Applicble Anlysis nd Discrete Mthemtics, 27, 3 323. Avilble electroniclly t http://pefmth.etf.bg.c.yu Presented t the conference: Topics in Mthemticl Anlysis nd Grph Theory, Belgrde, September 4, 26. FRACTONAL NTEGRALS AND DERVATVES N -CALCULUS Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković We generlize the notions of the frctionl -integrl nd -derivtive by introducing vrible lower limit of integrtion. We discuss some properties nd their reltions. Finlly, we give -Tylor-like formul which includes frctionl -derivtives of the function.. NTRODUCTON n the theory of -clculus see [5] nd [7], for rel prmeter R + \{}, we introduce -rel number [] by [] := R. The -nlog of the Pochhmmer symbol shifted fctoril is defined by: ; =, ; k = k i i= Also, the -nlog of the power b k is k N { }. b =, b k = k b i i= There is the following reltionship between them: k N;, b R. b n = n b/; n. 2 Mthemtics Subject Clssifiction. 4A5, 33D6. Key Words nd Phrses. Bsic hypergeometric functions, -integrl, -derivtive, frctionl clculus. 3
32 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković Their nturl expnsions to the rels re b = b/; b/;, ; = ; ; R. Notice tht b = b/;. The following formuls see, for exmple, [5] nd [4] will be useful: ; n = n /; 2 n n n n 2 ; n ; n b n = /; n n; 3 ; n /b; n b [ ] b = k k 2 b k. 4 k k= The -gmm function is defined by 5 Γ x = ; x ; x = x x, where x R \ {,, 2,...}. Obviously, Γ x + = [x] Γ x. We cn define -binomil coefficients with [ ] Γ + = Γ + Γ + = + ; + ; ; + ;,, R \ {, 2,...}. Prticulrly, [ ] 6 = ; k k k k 2 k N. k ; k The hypergeometric function is defined s 2φ, b c ; x = n= The fmous Heine trnsformtion formul [5] is 7 2φ, b c ; x = bx/c; x; ; n b; n c; n ; n x n. 2φ c/, c/b c We define -derivtive of function fx by ; bx/c. D f x = fx fx x x x, D f = lim x D f x
Frctionl integrls nd derivtives in -clculus 33 nd -derivtives of higher order: 8 D f = f, Dn f = D D n f n =, 2, 3,.... For n rbitrry pir of functions ux nd vx nd constnts, R, we hve linerity nd product rules D ux + vx = D u x + D v x, D ux vx = ux D v x + vx D u x. The -integrl is defined by, f x = x ftd t = x fx k k <, k= nd 9, f x = x ftd t = x ftd t ftd t. However, these definitions cuse troubles in reserch s they include the points outside of the intervl of integrtion see [6] nd ]. n the cse when the lower limit of integrtion is = x n, i.e., when it is determined for some choice of x, nd positive integer n, the -integrl 9 becomes x ftd t = x n fx k k. x n k= As for -derivtive, we cn define n opertor n, by,f = f, n,f =, n, f n =, 2, 3,.... For opertors defined in this mnner, the following is vlid: D, f x = fx,, D f x = fx f. The formul for -integrtion by prts is b ux D v xd x = [ uxvx ] b b vx D u xd x. W. A. Al-Slm [2] nd R. P. Agrwl [] introduced severl types of frctionl -integrl opertors nd frctionl -derivtives. Here, we will only mention the frctionl -integrl with the lower limit of integrtion =, defined by η, f x = x η+ Γ x t t η ft d t η, R +.
34 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković On the other hnd, the solution of nth order -differentil eution D n y x = fx, D k y = k =,,..., n, cn be written in the form of multiple -integrl yx = n, f x = x d t t t n d t n t 2 d t n 2 ft d t. The reduction of the multiple -integrl to single one ws considered by Al- Slm [3]. He thought of it s -nlog of Cuchy s formul: 2 yx = n,f x = [n ]! x t n ft d t n N. n this pper, our purpose is to consider frctionl -integrls with the prmetric lower limit of integrtion. After preliminries, in the third section we define the frctionl -integrl in tht sense. On the bsis of tht, the frctionl -derivtive is introduced in the fourth section. Finlly, in the lst section, we give -Tylor-like formul using these frctionl -derivtives. 2. PRELMNARES We will first specify some results which re useful in the seuel nd which cn be proved esily. Lemm. For, b, R + nd k, n N, the following properties re vlid: 3 4 5 b k = k b/, b k b = b/; k b/; k, n k = k n. The next result will hve n importnt role in proving the semigroup property of the frctionl integrl. Lemm 2. For µ,, R +, the following identity is vlid µ n +n 6 n µ+ =. + n= Proof. According to the formuls nd 3, we hve µ n = µ n ; µ n ; = µ n ; n µ; µ n ; n µ ; = µ µ ; n µ ; n n.
