tamil nadu tourism development corporation ( tamil : தமிழ்நாடு சுற்றுலா கழகம் ( வரையறுக்கப்பட்டது ) ) is a state-government public sector undertaking of the government of tamil nadu located in the indian state of tamil nadu .

tamil nadu has been entrusted with the promotion of tourism and development of tourist related infrastructure .

get quick answers from ttdc tamil nadu tourism development corporation - day tours staff and past visitors .
tamil nadu tourism complex , no . 2 wallajah road , chennai - 600002 .
the tamil nadu tourism development corporation ( ttdc ) ' s tirupati tour remains the favourite destination among the 30 - odd packages offered by it , seeing a 15 . 32 % increase in 2017 - 18 over the previous year ' s bookings .
employees of chandigarh industrial and tourism development corporation ( citco ) are up in arms against the decision of the management to privatise outlets .
tamilnadu tourism development corporation also operates various hotels , youth hostels , restaurants , tours , boat houses and telescope houses for the benefit of travellers .
contents owned and updated by concerned departments and coordinated by information technology department secretariat , fort st . george , chennai 600 009 , tamil nadu , india
the tourism development project , concentrating on the area based development ( abd ) zone comprising rockfort and vintage trichy , will spruce up six tourist spots including historical places .
ttdc or tamilnadu tourism development offers comfortable accommodation at reasonable cost at all tourist important centers in tamilnadu . basic facilities include
ttdc takes initiative to expose the abundant charms of tamil nadu to visitors from both india and abroad . it highlights the special tourist spots in tamilnadu that make the state so very popular with visitors . along with it also introduces new areas of exploration every now and then . for this ttdc or tamilnadu tourism development corporation offers numerous packages to travellers including sightseeing tours , inter state tours , ltc tours and train tours .
deputy chief minister sushil kumar modi said on monday hotel patliputra ashok of india tourism development corporation ( itdc ) would soon be handed over to state tourism department , which will either run it with its own resources or in public - private partnership ( ppp ) mode . the state would get 1 . 5 acre land from itdc as well as the hotel building .
tamilnadu tourism participated in patwa from 07 . 03 . 2018 - 11 . 03 . 2018
i am a great believer in the government tourism corporations and have always explored the conducted . . .
department of tamilnadu tourism ( t . t . d . c ) - offer the travelers a spectacular destination of ancient wisdom bestowed with eternal grace of nature ! come and explore the true beauty of this state with the tamilnadu department of tourism .
- - head of income - - business income or income from house property special right or privilege to franchisees - - the assessee was a wholly owned government of tamil nadu undertaking engaged in the business of development of tourism in the state . it was stated by the assessee before the ao that the assessee in its meeting , had resolved to lease / franchise uneconomic units to avoid further cash loss and to improve the existing units significantly . based on the government order , the assessee had called for open tenders for leasing the hotels . hence , the assessee claimed it as a lease income from house property . the ao rejected the submissions of the assessee and disallowed the deduction claimed at 30 per cent , by the assessee under the head
income from house property
holding that the assessee had given the hotel premises to private parties on franchise only , which was one of the activities of the regular / ordinary business of the assessee . while completing the assessment , the ao disallowed the statutory deduction of 30 per cent . the cit ( a ) granted the benefit in favour of the assessee and allowed the appeals . the tribunal came to the conclusion that what was derived as income out of the property was business income and not income out of house property and , hence , the assessee was not entitled to 30 per cent , deduction . accordingly , the tribunal allowed the appeals filed by the revenue . held : the assessee corporation continues to be in the business of tourism activities , though not directly but through the franchisees and has received income as franchisee fee and that cannot be lost sight of while determining the nature of income , hence assessable as business income , and not income from house property .
odisha chief secretary a p padhi held talks with the additional member of railways ( tourism and catering ) , sanjiv garg yesterday about the possibility of launching a luxury train service , the official said .
from the various clauses in the agreement , the assessee has not leased out the property merely as a land and building but also with further conditions as to how the business of the lessees / franchisees should be conducted with regard to the hotel industry only . the special conditions of contract make it clear that the name of the assessee should clearly , in fact , prominently be indicated in the name board and the name of the franchisee should be below the name of ttdc , thereby , making it clear that the ttdc continues to be operating their business through their franchisees . thus , the special conditions make it clear that the assessee - corporation continues to be in the business of tourism activities , though not directly but through the franchisees and has received income as franchisee fee and that cannot be lost sight of while determining the nature of income . [ para 13 ] there is also much force in the findings of the tribunal that the properties let out by the assessee are business assets of the assessee corporation , as they have not treated the leased out properties as non - business assets of the assessee - corporation . [ para 14 ] the assessee has given a special right or privilege to the franchisees to undertake a particular business in the property of the assessee on receipt of franchisee fee . therefore , the income is clearly in the nature of business income and not income from house property . [ para 16 ] the reasoning of the tribunal is justified on the facts of the case . accordingly , there is no question of law , much less any substantial question of law arises for consideration in the above tax case ( appeals ) . [ para 17 ]
well i would say that this is a really good tour considering it is a government organised one . i would say the tourism office staff is very helpful and i appreciated their help alot . i had booked online a half day tour of chennai and . . .
the one day tirumala tirupati tour organized by tamilnadu tourism by ac volvo bus is one of the best way to have darshan at tirumal tirupati . the tour starts at 5 . 30am to 6 . 00am boarding at wallajah road - ttdc office . breakfast is served on the . . .
i am a great believer in the government tourism corporations and have always explored the conducted tour packages offered by them . last december my bigger family ( parents , in - laws , wife , kids and self ) booked for the 8 days tamilnadu tour organized by ttdc . we made the . . .
is a tourist ' s paradise . a rich culture . a regal tapestry of history . a nature ' s bliss of blue beaches and clear skies . . . . . a modern state . to get a full view of this enchanting state , you are most welcome to tamilnadu any time . the tamilnadu department of tourism is here to help you and to make your holidays very special .
the one day tirumala tirupati tour organized by tamilnadu tourism by ac volvo bus is one of the best way to have darshan at tirumal tirupati . the tour starts at 5 . 30am to 6 . 00am boarding at wallajah road - ttdc office . breakfast is served on the way to the tirumala tirupati and there is a dedicated guide in the bus . we are lucky since we had a nice gentleman who coordinated the whole process of lord balaji darshan well . we reached tirumal by 10 . 30am by transferring into tirumala bus at tirupati . the transfer was smooth as the ghat vehicle was made ready exactly . upon reaching tirumala , time was given to the passengers for tonsuring and the whole group went for darshan around 11 . 30am . the guide made all the necessary arrangements and was very good . we entered the vaikumtum complex and lord balaji darshan was completed in 1 hour since we are lucky due to less crowd . lunch was offered the darshan at a decent hotel and then bus went the lord padmavathy thayar temple for darshan . after completing the darshan , we proceed to chennai and arrived back at chennai around 9pm . the whole tour was organized in a excellent manner .
well i would say that this is a really good tour considering it is a government organised one . i would say the tourism office staff is very helpful and i appreciated their help alot . i had booked online a half day tour of chennai and it covered fort st . george , govt . museum , valluvarkottam , snake park , kapaleeswarar temple and marina beach . i am glad i booked it since in half day on our own we could not have covered all these areas . the guide did not even rush us and we over all had a good experience . yes i agree that the guide can brush up the skills a bit and can tell us about areas which we cross while travelling rather than we having to ask him but still over all very good . we then booked a day tour to kanchipuram & mamallapuram via taxi as group tour on our date was not available . again the driver was driving safely and covered all areas and there was no problem except that if he spoke a bit more of english few avoided hassles like locating him could be avoided and he could have been helpful with giving us information . but over all there were no usual problems of over charging or going to places of commissions or anything like that . we had a good safe trip .
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; кzињa э + ф9 љqn < ѓае9яй ? p ' юыьlыў ў {  ; b ќnnbљі2фx > n ggq3њ & m ; о5ffќюd2w2iеижтн & фєj эћхфdъg , ђ >  * гвj ) r $ ak hhјщ бдфsg ~ # - giz0 - qгуzлf ) l№ ^ ь3фш @ иrшы \ тьіљэsю ? pohиѕѕнmnфюми
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 * kt аѓг ѕhа uв ] \ иiьgvѕаa \ ём ѓ6тл *  < qj3t
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we really had good service by ttdc while visiting thrupiti . bus is fine . well maintained . the guide is . . .
we really had good service by ttdc while visiting thrupiti . bus is fine . well maintained . the guide is good and responsible .
we appreciated the bus ride and the trips to places in chennai we never would have visited otherwise . very cheap and fun to travel with locals . the guide is one star .
this is the university ; this is the war memorial ( which war we never found . . .
ttdc tours are for those who are looking to save some money . we undertook 3 day trips - - tirupati , chennai - mahabalipuram and half day chennai tour . some areas that need immediate improvement are - - guides should be better trained , food served ( wherever included in ticket ) should . . .
this is a one day ex - chennai tour which covers only a few tourist points in kanchipuram and mamallapuram as enumerated below : kanchipuram 1 . enkaleswar temple 2 . kanchipuram kamatchi temple 3 . varadarajan temple mamallapuram 4 . shore temple 5 . five chariots temple 6 . muttukadu boating due to . . .
the rooms were spacious and clean . . . the staff courteous . . . the fans , ac , water coolers , etc all working . . . nice cozy bed . . . what else do you need in a room ! ? 10 / 10 recommended . . . and made a point to book these before locating any oyo or goibibo . . . . these are cheaper and better !
1 day divya desam tour is surprising me . the tour is performed saturday .
the tour guide explained each temple history which is quite interesting . with in this short period of time 1 day , we visited more perumal temples around kancheepuram with bonus visit to shiva temple egambareswarar
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hello , i wanted to know what are the pick up points for the ttdc one day chennai mamallapuram tour . kindly help . .
the pick up point are two - one originating point at ttdc office and other near to motel highway near poonamallee high road . please check with ttdc office .
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tamilnadu hotel ooty room is fine & clean . room service is best . no problem for hot water drinking & both
we took a one day package tour to thirumala on 12th aug ' 16 . we were into a pleasant surprise right from the beginning 1 . the staff in the counter in were very courteous and helpful 2 . the ac bus was very clean with comfortable seats 3 . the best part of the tour was the tour guide mr . karthick who took care of all our needs .
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tax case ( appeal ) nos . 341 & 342 of 2014 & m . p . no . 1 of 2014
k vijayaraghavan for the appellant . g . vardhini karthick for t . raxrikumar , standing counsel for income - tax , for the respondent .
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1 . avail 15 % discount upon continuous stay in a minimum of 3 tamilnadu hotels .
2 . avail reimbursement of one way auto fare for one day stay & up & down auto fare for 2 and more days stay at any of the ttdc hotels .
4 . honey moon couples may avail special honey moon packages with maximum discount at all hill station hotels and selected other hotel tamilnadu .
no . 2 , dr . nanjappa road ( opp . central bus stand ) ,


in mathematics , katugampola fractional operators are integral operators that generalize the riemann – liouville and the hadamard fractional operators into a unique form .

the katugampola fractional integral generalizes both the riemann – liouville fractional integral and the hadamard fractional integral into a single form and it is also closely related to the erdelyi – kober operator that generalizes the riemann – liouville fractional integral .

katugampola fractional derivative has been defined using the katugampola fractional integral and as with any other fractional differential operator , it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators .

katugampola fractional operators

katugampola , un : new approach to generalized fractional integral . appl . math . comput .
this manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives . we introduce new fractional integration and differentiation operators . we define spaces and present some theorems related to these operators .
hua chen , and udita n . katugampola , “hermite–hadamard and hermite–hadamard–fejér type inequalities for generalized fractional integrals , ”
katugampola , un : a new approach to generalized fractional derivatives . bull . math . anal . appl .
giorgio s . taverna , and delfim f . m . torres , “generalized fractional operators for nonstandard lagrangians , ”
in this paper we present a generalization to two existing fractional integrals and derivatives , namely , the riemann - liouville and hadamard fractional operators . the existence and uniqueness results for single term fractional differential equations ( fde ) have also been established . we also obtain the mellin transforms of such generalized fractional operators which are important in solving fractional differential equations .
abdeljawad , t , baleanu , d : monotonicity results for fractional difference operators with discrete exponential kernels . adv . differ . equ .
ryan m . evans , udita n . katugampola , and david a . edwards , “applications of fractional calculus in solving abel - type integral equations : surface–volume reaction problem , ”
this paper was devoted to an investigation of the fractional derivatives and integrals obtained by iterating conformable integrals . we obtained left - and right - fractional conformable integrals . with a standard fractional procedure we found left - and right - fractional conformable derivatives in the sense of riemann - liouville and caputo . we proved that these fractional integrals and derivatives have properties similar to the standard fractional integrals and derivatives . we also define the fractional derivatives of functions belonging to specific spaces in order to find the relation between these new fractional differential operators . the presented left - and right - fractional integrals are different from those defined by katugampola since their kernels depend on the end points a and b and hence need a different convolution theory when the conformable laplace is applied .
gao , f , yang , xj : fractional maxwell fluid with fractional derivative without singular kernel . therm . sci .
respectively . we suggest that iterating the above integral will end up with new fractional operators with two parameters and kernels different from the usual kernels of usual fractional derivatives and integrals . from the data analysis point of view we suppose that this new type of calculus will provide better understanding of the complexity of the dynamics of the phenomena from porous media .
] . as is well known , systems possessing a memory effect often appear in real world phenomena . however , for each type of data we always ask what is the optimal corresponding nonlocal model to be applied . moreover , many authors studied new fractional operators with local , nonlocal , singular and non - singular kernels ( see [
for the right - fractional conformable derivatives , we can state a similar lemma .
podlubny , i : fractional differential equations . academic press , san diego ( 1999 )
kilbas , aa : hadamard type fractional calculus . j . korean math . soc .
, we define the left - and right - fractional conformable integrals and derivatives . in section
magin , rl : fractional calculus in bioengineering . begell house publishers , redding ( 2006 )
abdeljawad , t : on conformable fractional calculus . j . comput . appl . math .
we study incommensurate fractional variational problems in terms of a generalized fractional integral with lagrangians depending on classical derivatives and generalized fractional integrals and derivatives . we obtain necessary optimality conditions for the basic and isoperimetric problems , transversality conditions for free boundary value problems , and a generalized noether type theorem .
] , in what follows , we recall some basic definitions and tools about classical fractional calculus .
hilfer , r : applications of fractional calculus in physics . word scientific , singapore ( 2000 )
, we define the fractional conformable derivatives of functions belonging to certain spaces and state their properties . in section
matheus j . lazo , and delfim f . m . torres , “variational calculus with conformable fractional derivatives , ”
in this section , we consider the fractional conformable derivatives of functions belonging to spaces stated in the following definitions .
] and the references therein ) . the standard fractional calculus may not provide us the required kernel in order to extract important information from these types of systems . at this stage , we ask the following question . can we generalize the standard fractional riemann - liouville integrals in a way such that we obtain unification to riemann - liouville , hadamard and other fractional derivatives [
before we state the inverse properties and the composition of two caputo fractional conformable derivatives , we shall consider the following lemmas .
el - nabulsi , ra , torres , dfm : fractional action - like variational problems . j . math . phys .
in this section we define the left - and right - fractional conformable derivatives in the sense of caputo and present their properties .
a . b . malinowska , and d . f . m . torres , “fractional calculus of variations of several independent variables , ”
shakoor pooseh , ricardo almeida , and delfim f . m . torres , “discrete direct methods in the fractional calculus of variations , ”
dina tavares , ricardo almeida , and delfim f . m . torres , “caputo derivatives of fractional variable order : numerical approximations , ”
atangana , a , baleanu , d : new fractional derivative with non - local and non - singular kernel . therm . sci .
a generalized fractional calculus of variations - control and cybernetics - tom vol . 42 , no . 2 ( 2013 ) - baztech - yadda
tatiana odzijewicz , agnieszka b . malinowska , and delfim f . m . torres , “variable order fractional variational calculus for double integrals , ”
tatiana odzijewicz , agnieszka b . malinowska , and delfim f . m . torres , “green ' s theorem for generalized fractional derivatives , ”
michael a . zhuravkov , and natalie s . romanova , “review of methods and approaches for mechanical problem solutions based on fractional calculus , ”
roberto garra , giorgio s . taverna , and delfim f . m . torres , “fractional herglotz variational principles with generalized caputo derivatives , ”
we present the fractional conformable derivatives in the caputo setting and state their properties . finally , the last section is devoted to our conclusion .
caputo , m , fabrizio , m : a new definition of fractional derivative without singular kernel . prog . fract . differ . appl .
losada , j , nieto , jj : properties of a new fractional derivative without singular kernel . prog . fract . differ . appl .
in the area of fractional calculus and its applications in many branches of science and engineering , several fractional derivatives were mainly utilized . the most common used were caputo and riemann - liouville derivatives , which were successfully utilized in modeling complex dynamics appearing in physics , biology , engineering and many other fields [
jarad , f , abdeljawad , t , baleanu , d : caputo - type modification of the hadamard fractional derivative . adv . differ . equ .
gastao s . f . frederico , and delfim f . m . torres , “fractional noether ' s theorem with classical and riemann - liouville derivatives , ”
samko , sg , kilbas , aa , marichev , oi : fractional integrals and derivatives : theory and applications . gordon & breach , yverdon ( 1993 )
yang , xj , gao , f , machado , jat , baleanu , d : a new fractional derivative involving the normalized sinc function without singular kernel .
jarad , f , abdeljawad , t , baleanu , d : on the generalized fractional derivatives and their caputo modification . j . nonlinear sci . appl .
mohammad maleki , ishak hashim , saeid abbasbandy , and a . alsaedi , “direct solution of a type of constrained fractional variational problems via an adaptive pseudospectral method , ”
abdeljawad , t , baleanu , d : on fractional derivatives with exponential kernel and their discrete versions . j . rep . math . phys . ( to appear )
gambo , yy , jarad , f , abdeljawad , t , baleanu , d : on caputo modification of the hadamard fractional derivative . adv . differ . equ .
kilbas , a , srivastava , hm , trujillo , jj : theory and application of fractional differential equations . north - holland mathematics studies , vol . 204 ( 2006 )
hegagi m . ali , fernando lobo pereira , and sãlvio m . a . gama , “a new approach to the pontryagin maximum principle for nonlinear fractional optimal control problems , ”
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] . the core of this procedure is to decide which differentiation operator should be used as a starting point for the iteration procedure . for the standard fractional calculus , we iterate the usual integral of a function and using the cauchy formula we obtain the integral of higher integer orders and then replace this integer by any complex number . in [
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tatiana odzijewicz , 1 agnieszka b . malinowska , 2 and delfim f . m . torres 1
copyright © 2012 tatiana odzijewicz et al . this is an open access article distributed under the creative commons attribution license , which permits unrestricted use , distribution , and reproduction in any medium , provided the original work is properly cited .
the following is the list of published articles that have cited the current article .
delfim f . m . torres , and agnieszka b . malinowskapp . 1–275 , 2012 .
rajesh k . pandey , and om p . agrawal , “numerical scheme for generalized isoperimetric constraint variational problems with a - operator , ”
rajesh k . pandey , and om p . agrawal , “numerical scheme for a quadratic type generalized isoperimetric constraint variational problems with a - operator , ”
] , it was suggested that the conformable integral should be fractionalized properly . we recall that an integral type like the one from [
$ $ \ bigl ( \ mathbf { i } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( \ alpha ) } \ int _ { t } ^ { b } ( y - t ) ^ { \ alpha - 1 } f ( y ) \ , dy . $ $
$ $ \ bigl ( { } _ { a } d ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( { } _ { a } \ mathbf { i } ^ { n - \ alpha } f \ bigr ) ( t ) , \ quad n = [ \ alpha ] + 1 . $ $
$ $ \ bigl ( d _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( - \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( \ mathbf { i } _ { b } ^ { n - \ alpha } f \ bigr ) ( t ) . $ $
$ $ \ bigl ( { } ^ { c } d _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ bigl ( \ mathbf { i } _ { b } ^ { n - \ alpha } ( - 1 ) ^ { n } f ^ { ( n ) } \ bigr ) ( t ) . $ $
$ $ \ bigl ( \ mathfrak { j } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( \ alpha ) } \ int _ { t } ^ { b } ( \ ln y - \ ln t ) ^ { \ alpha - 1 } f ( y ) \ frac { dy } { y } . $ $
$ $ \ bigl ( \ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( - t \ frac { d } { dt } \ biggr ) ^ { n } \ bigl ( i _ { b } ^ { n - \ alpha } f \ bigr ) ( t ) . $ $
$ $ \ bigl ( { } ^ { c } \ mathfrak { d } _ { b } ^ { \ alpha } f \ bigr ) ( t ) = \ biggl ( \ mathfrak { j } _ { b } ^ { n - \ alpha } \ biggl ( - t \ frac { d } { dt } \ biggr ) ^ { n } f \ biggr ) ( t ) . $ $
$ $ x _ { c } ^ { p } ( a , b ) = \ biggl \ { f : [ a , b ] \ rightarrow \ mathbb { r } : \ | f \ | _ { x _ { c } ^ { p } } = \ biggl ( \ int _ { a } ^ { b } \ bigl \ vert t ^ { c } f ( t ) \ bigr \ vert ^ { p } \ frac { dt } { t } \ biggr ) ^ { 1 / p } < \ infty \ biggr \ } . $ $
$ $ \ bigl ( { } ^ { c } d _ { b } ^ { \ alpha , \ rho } f \ bigr ) ( t ) = \ bigl ( { } _ { a } i ^ { n - \ alpha , \ rho } ( - \ gamma ) ^ { n } f \ bigr ) ( t ) = \ frac { 1 } { \ gamma ( n - \ alpha ) } \ int _ { t } ^ { b } \ biggl ( \ frac { y ^ { \ rho } - t ^ { \ rho } } { \ rho } \ biggr ) ^ { n - \ alpha - 1 } ( - \ gamma ) ^ { n } f ( y ) \ frac { dy } { y ^ { 1 - \ rho } } . $ $
$ $ \ begin { aligned } \ begin { aligned } [ b ] { } _ { a } i ^ { n , \ alpha } f ( x ) & = \ int _ { a } ^ { x } \ frac { dt _ { 1 } } { ( t _ { 1 } - a ) ^ { 1 - \ alpha } } \ int _ { a } ^ { t _ { 1 } } \ frac { dt _ { 2 } } { ( t _ { 2 } - a ) ^ { 1 - \ alpha } } \ cdots \ int _ { a } ^ { t _ { n - 1 } } \ frac { f ( t _ { n } ) \ , dt _ { n } } { ( t _ { n } - a ) ^ { 1 - \ alpha } } \ \ & = \ frac { 1 } { \ gamma ( n ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } f ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . \ end { aligned } \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } f ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . $ $
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } f ( t ) \ frac { dt } { ( b - t ) ^ { 1 - \ alpha } } . $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigr ) f ( x ) = { } _ { a } ^ { \ beta + \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) , \ qquad { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigr ) f ( x ) = { } ^ { \ beta + \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ z = \ frac { ( t - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { ( x - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } . $ $
the second formula can be proved in a similar way or by using the action of the q - operator . □
$ $ \ begin { aligned } & \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } ( b - t ) ^ { \ alpha \ nu - \ alpha } \ bigr ) ( x ) = \ frac { 1 } { \ alpha ^ { \ beta } } \ frac { \ gamma ( \ nu ) } { \ gamma ( \ beta + \ nu ) } ( b - x ) ^ { \ alpha ( \ beta + \ nu - 1 ) } . \ end { aligned } $ $
$ $ \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } ( t - a ) ^ { \ nu - 1 } \ bigr ) ( x ) = \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } ( t - a ) ^ { \ alpha \ nu - \ alpha } \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } . $ $
$ $ \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } ( t - a ) ^ { \ nu - 1 } \ bigr ) ( x ) = \ frac { ( x - a ) ^ { \ alpha ( \ beta + \ nu - 1 ) } } { \ gamma ( \ beta ) \ alpha ^ { \ beta } } \ int _ { 0 } ^ { 1 } ( 1 - u ) ^ { \ beta - 1 } u ^ { \ nu - 1 } \ , du = \ frac { \ gamma ( \ nu ) } { \ alpha ^ { \ beta } \ gamma ( \ beta + \ nu ) } ( x - a ) ^ { \ alpha ( \ beta + \ nu - 1 ) } . $ $
$ $ \ begin { aligned } & \ bigl [ { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } ( b - t ) ^ { \ alpha \ nu - \ alpha } \ bigr ] ( x ) = \ alpha ^ { \ beta } \ frac { \ gamma ( \ nu ) } { \ gamma ( \ nu - \ beta ) } ( b - x ) ^ { \ alpha ( \ nu - \ beta - 1 ) } . \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { - \ alpha } f , \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } = { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { - \ alpha } f . $ $
$ $ \ frac { d } { dx } \ bigl ( { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( x ) \ bigr ) = \ frac { 1 } { ( x - a ) ^ { 1 - \ alpha } } \ int _ { a } ^ { x } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } + \ frac { { } _ { a } ^ { n - 1 } t ^ { \ alpha } f ( a ) } { ( x - a ) ^ { 1 - \ alpha } } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( x ) = & \ int _ { a } ^ { x } \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + { } _ { a } ^ { n - 1 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { n - 3 } t ^ { \ alpha } f ( x ) = & \ frac { 1 } { 2 ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { 2 } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { 2 \ alpha } } { 2 \ alpha ^ { 2 } } \ \ & { } + { } _ { a } ^ { n - 2 } t ^ { \ alpha } f ( a ) \ frac { ( x - a ) ^ { \ alpha } } { \ alpha } + { } _ { a } ^ { n - 3 } t ^ { \ alpha } f ( a ) . \ end { aligned } $ $
$ $ \ begin { aligned } f ( x ) = & \ frac { 1 } { ( n - 1 ) ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ psi ( t ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k } k ! } ( x - a ) ^ { \ alpha k } . \ end { aligned } $ $
$ $ \ begin { aligned } f ( x ) = & \ frac { 1 } { ( n - 1 ) ! } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ frac { ( { } ^ { n } t _ { b } ^ { \ alpha } f ) ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) } { k ! } \ frac { ( b - x ) ^ { \ alpha k } } { \ alpha ^ { k } } . \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) = & \ frac { 1 } { \ gamma ( n - \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) ) ( b - x ) ^ { \ alpha ( k - \ beta ) } } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . \ end { aligned } $ $
$ $ f ( x ) = \ frac { 1 } { ( n - 1 ) ! } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - 1 } \ frac { ( { } _ { a } ^ { n } t ^ { \ alpha } f ) ( t ) } { ( t - a ) ^ { 1 - \ alpha } } \ , dt + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { k ! } \ frac { ( x - a ) ^ { \ alpha k } } { \ alpha ^ { k } } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) = & \ frac { { } _ { a } ^ { n } t ^ { \ alpha } } { ( n - 1 ) ! \ gamma ( n - \ beta ) } \ int _ { a } ^ { x } \ int _ { a } ^ { t } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ biggl ( \ frac { ( t - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n } \ \ & { } \ times \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( u ) \ bigr ) \ frac { du } { ( u - a ) ^ { 1 - \ alpha } } \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { k ! \ alpha ^ { k } } \ alpha ^ { \ beta } \ frac { \ gamma ( k + 1 ) } { \ gamma ( k + 1 - \ beta ) } ( x - a ) ^ { \ alpha ( k - \ beta ) } . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) = & \ biggl ( \ frac { \ gamma ( n - \ beta ) \ gamma ( n ) { } _ { a } ^ { n } t ^ { \ alpha } } { ( n - 1 ) ! \ gamma ( n - \ beta ) \ gamma ( 2n - \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( u - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { 2n - \ beta - 1 } \ frac { ( { } _ { a } ^ { n } t ^ { \ alpha } f ) ( u ) \ , du } { ( u - a ) ^ { 1 - \ alpha } } \ biggr ) \ \ & { } + \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } \ gamma ( k + 1 - \ beta ) } ( x - a ) ^ { \ alpha ( k - \ beta ) } . \ end { aligned } $ $
$ $ { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta - m } \ mathfrak { i } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { m } t _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = { } ^ { \ beta - m } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = ^ { \ beta - \ gamma } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ gamma } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { m - \ gamma } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) \ bigr ) \ \ & = { } _ { a } ^ { m } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta + m - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) \ \ & = { } _ { a } ^ { \ beta - \ gamma } \ mathfrak { i } ^ { \ alpha } f ( x ) . \ end { aligned } $ $
$ $ { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) ; \ qquad { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) . $ $
and utilized the properties of the gamma and beta functions . the last step in the proof is to use lemma 2 . 1 in [
$ $ { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { ( - 1 ) ^ { n - j } \ , { } ^ { \ beta - j } \ mathfrak { d } _ { b } ^ { \ alpha } f ( b ) } { \ alpha ^ { \ beta - j } \ gamma ( \ beta - j + 1 ) } ( b - x ) ^ { \ alpha \ beta - \ alpha j } . $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = & \ frac { 1 } { \ gamma ( \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta - 1 } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( t ) \ bigr ) \ bigr ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ \ = & \ frac { { } _ { a } ^ { 1 } t ^ { \ alpha } } { \ gamma ( \ beta + 1 ) } \ biggl [ \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { \ beta } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( t ) \ bigr ) \ bigr ) \ frac { dt } { ( t - a ) ^ { 1 - \ alpha } } \ biggr ] . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = & { } _ { a } ^ { 1 } t \ biggl [ { } _ { a } ^ { 1 } \ mathfrak { i } ^ { \ alpha } f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { ( { } _ { a } ^ { n - j } t ^ { \ alpha } ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } f ( a ) ) ) } { \ gamma ( \ beta + 2 - j ) \ alpha ^ { \ beta - j + 1 } } ( x - a ) ^ { \ alpha \ beta - \ alpha j + \ alpha } \ biggr ] \ \ = & f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { \ beta - j } \ mathfrak { d } ^ { \ alpha } f ( a ) } { \ alpha ^ { \ beta - j } \ gamma ( \ beta - j + 1 ) } ( x - a ) ^ { \ alpha \ beta - \ alpha j } . \ end { aligned } $ $
$ $ { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) = { } ^ { \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ biggl [ f ( t ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) } { k ! \ alpha ^ { k } } ( b - t ) ^ { \ alpha k } \ biggr ] ( x ) . $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) & = \ frac { ( - 1 ) ^ { n } } { \ gamma ( n - \ beta ) } \ int _ { x } ^ { b } \ biggl ( \ frac { ( b - x ) ^ { \ alpha } - ( b - t ) ^ { \ alpha } } { \ alpha } \ biggr ) ^ { n - \ beta - 1 } \ frac { ^ { n } t _ { b } ^ { \ alpha } f ( t ) } { ( b - t ) ^ { 1 - \ alpha } } \ , dt \ \ & = { } ^ { n - \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( ^ { n } t _ { b } ^ { \ alpha } f ( x ) \ bigr ) . \ end { aligned } $ $
$ $ \ begin { aligned } { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } k ! } \ frac { \ gamma ( k + 1 ) } { \ gamma ( k - \ beta + 1 ) } ( x - a ) ^ { k \ alpha - \ beta \ alpha } \ \ & = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ^ { k } t _ { a } ^ { \ alpha } f ( a ) } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } ( x - a ) ^ { k \ alpha - \ beta \ alpha } \ \ & = \ frac { 1 } { \ gamma ( n - \ beta ) } \ int _ { a } ^ { x } \ biggl ( \ frac { ( x - a ) ^ { \ alpha } - ( t - a ) ^ { \ alpha } } { \ rho } \ biggr ) ^ { n - \ beta - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( t ) } { ( t - a ) ^ { 1 - \ alpha } } \ , dt \ \ & = { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) . \ end { aligned } $ $
$ $ \ bigl \ vert { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr \ vert \ leq \ frac { \ vert f \ vert _ { c } } { \ vert \ gamma ( \ beta - k ) \ vert ( \ operatorname { re } ( \ beta ) - k ) } \ frac { ( x - a ) ^ { \ alpha ( \ operatorname { re } ( \ beta ) - k ) } } { \ alpha ^ { \ operatorname { re } ( \ beta ) - k } } . $ $
$ $ \ bigl \ vert ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr \ vert \ leq \ frac { \ vert ^ { n } t _ { b } ^ { \ alpha } f \ vert _ { c } } { \ vert \ gamma ( n - \ beta ) \ vert ( n - \ operatorname { re } ( \ beta ) ) } \ frac { ( b - x ) ^ { \ alpha ( n - \ operatorname { re } ( \ beta ) ) } } { \ alpha ^ { n - \ operatorname { re } ( \ beta ) } } . $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) ; \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) . $ $
$ $ \ begin { aligned } & { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ frac { { } ^ { \ beta + 1 - n } \ mathfrak { i } _ { b } ^ { \ alpha } f ( b ) } { \ alpha ^ { n - \ beta } \ gamma ( n - \ beta ) } ( b - x ) ^ { \ alpha n - \ alpha \ beta } . \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { k } t ^ { \ alpha } ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( a ) ) ( x - a ) } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { \ beta - k } \ mathfrak { i } ^ { \ alpha } f ( a ) ( x - a ) ^ { \ alpha - k - \ beta \ alpha } } { \ alpha ^ { k - \ beta } \ gamma ( k - \ beta + 1 ) } . $ $
) is proved . the second identity can be proved by using the same arguments .
$ $ \ begin { aligned } & { } ^ { \ beta } \ mathfrak { i } _ { b } ^ { \ alpha } \ bigl ( ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { ( - 1 ) ^ { k } \ , { } ^ { k } t _ { b } ^ { \ alpha } f ( b ) ( b - x ) ^ { \ alpha k } } { \ alpha ^ { k } k ! } . \ end { aligned } $ $
$ $ \ begin { aligned } { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) & = { } _ { a } ^ { \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n - \ beta } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) \ bigr ) \ \ & = { } _ { a } ^ { n } \ mathfrak { i } ^ { \ alpha } \ bigl ( { } _ { a } ^ { n } t ^ { \ alpha } f ( x ) \ bigr ) \ \ & = f ( x ) - \ sum _ { j = 1 } ^ { n } \ frac { { } _ { a } ^ { n - j } d ^ { \ alpha } f ( a ) } { \ alpha ^ { n - j } \ gamma ( n - j + 1 ) } ( x - a ) ^ { ( n - j ) \ alpha } \ \ & = f ( x ) - \ sum _ { k = 0 } ^ { n - 1 } \ frac { { } _ { a } ^ { n } t ^ { \ alpha } f ( a ) } { \ alpha ^ { k } k ! } ( x - a ) ^ { k \ alpha } . \ end { aligned } $ $
$ $ { } _ { a } ^ { c \ beta } \ mathfrak { d } ^ { \ alpha } \ bigl ( { } _ { a } ^ { c \ gamma } \ mathfrak { d } ^ { \ alpha } f ( x ) \ bigr ) = { } _ { a } ^ { c ( \ beta + \ gamma ) } \ mathfrak { d } ^ { \ alpha } f ( x ) ; \ qquad { } ^ { c \ beta } \ mathfrak { d } _ { b } ^ { \ alpha } \ bigl ( ^ { c \ gamma } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) \ bigr ) = ^ { c ( \ beta + \ gamma ) } \ mathfrak { d } _ { b } ^ { \ alpha } f ( x ) . $ $
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jarad , f . , uğurlu , e . , abdeljawad , t . et al . adv differ equ ( 2017 ) 2017 : 247 . urltoken