Fractional Integrals Involving Generalized Polynomials And Multivariable Function

Transcription

1 IOSR Joual of ateatcs (IOSRJ) ISSN: Volue, Issue 5 (Jul-Aug 0), PP 05- wwwosoualsog Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto D Neela Pade ad Resa Ka Deatet of ateatcs APS uvest Rewa ada Pades,Ida Abstact:Ou a of ts ae s to fd a Eulea Itegal ad a a teoe based o te factoal oeato assocated wt geealzed oloal ad a ultvaable I-fucto avg geeal aguets Te teoe ovdes exteso of vaous esults Soe secal cases ae also gve Kewods-Factoaltegal,Eulea tegal,ultvaable I-fucto,Rea-Louvlle oeato,laucella fucto I Itoducto Te Rea-Louvlle oeato of factoal tegato R f of ode s defed b xd f = Γ x t f t dt A equvalet fo of Beta fucto s t a t b dt = a+b B a, b Wee,R (x<),re (a)>0, Re(b)>0 To ove te Eulea tegals, we use te followg foula () () (t x) a ( t) b ( t + q ) ρ ( t + q ) ρ dt = x a+b B a, b ( x + q ) e ( x + x q)e F () D [a, ρ ρ ; a + b, x ρ x ρ ] (3) x+q x+q Wee F D s Laucella fucto Wee x, R (x<) ;, q ρ ε C = R e a, R e b > 0 ad ax [ x ρ P x+q Te tegal eesetato of F D s defed as, x ρ P x+q ] < (π) ( x ) ξ ( x ) ξ dξ dξ Γ a Γ b Γ b Γ c F D Γ a+ξ + ξ Γ b+ξ Γ(b+ξ ) Γ(c+ξ + ξ ) Now ( t + q) α = (x + q) α + (t x) x +q Sce Γ A Γ(B) Γ(c), (t + q) α = (x +q) α Γ( α) π F A, B, C; Z = Γ π Γ (, q, αεc, x, tεr ad ag x +q a, b b, c, x x =, Γ ξ Γ ξ α = (x + q) α F α,,, (t x) Γ A+ Γ(B+ ) Γ(c+ ) α) (t x) x +q d te oles of Γ fo tose f Γ α Foula (3), ca be ovded wt te el of (), (4) ad (5) Te geealzed oloal defed b Svastava[8] s as follows x +q Γ Z d (5) < π ad at of tegato s ecessa suc a ae so as to seaate wwwosoualsog 5 Page

2 Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto S N N x x N = α =0 N α =0 ( N ) α ( N ) α B N, α,, N, α x α x α (6) Wee, N = 0,,, =, ae abta ostve teges ad te coeffcets B N, α,, N, α ae abta costats Hee s a ostve tege ad 0 0 would ea zeos II ULTI-VARIABLE I-FUNCTION It s defed ad eeseted te followg ae:- 0, I z z = I : 0, 3,,0,,,,,q : 3,q 3,,,q,,q,q Z a ; α,α a 3 ; α, 3,α 3 a ; α :,,α 3,, a,α a,α Z b ; β,β : β 3 ; β 3,β 3 β ; β,q,q3,β,q, b,β q b,β Wee, φ (ξ )= q = q ( b ),,q L L ξ ξ Ψ ξ ξ z ξ z ξ dξ dξ ( a ( b ) ( b ) ( b ( a (7) ) ) 3 ( a ) ( a3 3 ) ( a 3 ( a ) ( a3 3 ) ( a 3 q ) Te covegece ad ote detals of ultvaable I-fucto, see Pasad[4] 3 3 ad ) ) II a Itegal Te a Itegal to be establsed ee s λ μ x t t S N N W N α =0 x t λ t μ t a t + q t b ς t + q ς N N α N α α =0!α!α I wwwosoualsog + q ρ z t γ t t + q c z t t t + q c dt = 6 Page

3 Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto α B N, α,, N, α x α 0, x W I :0, 3 0, ++,,,,0,0,q : 3,q 3 ++,q ++:,q,q, 0, 0, R X,X,X 3, a α,α,, a 3 α 3,α 3,α 3, 3, a,α,α,0, 0, a,,α, a,α, ; ; ; R b β,β,q, b 3 β 3,β 3,β 3,q3 (), b,β,β,0, 0 X 4,X 5,(b,β ),q b,β,q,q, 0, (0,) Wee W = ( ) a+b ( + q ) (8) W = ( ) ) ( ( + q ) () ς α X = [ a λ α ; γ γ, ] X = b μ α ;,0 0 X 3 = + ρ + ς () α ; C C (), 0 0 X 4 = + ρ + ς () α ; C C (), 0 0 X 5 = [ a b λ + μ α ; γ +, γ +, ],, Z ( ) γ + ( + q ) c R = Z ( ) γ + ( + q ) c () R = ( ) +q ( ) +q Te followg ae te codtos of valdt of tegal (8) ), ε R <, γ, ; c (), λ, μ, ς () ε R +, ρ εr, q ε C, z εc ( =, = ) ) ax q ( - ) < 3) R e a + γ b > 0, = b, R e b + > 0, = B wwwosoualsog B 7 Page

4 Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto 4) ag z ( + q ) c < T π Wee T = () α () () = () + 3 α () + + α 3 () q ( < t <, = ) () + β () + β () 3 = 3 + q 3 q () β () + α () = () + α 3 () β () α () q () β = + = + III PROOF I ode to ove tegal (8),exad ultvaable I-fucto tes of ell-baes te of cotou tegal b (7), geealzed oloal b(6)now tecagg te ode of suato ad tegato (wc s essble ude te codtos of valdt stated above), We get te followg fo:- α () α () N N α =0 α =0 ( N ) α! α ( N ) α! α B N α,, N α x α x α X πw ξ ξ (ξ ) Z ξ Z t a+ λ α L L t b+ μ α + s = ξ s s = ( t + q ) + s = γ s ξ s () + σ α (s ) s = c ξs dt dξ dξ Now usg te foula (3) fo e tegal e, (t x) a ( t) b ( t + q ) ρ ( t + q ) ρ dt = x xa+b Ba,b(x+q)e (x+q)e FD() a, ρ ρ ;a+b, xx+q xx+q () Ad covetg te Laucella fucto F D tegal fo fo equato (4) ad afte slfcato,we get te equed esult: IV SPACIAL CASES:- If we set γ = 0 = = γ ad λ = 0 = = λ, te tegal (8) educes to N = E S N N t a t b x t μ t + q ς x t μ α =0 N N α N α!α!α 0, I :0, 3 0, ++,,,,0,0,q : 3,q 3 ++,q ++:,q,q, 0, 0, t + q ς I t + q ρ z t t + q c z t α =0 B N, α,, N, α x α x α E t + q c wwwosoualsog 8 Page

5 L A,A,A 3, a α,α, L b β,β, b 3 β 3,q Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto, a 3 α 3,β 3,β 3,α 3,α 3, 3, b,β,q3 (), a,α,α,0, 0,(a,α ), a,,α (),β,0, 0 A 4,A 5,(b,q,β ),q b,β, (); ; ;,q, 0, (0,) Wee, W = a+b W = μ α + q A = [ a; 0 0, ] + q ρ ς α A = b μ α ; ζ ζ,0 0 A 3 = + ρ + ς () α ; C C (), 0 0, A 4 = + ρ + ς α ; C C, 0 0 A 5 = [ a b L = Z ( ) ζ Z ( ) ζ μ α ; ζ ζ, ( + q ) c ( + q ) c (), L = ( ) +q ( ) +q Fo ζ = 0 = = ζ ad μ = 0 = μ, te tegal (8) educes to λ x t S N N x t λ t + q t a ς t + q ς N N α N α!α!α 0, I :0, 3 0, ++,,,,0,0,q : 3,q 3 ++,q ++:,q,q, 0, 0, I t b + q ρ z t γ t + q c z t γ t + q c α =0 B N, α,, N, α x α x α F N = Γ(b)F α =0 wwwosoualsog 9 Page

6 B,B, a α,α, b β,β,q, b 3 β 3 Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto, a 3 α 3,β 3,β 3,α 3,α 3, 3,q3 (), a,α,α,0, 0,(a,,α ), a,α ; ; ; Q, b,β,β (),0, 0 X 4,X 5,(b,q,β ),q b,β, 0, (0,) Q Wee, F = ( ) a+b ( + q ) ρ F = λ α + q ς α B = a λ α ; γ γ, B = + ρ + ς α ; C C, 0 0, B 3 = + ρ + ς () α ; C C (), 0 0, B 4 = [ a b Q = Z ( ) γ Z ( ) γ λ α ; γ γ, ( + q ) c ( + q ) c () Q = ( ) +q ( ) +q 3 We ζ = 0 = = ζ = 0 = γ = γ ad λ = 0 = μ, =, te tegal (8) educes to x S N N t + q ς t + q ς I t a t b z t + q c t + q c x z N N α N α!α!α 0, I :0, 3 0, ++,,,,0,0,q : 3,q 3 ++,q ++:,q,q, 0, 0, + q ρ α =0 B N, α,, N, α x α x α G D,D, a α,α, R R b β,β,q, b 3 β 3, a 3 α 3,α 3,α 3, 3,β 3,β 3,q3, b,β,β N = Γ(b)G α =0 (), a,α,α,0, 0,(a,,α ), a,α (),0, 0 X 3,X 4,(b,q,β ) b,β, (); ; ;,q, 0, (0,) wwwosoualsog 0 Page

7 Wee, G = ( ) a+b Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto ( + q ) ρ G = + q ς α D = a; 0 0, D = + ρ + ς α ; C C, 0 0 D 3 = + ρ + ς () α ; C C (), 0 0 D 4 = [ a b; 0 0, Z ( + q ) c R = R = Z ( + q ) c ( ) +q ( ) +q Let f t = t a S N N Te, D b f = D b f Γ b S N N Γ b (),, V AIN THEORE t + q ρ x t λ t + q ς x t λ b t f t dt t a t b t + q ς I t + q ρ x t λ t + q ς x t λ t + q ς I z t γ t + q c z t γ t + q c z t γ t + q c z t γ t + q c wwwosoualsog Page

8 N = I Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto α =0 N N α N α!α!α 0, I :0, 3 0, ++,,,,0,0,q : 3,q 3 ++,q ++:,q,q, 0, 0, α =0 B N, α,, N, α x α x α I H K,H, a α,α K b β,β,q, a 3,α 3, b 3,β 3,α 3,α 3, 3,β 3,β 3,q3 (), a,α,α,0, 0,(a,,α ), a,α (),0, 0 H 3,H 4,(b,q,β ),q b,β, b,β,β, (); ; ;,q, 0, (0,) Wee, G = ( ) a+b G = H = a λ α + q ( + q ) ρ λ α ; γ γ, ς α H = + ρ + ς α ; C C, 0 0 H 3 = + ρ + ς () α ; C C (), 0 0 H 4 = [ a b λ α ; γ γ, K = Z ( ) γ Z ( ) γ ( + q ) c ( + q ) c () ( ) +q K = ( ) +q Codtos of valdt of te teoe ae te sae as stated Eulea Itegal Secal case B secalzg te vaous aaetes, we get soe ow ad uow esults Refeces [] YN Pasad, ultvaable I-Fucto, Vaa Pasad Ausada Pata 9(986) 3 35 [] H Svastava ad A Hussa, Factoal tegato of te H-Fucto of seveal vaable, coute, at, Al 30(995),73-85 [3] VBL Cauasa ad VK Sgal,Factoal tegato of ceta secal fuctos,taag Jat35(004),3- [4] APPudov,YuA Bcov ad OIacev,Itegals ad sees,voli,eleeta Fuctos,Godo ad Beac,Newo- Lodo-Pas-oteux-Too,986 [5] Sago ad RKSaxea,Ufed factoal tegal foula fo te ultvaable H-fucto,JFactCalc5 (9999),9-07 [6] RK Saxea ad KNsoto,Factoal tegal foula fo te H-fucto,JFactCalc 3 (994),65-74 [7] RK Saxea ad Sago, Factoal tegal foula fo te H-fucto II,JFactCalc 6 (994),37-4 [8] HSvastava ad CDaoust,Ceta geealzed Neua exasos assocated wt te Kae de Feet fucto,nedelacadwe-tecidagat 3 (969), [9] H Svastava, KC Guta ad SP Goal,Te H-fuctos of Oe ad Two Vaables wt Alcatos,Sout Asa Publses,New Del-adas,98,, wwwosoualsog Page