A Study on New Sequence of Functions Involving the Generalized Contour Integral

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Globl Jourl of Scece Froter Reerch Mthetc d Deco Scece Volue 3 Iue Vero.

(USA Ole ISS: 2249-4626 & Prt ISS: 975-5896 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl By Mehr Chd Mlw College of IT d

5 whch volvg wth the geerlzed PQ;R.By ug opertol techque oe teretg geertg relto d fte uto forule re obted ecto 2 d 3.

For the ke of llutrto we record here oe pecl ce of our forul ecto 4 whch re lo ew d kow.

1 Globl Jourl of Scece Froter Reerch Mthetc d Deco Scece Volue 3 Iue Vero. Yer 23 Type : Double Bld Peer Revewed Itertol Reerch Jourl Publher: Globl Jourl Ic. (USA Ole ISS: & Prt ISS: A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl By Mehr Chd Mlw College of IT d Mgeet Bthd Abtrct - The of the preet pper ttept to troduce ew equece of fucto ( { ;PQR; V ( ; k / 2... } cotour tegrl I [ ] = equto (.5 whch volvg wth the geerlzed PQ;R.By ug opertol techque oe teretg geertg relto d fte uto forule re obted ecto 2 d 3. Thee geertg relto d fte uto forule re ufed ture d ct key forule fro whch we c obt ther pecl ce. For the ke of llutrto we record here oe pecl ce of our forul ecto 4 whch re lo ew d kow. Keyword : geertg fucto geerlzed cotour tegrl opertol techque. GJSFR-F Clfcto : MSC 2: B34 R33 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Strctly per the coplce d regulto of : 23. Mehr Chd. Th reerch/revew pper dtrbuted uder the ter of the Cretve Coo Attrbuto- ocoercl 3. Uported Lcee perttg ll o coercl ue dtrbuto d reproducto y edu provded the orgl work properly cted.

Ref.. Chk A. M. (956 A cl of polyol d geerlzto of trlg uber Duke J. Mth. 23 45-55.

equece of fucto ( { V ;PQR; ( ; k / 2... } [ ] IPQ;R = equto (.5 whch volvg wth the geerlzed cotour tegrl.

Thee geertg relto d fte uto forule re ufed ture d ct key forule fro whch we c obt ther pecl ce.

Keyword : geertg fucto geerlzed cotour tegrl opertol techque. I.

I 956 Chk [] defed cl of polyol k+ k ( k ( ( G = e D

2 Ref.. Chk A. M. (956 A cl of polyol d geerlzto of trlg uber Duke J. Mth A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Abtrct - Mehr Chd The of the preet pper ttept to troduce ew equece of fucto ( { V ;PQR; ( ; k / 2... } [ ] IPQ;R = equto (.5 whch volvg wth the geerlzed cotour tegrl. By ug opertol techque oe teretg geertg relto d fte uto forule re obted ecto 2 d 3. Thee geertg relto d fte uto forule re ufed ture d ct key forule fro whch we c obt ther pecl ce. For the ke of llutrto we record here oe pecl ce of our forul ecto 4 whch re lo ew d kow. Keyword : geertg fucto geerlzed cotour tegrl opertol techque. I. Itroducto Opertol techque hve drw the tteto of everl reercher the tudy of equece of fucto d polyol. I 956 Chk [] defed cl of polyol k+ k ( k ( ( G = e D e (. d Where k cott d = 2... D. d Gould d Hopper [5] troduced geerlzed Herte polyol 962 ( = ( ep( ep( r r r H p p D p (.2 Chttere [4] tuded cl of polyol for geerlzed Lguerre polyol 964 ( ( r ep( ( 2 r Tr p = p D + ep( p! (.3 I 968 Sgh [6] tuded the geerlzed Truedell polyol defed ( r ( ep( ( r = ep( T r p p D p (.4 23 Jourl of Scece Froter Reerch Volue XIII Iue I V ero I 9Globl Yer F Author : Deprtet of Mthetc Mlw College of IT d Mgeet Bthd-5 Id. 23 Globl Jourl Ic. (US

