ON GENERALIZATION OF SISTER CELINE S POLYNOMIALS

Transcription

1 Palestie Joural of Mathematics Vol. 5() (6), 5 Palestie Polytechic Uiversity-PPU 6 ON GENERALIZATION OF SISTER CELINE S POLYNOMIALS Khursheed Ahmad, M. Kamarujjama ad M. Ghayasuddi Commuicated by Jose Luis López-Boilla MSC Classificatios: 33C6, 33C45. Keywords ad phrases: Sister Celie polyomials, Batema polyomials, Bessel polyomials, Laguerre polyomials, Geeralized Rice polyomials ad Pasterak polyomials. Abstract. This paper deals with a geeralizatio of Sister Celie s polyomials, their geeratig fuctios ad itegral represetatios. A set of relatios with other polyomials are give i the last sectio. Itroductio Sister Celie has itroduced the polyomial (x) a,, a p ; b,, b q ; x p+ F q+, +, a, a p ;,, b,, b q ; x, (.) which is defied by the followig geeratig fuctio (see 4, p.9) ( t) a,, a p ; 4xt a,, a p ; pf q b,, b q ; ( t) b,, b q ; x t, (.) where p F q deotes the geeralized hypergeometric fuctio 4. For p, q, a, b the followig itegral represetatio of Sister Celie polyomials is give ( ; ; x) y e y ( ; ; xy)dy. (.3) π Equatio (.) with o a s ad o b s deotes simply (x) F, + ;, ; x For the (x) the geeratig fuctio (.) becomes r ( ) r ()! () ( ) r( r)!. (.4) ( t) exp( 4xt ( t) ) (x)t. (.5) Geeralizatio of Sister Celie s Polyomials I the view of above results, we defie the geeralized Sister Celie polyomial i followig maer (α,β) a,, a p ; x ( + α + β) b,, b q ;, + α + β +, a,, a p ; p+f q+ + α,, b,, b q ; x. (.)

2 6 Khursheed Ahmad, M. Kamarujjama ad M. Ghayasuddi Equatio (.) with o a s ad o b s deotes simply (x) ( + α + β) F, + α + β + ; + α, ; x (.) ( + α + β) ( ) r ( + α + β + ) r ( + α) r ( ). (.3) r r Ideed f (,) (x) (x). (.4) 3 Geeratig fuctios The followig geeratig fuctio ca be easily obtaied, (x)t ( t) α β F +α+β (C) (α,β) (x)t ( t) C α β 3F 3 +α+β ; + α, ; ( t) 4xt +α+β +α+β C,, ; 4xt + α, + α + β ; ( t), (3.) (Obviously for C, α β, equatio (3.) reduces to the geeratig fuctio (.5)), ad +α+β +α+β ( t) α β a, a p,, ; 4xt p+f q+ b, b q, + α, ; ( t) a, a p ; b, b q ; x (3.) t. (3.3) Proof. Expadig left had member of (3.) with the help of (.), we get (α,β) (x)t ( + α + β) r r r ( ) r ( + α + β + ) r ( + α) r ( ) r ( ) r ( + α + β) +r ( r)! ( + α) r ( ) r t ( ) r ()! ( + α + β) r ( xt) r r ( + α) r ( ) r r ( t) α β ( + α + β) +r ( + α) r ( ) r t+r ( + α + β + r) t ( + α + β) r ( xt) r ( + α) r ( ) r( t) +α+β+r r ( t) α β r ( + α + β) r ( xt) r ( + α) r ( ) r ( t) r ( +α+β ) r ( +α+β ) r ( + α) r ( ) r ( 4xt ( t) ) r t Thus we arrive at the result (3.). Similarly we ca easily prove equatios (3.) ad (3.3).

3 ON GENERALIZATION OF SISTER CELINE S POLYNOMIALS 7 4 Itegral represetatios The followig itegral represetatios ivolvig Sister Celie s polyomials are obtaied as follows: (a). (α,β) (x) ( + α + β) ( + α) π For α β, equatio (4.) reduces to kow result (x) π t ( t) P (α,β) ( xt)dt. (4.) t ( t) P ( xt)dt, (4.) where P (α,β) (x) ( + α) F, + α + β + ; + α; x. (b). (c). Γ(β + γ) Γ(β)Γ(γ) t β ( t) γ P (α,β) ( + α) ( x) ( + α + β) π (x( t)) ( + α + β) 3F 3 e y y f (α,β) (xy)dy, for α β, equatio (4.4) reduces to kow result, p.9, γ, + α + β + ; + α, β + γ, ; x. (4.3) (4.4) P ( x) e y y f (xy)dy (4.5) π ad (α,β) ( ; ; x) π e y y f (α,β) ( ; ; xy)dy. (4.6) Proof. (a) (b) ( + α + β) π ( + α + β) π( + α) r ( + α + β) t ( t) P (α,β) ( xt)dt ( ) r ( + α + β + ) r ( + α) r r ( ) r ( + α + β + ) r ( + α) r ( ) r (α,β) (x). t r ( t) dt ( + α + β) Γ(β + γ) Γ(β)Γ(γ) r t β ( t) γ (α,β) (x( t))dt ( ) r ( + α + β + ) r ( + α) r ( ) r ( + α + β) r ( + α + β) 3F 3 ( ) r ( + α + β + ) r (γ) r ( + α) r ( ) r(β + γ) r, + α + β +, γ; + α, β + γ, ; x t β γ+r Γ(β + γ) ( t) Γ(β)Γ(γ) dt.

4 8 Khursheed Ahmad, M. Kamarujjama ad M. Ghayasuddi (c) ( + α) π ( + α) ( + α + β) π r ( + α) e y y f (α,β) (xy)dy ( ) r ( + α + β + ) r Γ( )Γ(r + ) r ( + α) r Γ(( + r) ( ) r ( + α + β + ) r ( + α) r P α,β) ( x). This completes the proof of (4.), (4.3) ad (4.4). 5 Relatio with other polyomials Batema s polyomial ( ; ; x) ( + α + β) Z (α,β) (x), (5.) where Z (α,β) (x) Geeralized Batema s polyomials 3. For α β, we get ad also we get f (,) ( ; ; x) Z (x), (5.) ( + α) ( + α + β) Z (α,β) (x) π for α β, equatio (5.3) reduces to kow result 4, pp.9 Bessel polyomials y e y ( ; ; xy)dy, (5.3) Z (x) y e y ( ; ; xy)dy. (5.4) π y (α,β) (; ; x) ( + α) (x), (5.5) where y (α,β) (x) (+α+β) F o (, +α+β+; ; x ), is the Bessel s polyomials 5, p.75. Obviously, y (, ) y (x) F o (, + ; ; x ). (5.6) Laguerre polyomials { ( + α) } f (α,α) ( ; ; 4 x ) L α (x)l α ( x), (5.7) obviously for α, the result (5.7) reduces to the kow result 4, p.9 Geeralized Rice polyomials ( + α) ( + α + β) ( ; ; x 4 ) L (x).l ( x). (5.8) e t t ξ (α,β) (xt)dt Γ(ξ)H (α,β) (ξ, p, v), (5.9)

5 ON GENERALIZATION OF SISTER CELINE S POLYNOMIALS 9 where H (α,β) (ξ, p, v) (+α+β) due to Khadekar. r ( ) r(+α+β+) r(ξ) r (+α) r(p) r v r is the geralized Rices polyomial (α,β) (,, ξ; p; v) H (α,β) (ξ, p, v) ( + α + β), + α + β +, ξ; 3 F + α, p; v. (5.) Geeralized Batema s polyomials (α,β) (,, + z ;, p; ) ( ) r ( + α + β + ) r ( +z ( + α) r (p) r r (+z), + α + β +, 3 F ; + α, p; which is a Jacobi type geeralized Batema s polyomial F (Z) ad (,, + Z ;, ; ) 3 F ) r F (α,β) (p, z), (5.) +Z, + α + β +, ; + α, ; For β α, equatio (5.) reduces to f (α,α) (,, + Z ;, ; ) 3 F F (α,β) (, Z). (5.), + α + ; + α, ; Pasterak s polyomials (,, + Z + m ; m + ; ) 3 F F (α,α) (, Z). (5.3), + α + β +, +Z+m ; + α, m + ;, (5.4) which is geeralized Pasterak s polyomial deoted by F m (Z), For α β, the above equatio (5.4) reduces to (α,α) (,, + Z + m, + α +, ;, m + ; ) 3 F ( + Z + m);, + α, m + ; (5.5) which is a ultra spherical type geeralized Pasterak s polyomial F,m (α,α) (Z). For α it reduces to Pasterak s polyomials F m (Z). Refereces Fasemyer, Sister, M. Celie, Some geeralized hypergeometric polyomials, Bull. Amer. Math. Soc. 53 (947) Khadekar, P. R., O a geeralizatio of Rice s polyomial, I. Proc. Nat. Acad. Sci. Idia, Sect. A34 (964), Kha, M. A. ad Nisar, K. S., O operator represetatio of various polyomials usig the operator of W. A. Al-Salam, Iteratioal Joural of computatioal ad applied Mathematics, (3), 5 (), pp

6 Khursheed Ahmad, M. Kamarujjama ad M. Ghayasuddi 4 Raiville, E. D., Special Fuctio, MaccMillo, New York, 96 Repriterd by Chelsea Publishig Compay, Brox, New York, Srivastava, H. M. ad Maocha, H. L., A Treatise o geeratig fuctio, Ellis Horwood Limited Publishers, Chichester Halsfed Press, a divisio of Joh Wiley ad Sos, New York (984). Author iformatio Khursheed Ahmad, Departmet of Mathematics, Vivekaad College of Techology ad Maagemet, Kher- Road, Aligarh., Idia. khursheedahmad8@rediffmail.com M. Kamarujjama, Departmet of Applied Mathematics, Faculty of Egieerig ad Techology, Aligarh Muslim Uiversity, Aligarh-, Idia. mdkamarujjama@rediffmail.com M. Ghayasuddi, Departmet of Applied Mathematics, Faculty of Egieerig ad Techology, Aligarh Muslim Uiversity, Aligarh-, Idia. ghayas.maths@gmail.com Received: February, 5. Accepted: May 9, 5