Int. J. Contemp. Math. Scences, Vol. 7, 202, no. 20, 995-004 Some Theoems on Multple A-Functon Tansfom Pathma J SCSVMV Deemed Unvesty,Kanchpuam, Tamlnadu, Inda & Dept.of Mathematcs, Manpal Insttute of Technology Manpal Unvesty, Manpal, Kanataka, Inda pamuthaa@yahoo.co.n T.M. Vasudevan Nambsan Head,Dept.of Mathematcs, Tkapu Eng. College,Kasaagod Dst., Keala, Inda tmvasudevannambsan@yahoo.com Abstact In the pesent pape we have ntoduced multple A-functon tansfom nvolvng A-functon of vaables as kenel. In the eale pape we have establshed the multple Melln tansfom of poduct of two A-functons. Wth the help of ths fomula, n the pesent pape we have dscussed some theoems on A-functon tansfom. Specal case ncludes the esults gven by V.C.Na. Mathematcs Subect Classfcaton: 44A20, 33E99 Keywods: Melln tansfom, A-functon of vables, multple A-functon tansfom, H-functon tansfom, H-functon of vaables.
996 Pathma J and T.M. Vasudevan Nambsan Intoducton The A-functon of vaables used n ths pape was ntoduced by Gautam and Goyal [2] n the followng manne. A[z,,z ]= A M,N:(m,n );...;(m,n ) P,Q:(p,q );...;(p,q ) () () () () z ( a,α,...,α ):c,γ ;...; c,γ M () () () () z ( b,β,...,β ):d,δ ;...; d,δ = 2π whee L L ζ ζ ϕ... θ ζ...θ ζ ζ,...,ζ z...z dζ...dζ (.) m () () n () () Γ( d -δ ζ) Γ -c +γ ζ () () () () ( ) ( ) = = q p θ ζ = Γ -d +δ ζ c -γ ζ =m + =n +, =,,, (.2) ϕ ( ζ,...,ζ ) = N () M () Γ -a + α ζ Γ b- β ζ = = = = () () Γ -b + β ζ Γ a- α ζ = =N+ = Q P =M+ (.3) Hee M, N, P, Q, m,n, p, q ae non-negetve nteges. a s, b s, c s,d s,α s,β s, γ s, δ s ae complex numbes. * π Integal (.) s absolutely convegent f μ =0, η >0 and ag( μ) z < η 2 P () whee { } () () { } () { } () { } α Q -β q δ p -γ μ = α β δ γ = = = = P () Q q () p * () () μ =Im α - β + δ - γ = = = =
Theoems on multple A-functon tansfom 997 N () P () M () Q () η =Re α - α + β - β = =N+ = =M+ m () q () n () q () + δ - δ + γ - γ, =, 2,...,. = =m + = =n + When () () () α s, β s, γ s, δ s ae all postve eal numbes and M=0,(.) educes to the H-functon of vaables gven by Svastava and Panda[6,p-25]. The multple A-functon tansfom means the tansfom n whch A-functon appeas s defned n the followng fom: λ λ (.4) φ p,..., p =... A c pt,...,c pt f t,...,t... The followng mpotant theoem whch s an analogue of the well-known Paseval-Goldsten theoem wll be equed n the sequel: T f x ;s =φ s { } If T f ( x );s =φ s and ϕ = ϕ then f x x dx f x x dx 2 2 { 2 } 2 (.5) Gupta and Mttal [3] have poved the followng. f x and f x ae contnuous fo x 0 and If 2 ( a;α ) m,n 0 ( b;β ) 0 ( a;α ) ( b;β ),p m,n,p Hp,q sx f (x)dx = H p,q sx f 2(x)dx,,q,q both ntegals beng convegent, then f x f x. 2 Wthout loss of genealty the above theoem can be extended to multple ntegals wth appopate convegent conons. Pathma and Nambsan [5] have poved the followng esult.... x η...x η A s x,...,s x ρ ρ A t x,...,t x dx...dx η η - - ρ ρ M+ N,M + N: m + n,m+ n ;...; m + n,m + n + + + + + + = t...t A ρ...ρ P Q,P Q : p q, p q ;...; p q,p q
998 Pathma J and T.M. Vasudevan Nambsan - s ρ t C :C ;...;C M (.6) - s ρ t D : D ;...;D Whee () () ( η () () ) η η C= a ;α,...,α, -b - β, β,..., β, a ;α,...,α = ρ ρ ρ () () () () η () () () C = ( c,γ ), -d -δ,δ, ( c,γ ), =,..., n ρ ρ n + p N N+ P Q q () () ( η () () ) η η D= b ;β,...,β, -a - α, α,..., α, b ;β,...,β = ρ ρ ρ M P M+ Q D = () () () () η () () () ( d,δ ), -c -γ,γ, d,δ ρ ρ povded that (),ρ >0, =,2,..., () d d - mn Re -ρ () mn Re <Re () ( η) m δ m δ -c -c < mn Re +ρ () mn Re, n () γ n γ =,...,. m m + q p, =,, and () the A-functons nvolved n (.6) exst. 2 Theoems on multple A-functon tansfom Theoem (2.) If λ λ φ p,...,p =... A c p t,...,c p t f t,...,t...
Theoems on multple A-functon tansfom 999 and ( v v ) ψ p,...,p =... A C pt,...,c pt f t,...,t h t,...,t... (2.2) then ψ p,...,p... g x,...,x,p,...,p φ x,...,x dx...dx = (2.3) whee v- v- v v v v v...v p...p h ( p,...,p ) A C( p k ),...,C( p k) λ λ =... A c ( p t ),...,c ( p t ) g t,...,t,k,...,k... (2.4) povded the ntegals nvolved ae absolutely convegent. Poof Usng (.5) fo the tansfom pas (2.) and (2.4),... g t,...t,k,...,k φ t,...t... = v- v- v v v v v...v... t...t h t...t A C t k,...,c t k f t...f t... ( x ) Hence... g,...x,p,...,p φ x,...x dx...dx v- v- v v v v = v...v... t...t h t...t A C t k,...,c t k f ( t,...,t)... Puttng t v = x, =,..., on the ght hand sde and usng (2.2) the esult follows. Theoem 2 v v λ λ If φ( p,...,p ) =... A c( pt ),...,c( pt) f ( t,...,t )... (2.5) and ρ ρ ψ p,...p... x...x A C p x,...,c p x φ x,...,x dx...dx then = (2.6)
000 Pathma J and T.M. Vasudevan Nambsan λ...λ -ρ vλ -ρ vλ -ρ v -ρ v ψ( p,...,p ) = c...c... t...t v...v ( ) ( ) M +N, N +M: m +n,n +m ;...; m +n,n +m A P +Q,Q +P: p +q,q +p ;...; p +q,q +p - v λ λ C p c t E:E ;...E f t,...,t M - vλ λ C p c t F:F ;...;F whee () () () () ( ) ρ... E= a ;α,...,α, -b - β,β,...,β, N = vλ vλ vλ M () ρ () () () -b - β,β,...,β, ( a ;α,...,α ) N+ P M+ = vλ vλ vλ Q () () () () ρ () E= c,γ, -d -δ,δ, n vλ vλ m ( ) () () ρ () () () -d -δ,δ, c,γ =,..., v n+ p λ m+ vλ q (2.7) povded () () () () ( ) ρ ρ F= b ;β,...,β, -a - α,α,...,α, M = vλ vλ vλ N ρ ρ ( ) () () () () -a - α,α,...,α, b ;β,...,β M+ Q N+ = vλ vλ vλ P () () () () ρ () F= d,δ, -c -γ,γ, m vλ vλ n ( ) () () ρ () () () -c -γ,γ, d,δ =,..., v m+ q λ n+ vλ p
Theoems on multple A-functon tansfom 00 >0, λ >0, Δ >0, Δ >0, μ >0, μ >0 π λ ( ) π ag C p < μ, =,..., ; ag c t <μ, =,..., 2 2 whee Q P Q P () () () () Δ = β - α ; Δ = β - α = = = = = = = = M N Q Q () () () () μ = β + α - β - α = = = = =M+ = =N+ = M N Q Q () () () () μ = β + α - β - α = = = = =M+ = =N+ = b b k Re ρ +λ + () > 0, () β β k =,...,M; k=,...,m ; =,..., ( -a ) ( -a k ) Re ρ-λ - () < 0, () α α k =,...,N; k=,...,n ; =,..., Poof On the R.H.S of (2.6) use (2.5) to get ρ - ρ - ψ( p,...,p ) =...... x...x A C( px ),...,C( px) v λ v λ } A c x t,...,c x t dx...dx f t,...,t... (2.8) In (2.8) use [.6] to get the R.H.S of (2.7).
002 Pathma J and T.M. Vasudevan Nambsan Theoem 3 λ λ If φ p,...,p =... A c p t,...,c p t f t,...,t... (2.9) v v μ ( ) ( ) μ and ψ p,...,p φ p,...,p =... A C p t,...,c p t g t,...,t... (2.0) g t,...,t... k x,...,x ;t,...,t f x,...,x dx...dx = (2.) then whee v v μ ψ p,...,p A c p x,...,c p x =... A C ( p t ),...,C ( p t ) λ λ μ povded the ntegals nvolved ae absolutely convegent. k ( x,...,x ;t,...,t )... (2.2) Poof Multplyng both sdes of (2.2) by f ( x,...,x) to x,...,x fom 0 to, and ntegatng wth espect v λ λ v ψ p,...,p... A c p x,...,c p x f x,...,x dx...dx =... f x,..., x... k x,...,x ;t,...,t μ μ } A C pt,...,c pt... dx...dx (2.3) Usng (2.9) L.H.S of (2.3) s ( v v ) ψ p,...,p φ p,...,p.
Theoems on multple A-functon tansfom 003 Theefoe fom (.6) and (2.0) we get (2.). The change n the ode of ntegaton s vald povded The A-tansfom of g t,...,t and k(x,...,x ;t,...,t exst and the ntegal (2.) s convegent by vtue of De la Vallee Pousen s theoem [, p-504]. Specal cases: When M=0, the above thee theoems educe to the theoems on multple H-functon tansfoms. Futhe when M=N=P=Q=0 and = n above thee theoems we obtan the esults gven by Na. Refeences [] T.J.I A Bownwch,: An ntoducton to the theoy of nfnte sees,macmllan, (955). [2] G. P. Gautam, Asgha Al, A. N. Goyal, A unfed fomal soluton of N-ntegal equatons, Vnana Pashad Anusandhan Patca 29, No., (986), 67-8. [3] K.C Gupta and Mttal,P.K(97): TheH-functon.tansfom,II J.Austal.Math.Unv.Nac.Tucuman Rev.Se.A-22,0-07. [4] V.C Na : Investgatons n tansfom calculus,ph.d thess,unvesty of Raasthan, (968). [5] J Pathma and T.M.Vasudevan Nambsan: Multple Melln tansfom of genealzed H-functons of vaables, Advances n Theoetcal and Appled Mathematcs Vol.5, No: (200), 9-96. [6] H.M.Svastava,Gupta K.C, S.P.Goyal:The functon of one and two vaables wth applcatons,south Asan Publshes,New Delh.
004 Pathma J and T.M. Vasudevan Nambsan Receved: Novembe, 20