Frctionl integrls nd derivtives in -clculus 35 Applying the identity 4 to the expression +n /, the sum on the left side of 6 cn be written s µ ; n µ ; n LS = ; n= n µ n n ; n µ µ, = 2φ ; µ. Using 7, we get LS = = µ + ; ; µ +, µ 2φ ; + µ n= According to 2 nd, the following is vlid: ; n µ ; n ; n µ ; n +n. µ ; n µ ; n = µ+ n ; n µ n ; n n = µ+ n ; µ + ; µ ; µ n ; n Hence LS = = µ ; µ + ; µ + n ; µ n ; n = µ ; µ + ; µ + n n. µ + + n= ; n ; n n µ + n. f we use formuls 6 nd 4 nd chnge the order of the summtion, the lst sum becomes ; n n µ + n ; n= n [ ] [ ] = n n n 2 n k k 2 µ + n k n n= k k= [ ] = k k 2 µ + k [ ] n n 2 k n k k= n n= [ ] = k k 2 µ + k k =. k k= The lst reltion is vlid becuse of k = for k =, 2,.... Finlly, the identity holds: LS = µ + + = µ+ +.
36 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković 3. THE FRACTONAL -NTEGRAL n ll further considertions we ssume tht the functions re defined in n intervl, b b >, nd, b is n rbitrry fixed point. Also, the reuired -derivtives nd -integrls exist nd the convergence of the series mentioned in the proofs is ssumed. Generlizing the formul 2, we cn define the frctionl -integrl of the Riemnn-Liouville type by 7, f x = Γ Using formul 4, this integrl cn be written s, f [ ] x = x k k+ 2 x k Γ k k= x t ft d t R +. t k ftd t R +. Lemm 3. For R +, the following is vlid:, f x =, + D f x + f Γ + x < < x < b. Proof. Since the -derivtive over the vrible t is D x t = [] x t, nd using the -integrtion by prts, we obtin, f x = [] Γ = D x t ftd t x f + Γ + =, + D f x + f Γ + x. x t D f td t Lemm 4. For, R +, the following is vlid: x t, f td t = < < x < b. Proof. Using Lemm nd formul, for n N, we hve, f n = Γ n = Γ n u fud u n n j+ f j j =. j=
Frctionl integrls nd derivtives in -clculus 37 Then, ccording to the definition of -integrl, it follows x t,f td t = x n+,f n n =. n= Theorem 5. Let, R +. The -frctionl integrtion hs the following semigroup property,,f x = +, f x < < x < b. Proof. By previous lemm, we hve i.e.,,,f x = Γ x t,f td t,,,f x = Using the result from [], we conclude tht Γ Γ Γ Γ t x t t u fud u x t t u fud u.,, f x = +, f x,,, f x = +, f x Γ Γ Furthermore, we cn write,, f x =, + f x + Γ + wherefrom it follows Γ Γ x t x t t u fud u. x t + ftd t t u fud u,,, f x =, + f x + c j f j j, j=
38 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković with c j = x j+ + Γ + x Γ Γ x x n+ x n j+ n. n= By using the formuls from Lemm nd 5, we get c j = x + + x j+ + n= n+ x j+ n n. Putting µ = j /x into 6, we see tht c j = for ll j N, which completes the proof. Lemm 6. For R +, λ,, the following is vlid 8, x λ = Γ λ + x +λ < < x < b. Γ + λ + Proof. For λ, ccording to the definition 7, we hve, x λ = x t t λ d t x t t λ d t. Γ Also, the following is vlid: x t t λ d t = λ+ x k+ k λ k =. Therefrom, by using 6, we get x x t t λ d t = x +λ k= +k x k λ λ+k k= = λ +λ x +λ. Using 5, we obtin the reuired formul. Prticulrly, for λ =, using -integrtion by prts, we hve, x = x t d t = D x t d t Γ Γ [] = Γ + D x t d t = Γ + x.
Frctionl integrls nd derivtives in -clculus 39 4. THE FRACTONAL -DERVATVE We define the frctionl -derivtive by 9 D, f, f x, < x = fx, = D, f x, >, where denotes the smllest integer greter or eul to. Notice tht D,f x hs subscript to emphsize tht it depends on the lower limit of integrtion used in definition 9. Since is positive integer for R +, then for D f x we pply definition 8. Lemm 7. For R \ N, the following is vlid: D D,f x = D, + f x < < x < b. Proof. We will consider three cses. For, ccording to Theorem 5, we hve D D, f x = D, f x = D, f x = D,, f x =, + f x = D, + f x. n the cse < <, i.e., < + <, we obtin D D,f x = D, f x = D, + f x = D, + f x. For >, we get D D, fx = D D, fx = D +, fx = D, + fx. Theorem 8. For R \ N, the following is vlid: D D, fx D, D fx = f Γ x < < x < b. Proof. We will use formuls, Theorem 5, nd Lemm 6, to prove the sttement. Let us consider two cses. f <, then D D, fx = D, fx = D,, D fx + f = D,,D fx + fd, x x = D, + D fx + fd Γ + = D,, D fx + f [ ] x Γ + = D, D fx + f Γ x.