3 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Srvtv d Sgh [9] troduced geerl cl of polyol 97 ( k r k + ( ep( ( r G r p k = p D ep( p! (.5 23 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer I 97 the Rodrgue forul for geerlzed Lgurre polyol gve by Mttl [8] ( Tk ( = ep pk ( D + pk! Where pk ( polyol of degree k. Mttl [9] lo proved followg relto for (.6 ( ep( ( ( ep( ( ( k ( = ep k ( k + T p T p! Where cott d T ( D +. Recetly Shukl d Prpt [5] obted everl properte of (.7. Chdel [2] lo tuded cl of polyol defed 973 ( = ep( ( ep( k r k r T r p p D p I 974 Chdel [3] etblhed geerlzto of polyol yte hg( ( ( k hg( = ( G hg k e D e (.6 (.7 (.8 (.9 Where g( fucto of h d k re cott. I the e yer Srvtv [4] dcued oe opertol forul geerlzed fucto ( r ( F k p the for ( r r k ( ep( ( + k r = ep( F k p p D p I 975 Joh d Prpt [6] troduced the cl of polyol k ( k Mν ( k = ep( p ( ν ( + D ep( pν (! (. (. Subequetly 975 Ptl d Thkre [] hve obted everl forule d geertg relto for Ref. 8. Mttl H. B. (97 A geerlzto of Lguerre polyol Publ. Mth. Debrece ( r k ( ep( ( + r λ = λ+ ep( P r k p D p (.2 23 Globl Jourl Ic. (US

4 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl defed I 979 Srvtv d Sgh [7] tuded equece of fucto ( V ( k ; ( V ( ; k = ep{ pk ( } θ ep{ pk ( }! (.3 Ref. 7. Srvtv A.. d Sgh S.. (979 Soe geertg relto coected wth fucto defed by Geerlzed Rodrgue forul Id J. Pure Appl. Mth. ( By eployg the opertor θ ( + D where cott d p ( k polyol of degree k. J.C. Prpt d.k. Ahud [2] troduced the equece of fucto defed ( βδ β β V ( ; k = W( δ ; pk ( ( T W( δ ; pk (! A ew equece of fucto pper ( ; PQ ; { V ( k ; / 2... } (.4 = troduced th ( ; ; V ( ; = ; ( ( ; ( k I! pk T I pk d Where T ( + D D d re cott β d o-egtve teger pk ( polyol of degree k d I [ ] defed d repreeted the followg er [3]: d I z ( ( ( b β ( b β + P φξ ξ = ( z dξ 2π + M M L Q M Γ( bβξ Γ( + ξ = = φ( ξ = R Q P Γ + Γ = = M+ = + ( b βξ ( ξ PQ;R M PQ re teger tfy ( =... (.5 k fte d I-fucto (.6 (.7 P M Q R R fte β β re potve uber d b b re cople uber. I-fucto whch geerlzed for of the well kow Fo H-fucto [8]. I the equel the I- fucto wll be tuded uder the followg codto of etece: Aπ (I A > rg < z ( Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 23 Globl Jourl Ic. (US

5 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Aπ (II A rg z d Re( B + < (.9 2 P Where A = + β β = ( 2... R (.2 = = + = = M+ M Q 23 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 2 M Q P B b b P Q R (.2 Ad = + + ( = ( 2... = = M+ = = + 2 Soe geertg relto d fte uto forule of cl of polyol or equece of fucto hve bee obted by ug the properte of the dfferetl d opertor. T ( + D T ( + D where D bed o the work of d Mttl [] Ptl d Thkre [] Srvtv d Sgh [7]. Soe ueful Opertol Techque re gve below: β + ( β β ( tt ( f ( ( t ( f t ep = (.22 ( + ( tt ( ( ( f t f ( t / ep = + + (.23 = = (.24 ( T ( uv ( T ( v( T ( u + D + + D + + D + + D + + D β = β ( ( ( 2 ( 3...( ( β M = ( t ( t ( t (.25 β β = =! (.26 II. Geertg Relto ( ; ; ( ; = ( ( k( k ( ( ( V k t t I p I p t (2. ( ( ( M ; P Q R; + ( ; ( M M V k t = + t I ( pk( I pk / ( + t (2.2 = = + ( ; ; V M ( k ; t ( ( ( ( k( ( ( + M I pk ( ; ; = ( t V t ; k I p t (2.3 Ref. 7. Srvtv A.. d Sgh S.. (979 Soe geertg relto coected wth fucto defed by Geerlzed Rodrgue forul Id J. Pure Appl. Mth. ( Globl Jourl Ic. (US

