Modified H-Tran sform and Pathway Fractional Integral Operator
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![ractional integral operator eine by Nair [8]. Three corollarie o the main theorem have been erive.](/docs-images/77/75499680/images/1-2.jpg "Our ining provie intereting uniication an etenion o number o (new an known reult.")
1 Global Journal o Science Frontier Reearch Mathematic an Deciion Science Volume Iue 8 Verion. Type : Double Blin eer Reviewe International Reearch Journal ubliher: Global Journal Inc. (USA Online ISSN: & rint ISSN: Moiie -Tran orm an athway Fractional Integral Operator By Neeti Ghiya R.V. College o Engineering, Inia Abtract - In thi paper we have etablihe a theorem wherein we have obtaine the image o moiie -tranorm uner the pathway ractional integral operator eine by Nair [8]. Three corollarie o the main theorem have been erive. Our ining provie intereting uniication an etenion o number o (new an known reult. Keywor : athway ractional integral operator, moiie -unction tranorm, -unction o one variable, Whittaker unction, Wright generalize Beel unction. GJSFR - F Claiication : MSC : 6A33 Moiie -Tran orm an athway Fractional Integral Operator Strictly a per the compliance an regulation o :. Neeti Ghiya. Thi i a reearch/review paper, itribute uner the term o the Creative Common Attribution- Noncommercial 3. Unporte Licene permitting all non commercial ue, itribution, an reprouction in any meium, provie the original work i properly cite.
2 Re. Moiie -Tranorm an athway Fractional Integral Operator 7. M. Saigo, R. K. Saena an J.Ram, On the two imenional generalize Weyl ractional calculu aociate with two imenional -tranorm, J.Fractional Calculu, 8(995, Neeti Ghiya Abtract - In thi paper we have etablihe a theorem wherein we have obtaine the image o moiie - tranorm uner the pathway ractional integral operator eine by Nair [8]. Three corollarie o the main theorem have been erive. Our ining provie intereting uniication an etenion o number o (new an known reult. Keywor : athway ractional integral operator, moiie -unction tranorm, - unction o one variable, Whittaker unction, Wright generalize Beel unction. I. INTRODUCTION The moiie -unction tranorm wa introuce by Saigo, Saena an Ram [7] an i eine in the ollowing manner: h( h M, N, Q [ F( ;, ] (c,, M, N k (, Q ( F(, or k >, ( (,, Q F( ( a U(, > >, ( where U(- i the well- known eaviie unit unction. Further we aume that h( eit an belong to U. where U i the cla o unction ( on R (,, which i ininitely ierentiable with partial erivative o any orer uch that ( ( ( w a. (3 a - w The tranorm eine by ( eit provie that ollowing (uicient conition are atiie: (i arg < π Ω/k, 35 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I Author : Department o Mathematic, R.V. College o Engineering, Bangalore-5659, Karnataka. Inia. neeti.varaam@gmail.com Global Journal Inc. (US
3 Moiie -Tran orm an athway Fractional Integral Operator where Ω N N M Q M 36 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I (ii Re(w >, c (iii Re( w k ma Re N <. The Fo -unction or imply -unction wa introuce by Charle Fo [5]. Thi unction i eine an repreente by mean o the ollowing Mellin-Barne type o contour integral: where i ( /, z (a, M, N N, [ ] M,, Q z, Q z θ( z, (4 L (b,β, Q π i θ( an Γ(b Q M M β Γ( b β N Γ( a N Γ(a. (5 The nature o the contour L in (4, the conition o convergence o the integral (4, the aymptotic epanion o the -unction an ome o it pecial cae can be reerre to the work o Srivatava, Gupta an Goyal [6] an Mathai an Saena [4]. The athway ractional integral operator introuce by Nair [8] an i eine in the ollowing manner: (, ( where ( b, C, R(, a a ( a ( t ( (t t L(a, > > an pathway parameter <., (6 The pathway moel i introuce by Mathai [] an tuie urther by Mathai an aubol ([], [3]. For real calar, the pathway moel or calar ranom variable i repreente by the ollowing probability enity unction (p...: < β [ ] a ( ( c, (7 <, >, β, ( a( >, >, where c i the normalizing contant an i calle the pathway parameter. For real, the normalizing contant i a ollow: β [ a( ] Γ c, or <, (8 β Γ Γ Re. 5. Fo, The G an -unction a ymmetrical Fourier kernel,tran.amer.math. Soc., 98(96, Global Journal Inc. (US
4 Moiie -Tran orm an athway Fractional Integral Operator β [ a( -] Γ - β Γ Γ -, or >, >, (9 Note (a β or. Γ For <, it i a inite range enity with - a(- > an (7 remain in the etene generalize type- beta amily. The pathway enity in (7, or <, inclue the etene type- beta enity, the triangular enity, the uniorm enity an many other p... For >, we have [ ] β a ( - ( ( c, ( < <, >, β, >, which i the etene generalize type- beta moel or real. It inclue the type- beta enity, the F enity, the Stuent-t enity, the Cauchy enity an many more. ere it i coniere only the cae o pathway parameter <. For (7 an ( take the eponential orm, ince Lim c [ a ( ] - Lim c [ a ( - ] c e a Thi inclue the generalize Gamma-, the Weibull-, the Chi-quare, the Laplace-, the Mawell- Boltzmann an other relate enitie. When, a ( t e a t integral tranorm o with parameter a : a - t. (, ( e (t t L (, then operator (6 reuce to the Laplace a When, a, then replacing by in (6 the integral operator reuce to the Riemann-Liouville ractional integral operator. (3 37 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I Global Journal Inc. (US
5 Moiie -Tran orm an athway Fractional Integral Operator II. MAIN TEOREM I h( (c,, M, N k (, Q ( F(, (4 (,, Q 38 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I an then (, ( a ( (, [h (] Γ [a( ] a ( t M, N, Q ( (t t a(- k, (5 ( (c,,,, Q, (, k F(,, (, k where F( ( a U(, > > a eine in ( provie that (i Ω >, arg < π Ω/k, c (ii Re(w >, Re( w k ma Re N <, (iii ( L(a,b, C, R( >, a >, <, (iv R( >. roo: In orer to prove the main theorem, ubtituting the value o h( rom (4 in the let han ie o (5, we in that (, ( a ( a ( t [h(] t (t M, N, Q (t k (c, (,,, Q F( t Now interchanging the orer o an t integral which i permiible uner given conition, we get (6 (7 Re.. A. M. Mathai,. J. aubol, athway moel, Super tatitic, Talle tatitic an a generalize meaure o entropy, hyica A, 375(7, -. Global Journal Inc. (US
6 Moiie -Tran orm an athway Fractional Integral Operator (, [h(] F( - a ( t a ( t ( t - M, N, Q (t k (c, (,,, Q t (8 Re. 4. A.M. Mathai an R. K. Saena, The -unction with Application in Statitic an other Dicipline, John Wiley an Son (974. To evaluate the t-integral, we epre the - unction in term o Mellin-Barne contour integral with the help o (4 an change the orer o contour integration an t- integral. Ater evaluating the t- integral an reinterpreting the reult thu obtaine in term o -unction, we eaily arrive at the right han ie o (6 ater a little impliication. When, then (6 ten to Lim (, [h (] (a III. M, N, Q a SECIAL CASES k (c,,, (, k F( (,, Q (I I we reuce the Fo -unction involve in (4 to Whittaker unction by uing a known reult [4, p.55], a little impliication will give the ollowing Corollary. I an then h( - (9 ( e W ( F(, ( (, ( a ( b, c (, [h (] Γ [a( ] a ( t,, 3 ( (t t, ( (,, ( b, F(. a(- (,, ( c,, ( c, ( The conition o valiity o the aorementione corollary can be eaily erive rom our main theorem. When then ( ten to Lim (, [h (] [a ],, a (,, ( b, F( ( c,, ( c,. (3 39 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I Global Journal Inc. (US
7 Moiie -Tran orm an athway Fractional Integral Operator (II Reucing Fo -unction in (4 to eponential unction uing known reult [6, p.8] then we have the ollowing I h( - ( e F(, (4 4 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I an then (, ( a ( a ( t (, [h (] B, [a ( ] ( (t t F, (5 ( ; ( ;- F(. a(-. (6 The conition o valiity o the aorementione corollary can be eaily erive rom our main theorem. When then (6 ten to Lim (, [h (] Γ( [a ] a F(. (7 (III I we reuce the Fo -unction involve in (4 to Wright generalize Beel unction by uing a known reult [6, p.9], ater a little impliication we have I an then h( ν ( J λ ( F(, (8 (, ( a ( a ( t (, [h (] Γ [a( ] ψ ( (t t, (9 (, ; ( λ, ν,(,; F(, a ( (3 Re. 6.. M. Srivatava, K. C. Gupta, S.. Goyal, The - unction o one an two variable with application, South Aian publiher, New Delhi (98. Global Journal Inc. (US
8 Moiie -Tran orm an athway Fractional Integral Operator where ψ i wright hypergeometric unction an the conition o valiity o the aorementione corollary can be eaily erive rom our main theorem. When then (3 ten to Lim (, [h (] [a ] ψ (, ; ( λ, ν; F( a. (3 Note Reerence Réérence Reerencia. M. Mathai, A pathway to matri variate gamma an normal enitie, Linear Algebra an it Application, 396(5, A. M. Mathai,. J. aubol, athway moel, Super tatitic, Talle tatitic an a generalize meaure o entropy, hyica A, 375(7, A. M. Mathai,. J. aubol, On generalize itribution an pathway, hyic Letter, 37(8, A.M. Mathai an R. K. Saena, The -unction with Application in Statitic an other Dicipline, John Wiley an Son ( Fo, The G an -unction a ymmetrical Fourier kernel,tran.amer.math. Soc., 98(96, M. Srivatava, K. C. Gupta, S.. Goyal, The - unction o one an two variable with application, South Aian publiher, New Delhi ( M. Saigo, R. K. Saena an J.Ram, On the two imenional generalize Weyl ractional calculu aociate with two imenional -tranorm, J.Fractional Calculu, 8(995, Seema S. Nair, athway ractional integration operator, Fractional calculu & Applie analyi, (3(9, Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I Global Journal Inc. (US
9 Moiie -Tran orm an athway Fractional Integral Operator Note 4 Global Journal o Science Frontier Reearch F Volume XII Iue VIII V erion I Thi page i intentionally let blank Global Journal Inc. (US
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