32 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković f >, there exists l N, such tht l, l +. Then, pplying similr procedure, we get D D,fx = D D l+ = D l+2, l+ l+, fx, D fx + f = D l+ D,, l+ D fx + = D, D fx + f Γ x. f Γ l + 2 Dl+ x l+ 5. THE FRACTONAL -TAYLOR-LKE FORMULA Mny uthors tried to generlize the ordinry Tylor formul in different mnners. The use of the frctionl clculus is of specil interest in tht re see, for exmple [] nd [8]. Here, we will present one more generliztion, bsed on the use of the frctionl -derivtives. Lemm 9. Let fx be function defined on n intervl, b nd R +. Then the following is vlid: D,, f x = fx < < x < b. Proof. For >, we hve D,, f x = D,, f x = D = D, f x = fx., + f x Lemm. Let,. Then, D, f x = fx + Kx < < x < b, where K does not depend on x. Proof. Let Ax =, D, f x fx. Applying D, to the both sides of the bove expression, nd using Lemm 9, we get D, Ax = D,, D, f x D, fx = D,,D,f x D,fx =. On the other hnd, ccording to Lemm 6, we obtin x = D x = D x =. D,,
Frctionl integrls nd derivtives in -clculus 32 Hence, we conclude tht Ax is function of the form Ax = Kx. Lemm. Let < c < x < b nd,. Then the following is vlid: +k,c D, +k x c+k fx = Γ + k + D+k, Proof. According to Lemm 3 nd Lemm 4, we hve fc + +k+,c D, +k+ fx, k N.,c +k D, +k fx =,c +k+ D D, +k fx + D+k, fc x c+k Γ + k + = D+k, fc Γ + k + x c+k +,c +k+ D, +k+ fx. Now, we re redy to prove Tylor type formul with frctionl -derivtives, which is the min result of this section. Theorem 2. Let fx be defined on, b nd,. For < < c < x < b, the following is true: 2 fx = n k= D +k, fc Γ + k + x c+k + R n f, with R n f = R f Kx + E n f, where R f = Γ c x t D,ftd t, nd E n f cn be represented in either of the following forms: 2 E n f =,c +n D, +n fx, 22 E n f = D+n, fξ x c+n c Γ + n + < ξ < x. Proof. We will deduce the proof of 2 by mthemticl induction. Since, D, fx = Γ using Lemm, we obtin c x t D, ftd t +,c D, fx, fx =,cd,fx + R f Kx.
322 Predrg M. Rjković, Sld - n D. Mrinković, Miomir S. Stnković According to Lemm, for k =, we hve,c D, fx = D,fc Γ + x c + +,c D+, fx = D, fc Γ + x c + E f, which completes the expression for R f nd proves 2 for n =. Assume tht 2 is vlid for ny n N. Then, gin from Lemm, the following holds: E n f = +n,c D +n, D+n, fx = fc Γ + n + x c+n +,c +n+ = D+n, fc Γ + n + x c+n + E n+ f. D, +n+ fx Hence the formul 2 is vlid for n +. So, it is vlid for ech n N. The second form of reminder, 22, cn be obtined by using men vlue theorem for integrls [9]. ndeed, there exists ξ c, x, such tht E n f =,c +n D, +n fx = = D+n, fξ Γ + n = c Γ + n D +n, f ξ Γ + n + x c+n. c x t +n D +n, f td t x t +n d t = D+n, fξ Γ + n +n,c x Acknowledgements. We re grteful to the referees for helpful remrks. This work ws supported by Ministry of Science, Technology nd Development of Republic Serbi, through the project No 4423 nd No 443. REFERENCES. R. P. Agrwl: Certin frctionl -integrls nd -derivtives. Proc. Cmb. Phil. Soc., 66 969, 365 37. 2. W. A. Al-Slm: Some frctionl -integrls nd -derivtives. Proc. Edin. Mth. Soc., 5 966, 35 4. 3. W. A. Al-Slm: -Anlogues of Cuchy s Formuls. Proc. Amer. Mth. Soc., 7, No. 3 966, 66 62. 4. W. A. Al-Slm, A. Verm: A frctionl Leibniz -formul. Pcific Journl of Mthemtics, 6, No. 2 975, 9.
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