6 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Proof of (2. Fro (.5 we coder M M ; M k k = ( ; ; ( ; ( ( ep( = ( ( V k t I p tt I p (2.4 ote Ug opertol techque (.22 bove equto (2.4 reduce to: Ad replcg t by Proof of (2.2 V M = ( ; ; ( ; k t ( / k k ( ( ( ( ( = I p t I p t Ag fro (.5 we hve ( ( ( M M / k k = ( t I p ( I p ( t (2.5 t th gve (2.. ( M ; P Q R ; ( ; ( V = M ; ( ep( M ; ( ( k t I pk tt I pk (2.6 = pplyg the opertol techque (.23 we get Th prove (2.2. Proof of (2.3 We c wrte (.5 ( M ; P Q R; V ( k ; t = + ; ( ( ( k k ( / ( = I p + t I p + t ( + / = ( + t I ; ( ( ; ( + pk I pk t ( ; ; ( T ( ( =! I pk V ( k ; ; (2.7 I ( pk( ( Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 3 Or ( ( ( { ; } ep t T T I( pk( 23 Globl Jourl Ic. (US

7 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl 23 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 4 ( ; ; ( M =!ep tt V ( ; k M I ;( pk( = t + ( T ;( (! I pk ( ; ; ( M =!ep T V ( k ; M I ;( pk( (2.9 Ug the opertol techque (.2 bove equto c be wrtte : = t + ( T ;( (! I pk ( k( ( ( ( ; ;! ( M = t V ( t ; k I p t ue of (2.8 gve Therefore ( +! ( ; ; ( ; t V + k!! I ( p( = k ( ( ; ; ( M = t V ( ; t k M I ; ( ( pk t = + ( ; ; V + ( kt ; ( k( ( ( I ;( ( ( ; ; ( pk M = t V ( t ; k I p t Ad replcg t by t th gve the reult (2.3. (2. (2. (2.2 ote III. Fte Suto Forule V k V k =! ( ; ; ( ; ; ( ; = ( ( ; (3. 23 Globl Jourl Ic. (US

8 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl β V k V k =! ( ; ; ( ; ; β ( ; = ( ( ; (3.2 ote Proof of (3. Fro equto (.5 we hve: M ( ( ( ; ; ( V ( ; k = I ( pk( T ;! I pk ( (3.3 Ug the opertol techque (.24 we hve: 23 V ( ; ; ( k ; = I p ( T I p ( T! ( ( k( k =! ( = I ; ( pk(!!! = ( ( ( ( 2... ( ( ( ( ( + D + + D + + D + + D ( k( ( ( ( 2... ( ( ( I p + D + + D + + D + + D M Ug the reult (.24 we hve: = I p + + D I p ; ( ( ( ; ( ( =!( M k! k = Put = d replcg by Th gve ( ( (3.3 we get: ( ; ; V ( ; = ; ( ( ( ; ( ( k I pk T! I pk T I p = V k! ( ( ( ( ;! M I ; ( pk( = k ( + + D I p ( ( ( k = ( ; ; ( ; V k I p ( k( ( ; ; ( ; (3.4 (3.5 (3.6 (3.7 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 5 Fro equto (3.5 d (3.7 we hve the the reult. 23 Globl Jourl Ic. (US

9 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Proof of (3.2 Equto (.5 c be wrtte : ( ( ( ; ; ( M ( ; = M ; ( ( ep M k k = ( V k t I p T I p (3.8 Applyg the (.2 equto (3.8 we hve: ote 23 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 6 V M = ( ; ; ( ; k t ( ( + k k ( ( ( ( = I p t I p t + ( M M ; ( ( ; ( k k ( ( / = t I p I p t (3.9 Applyg the reult fro equto (.26; equto (3.9 reduce to: ( ( t ( ( / β β k k =! ( ( ( = t I p I p t = = = = ( t β β β = I ; ( ( ep( ; ( ( pk T =! I pk ( + β t β β = I ; ( ( ( ; ( ( pk T!! I pk ( β t β β = I ( pk( ( T I!! ( ow equtg the coeffcet of t V we get: ( ; ; ( k ; ( k( p (3. ( β β β = I ; ( ( ( ; ( ( pk T =!! I pk ( (3. Ug the equto (.5 (3. we hve the reult ( Globl Jourl Ic. (US

10 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl IV. Specl Ce If we put R = I-fucto reduce to Fo H-fucto [8 p. Eq. (2..] the the equto (2. (2.2 d (2.3 tke the followg for: Ref. M PQ = ( ; ; ( ; V k t ( ( ( t H PQ ( pk( H PQ pk ( t = ( Meer C.S. O the G-fucto Proc. t. Acd. Wetech 49 p. 227 (946. = ( ; ; ( ; V k t M P Q ( ( + ( M M t H PQ ( pk( H PQ pk ( t / = + + = PQ + ( ; ; V M PQ ( k ; t ( ( ( ( k( ( ( + M H PQ pk ( ; PQ ; = ( t V t ; k H p t (4.2 (4.3 b If we put R = ; = β = the the I-fucto reduce to geerl type of G-fucto [7].e. ( ( ( ( I z G z ( ( + P ; = M p PQ b b b M M + Q q (2.3 tke the followg for: M PQ = ( ; ; ( ; V k t ( t GPQ ( pk( GPQ pk ( t the equto (2. (2.2 d ( ( = = ( ; ; ( ; V k t M P Q ( ( + ( M M t GPQ ( pk( GPQ pk ( t / = + + (4.4 (4.5 Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 7 23 Globl Jourl Ic. (US

