 3.5 s proved. Lemma 3.6. For ay a sum of the form (see [], (3.3)). ( ) ε P ( ) a ( ), Δ, (3.23) the coeffcets { a } may be determed from formulae / a L P(),,,...,. (3.24) / Proof. Let. Applyg the operator L to fucto P ( ) ad usg (3.5) (3.7), we obta / / ( ) / ( ) L P( ) al { ε ( )} al { ε ( x)} 3.25) / ( ) / / ( ) al { ε ( )} ae ( λ ;) al { ε ( )}. But sce E (;), the from (3.25) ad (3.7) we obta / / lm L P ( ) L P () a,. / The the operator L s appled to the fucto P ( ): L P ( ) a L { ( )} a L { ( )} a E ( ;). (3.26) / / ( ) / ( ) ε ε λ / / / a lm L P( ) L P(). ( ) C [; l( ϕ)], l, From here we obta Lemma 3.6 s proved. Now deote by < <, the set of fuctos f( ) H( ), satsfyg the followg codtos: / / ) the fuctos D f( ) D α D f( ),,,...,, are cotuous o [ ; l( ϕ )) ; 2) the fucto D ( )/ f( ) s cotuous o (; l( ϕ )) ad belogs to L(; l( ϕ )). It s easy to see that accordg to (3.), (3.2) ths fuctos / / L f( ) D L f( ),,,..., ; l( ϕ ) ad fucto L ( )/, are cotuous o [ ) f( ) s cotuous o [ ; l( ϕ )) ad belogs to L(; l( ϕ )). Theorem 3.. If ( f( ) C ) [ ; l( ϕ) ] >, the for ay / ( ) ε ϕ, (3.27) f( ) L f() ( ) R ( ; f), Δ (; l( ))

8 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p. 3. where ( ) ( )/ ϕ ϕ ε τ τ. (3.28) R (; f) ( t) L { f()} t dt, re, t e,< < r< l< Proof. Let We put / ( ) ε P (; f) L f() (). (3.29) f ( ) P( ; f) R( ; f). (3.3) / / L P ; f L f, It s easy to see that accordg to Lemma 3.6 ( ) ( ),,2,...,, ad cosequetly L / [ R ( ; f )],,,...,. Further, sce accordg to (3.5) L ( )/ [ ε ( ) ( )],,,2,...,, Δ, the ( )/ ( )/ L [ R ( ; f)] L f( ). Now cosder the problem: ( ) ( )/ ( )/ L R ; f L f( ), (3.3) / L R ( ; f ),,,...,. (3.32) Accordg to Lemma 3.4 ths problem (3.3) (3.32) has a uque soluto ( ) ( )/ R (; f) ε ( t) L f() t dt. It follows from formulae (3.28) (3.3) that / ( ) ( ) ( )/ ε ε f ( ) L f() ( ) ( t) L f( t) dt. Theorem 3. s proved. Receved REFERENCES. Dhrbashya M.M. ad Sahaya B.A. Iv. AN SSSR. Matematca., 975, v. 39,, p ( Russa). 2. Dhrbashya M.M. Itegral Trasforms ad Represetatos of Fuctos the Complex Doma. M.: Naua, 966 ( Russa). 3. Dhrbashya M.M. Iv. AN Arm. SSR. Matematca, 968, v. 3, 3, p ( Russa). 4. Dhrbashya M.M. ad Nersesya A.B. Iv. AN Arm. SSR. Phymat. au, 959, v. 2, 5, p ( Russa). 5. Dhrbashya M.M. ad Sahaya B.A. Iv. AN Arm. SSR. Matematca, 977, v. XII,, p ( Russa). 6. Sahaya B.A. Ucheye aps EGU, 988, 3, p. 3 4 ( Russa). 7. Sahaya B.A. Proceedgs of the YSU. Physcal ad Mathematcal Sceces, 2, 3, p

9 Proc. of the Yereva State Uv. Phys. ad Mathem. Sc., 2, 2, p. 3. Բ. Հ. Սահակյան Թեյլոր Մակլորենի տիպի բանաձևերի դասեր կոմպլեքս տիրույթում Աշխատանքում մուծվում են օպերատորների որոշ համակարգեր, որոնք ծնվում են Ռիման Լիուվիլի α (,) կարգի ինտեգրալով ու ածանցյալով, և ֆունկցիաներ, որոնք ծնվում են Միտագ Լեֆլերի տիպի ֆունկցիաներով: Ուսումնասիրվում են այդ համակարգերի և ֆունկցիաների որոշակի դասի մի շարք հատկություններ: Ստացվել են Թեյլոր Մակլորենի տիպի ընդհանրացված բանաձևեր: Б. А. Саакян. Классы формул типа Тейлора Маклорена в комплексной области В настоящей статье введены некоторые системы операторов, порожденные интегралом и производным Римана Лиувилля порядка α (,), и функции, порожденные функциями типа Миттага Леффлера. Изучены некоторые свойства этих систем и для определенного класса функций получены обобщения формул типа Тейлора Маклорена.