11 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 23 8 c If we put R = = P = PM = Q = Q + b = β = = b = b β = β the I-fucto reduce to the geerlzed wrght hypergeoetrc fucto [2 p.287].e. PQ = ( P ( ( b β PQ = ( ; ( ; V k t ( ( ( t PψQ( pk ( PψQ pk ( t = = ( ; ( ; V k t P Q ( ( + ( t PψQ( pk ( PψQ pk ( t / = + + = + ( ; V PQ ( k ; t ( p ( ( ( p ( t ( ( ψ P Q k ( PQ ; = ( t V t ; k ψ P Q k ( ( b β I z ; z the equto (2. (2.2 d p P + ; ψ pq = P Q Q Q (2.3 tke the followg for: + ( ; ; V M PQ ( k ; t ( ( ( ( k( (4.7 (4.8 (4.9 d The teretg pecl ce/reltohp betwee (.5 d cl of polyol (.-(.4 c lo be obted for pproprte vlue of preter k d thoe of volvg I-fucto. Referece Référece Referec. Chk A. M. (956 A cl of polyol d geerlzto of trlg uber Duke J. Mth Chdel R.C.S. (973 A ew cl of polyol Id J. Mth. 5( k 3. Chdel R.C.S. (974 A further ote o the cl of polyol T ( r p Id J. Mth.6( Chttere S. K. (964 O geerlzto of Lguerre polyol Red. Mt. Uv. Pdov ( ( + M G PQ pk ( ; PQ ; = ( t V t ; k G p t (4.6 ote 23 Globl Jourl Ic. (US

12 A Study o ew Sequece of Fucto Ivolvg the Geerlzed Cotour Itegrl ote 5. Gould H. W. d Hopper A. T. (962 Opertol forul coected wth two geerlzto of Herte polyol Duck Mth. J Joh C. M. d Prpt M. L. (975 The opertor T k d geerlzto of cert clcl polyol Kyugpook Mth. J Meer C.S. O the G-fucto Proc. t. Acd. Wetech 49 p. 227 ( Mttl H. B. (97 A geerlzto of Lguerre polyol Publ. Mth. Debrece Mttl H. B. (97 Opertol repreetto for the geerlzed Lguerre polyol Glk Mt.Ser III 26( Mttl H. B. (977 Bler d Blterl geertg relto Aerc J. Mth Ptl K. R. d Thkre. K. (975 Opertol forul for fucto defed by geerlzed Rodrgue forul-ii Sc. J. Shv Uv Prpt J. C. d Aud. K. O ew Sequece of Fucto d Ther MATLAB Coputto Itertol Jourl of Phycl Checl & Mthetcl Scece Vol. ; o. 2: ISS: X Accepted O: Se V.P. A Forl Soluto of Cert ew Pr of Dul Itegrl Equto Ivolvg H-Fucto Proc. t. Acd. Sc. Id Sect A52 ( Shrvtv P.. (974 Soe opertol forul d geerlzed geertg fucto The Mth. Educto Shukl A. K. d Prpt J. C. (27 O oe properte of cl of Polyol uggeted by Mttl Proyeccoe J. Mth. 26( Sgh R. P. (968 O geerlzed Truedell polyol Rvt de Mthetc Srvtv A.. d Sgh S.. (979 Soe geertg relto coected wth fucto defed by Geerlzed Rodrgue forul Id J. Pure Appl. Mth. ( Srvtv H.M. Gupt K.C. d Goyl S.P. The H-fucto of oe d two vrble wth pplcto South A Publher ew Dehl Mdr ( Srvtv H. M. d Sgh J. P. (97 A cl of polyol defed by geerlzed Rodrgue forul A. Mt. Pur Appl. 9( Wrght E.M. (935 The yptotc epo of the geerlzed hypergeoetrc fucto. J. Lodo Mth. Soc Globl Jourl of Scece Froter Reerch F Volue XIII Iue I V ero I Yer 9 23 Globl Jourl Ic. (US

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n 0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke