A Study of Bedient Polynomials

Transcription

1 Chapter 7 A Study of Bedient Polynomials 7. Introduction The current chapter is a study of Bedient polynomials and gives us a systematic analysis of various unknown results for Bedient polynomials such as generating integrals, operational representations, fractional integrals, fractional derivatives, hypergeometric representations, Laplace transforms, Mellin transforms and relationship with generalized Rice polynomials. 7. Generating Integrals In this section we have obtained various generating integrals for Bedient polynomials R n (β, ) and G n (α, β; ). These are as follows: Γ(β)Γ(β)Γ(γ β)γ(γ β) Γ(γ)Γ(γ) (α) n R n (β, )t n n (γ) n (uv) β ( u) γ β [ ut( ) vt( + ) ] α ( v) γ β dudv, Re(γ) > Re(β) >. (7..) (α + α ) n R n (β, )t n n (γ) n Γ(α)Γ(α )Γ( α α ) (πi) [:] p ( u) α (u ) α F 6 α, β; t( ) u

2 Chapter 7: A study of Bedient polynomials 6 F α, β; t( + ) du, (7..) u where ma { t( ) u Here we have [:] p n ( u) α (u ) β du, t(+ ) } < along the contour. u (πi) Γ( α)γ( β)γ(α + β) (α) n R n (β, )t n β; e u u α F ut( (γ) n Γ(α) ) β; F vt( + ) du, Re(t) < ; Re(α) >. (7..3) (α) n R n (β, )t n n (γ) n [ e u u β ψ α, β; γ, t( ), ut( + Γ(β) ) ] du, Re{t( + )} < ; Re(β) >. (7..4) n (γ) n Γ(γ) (uv) β ( u v) γ β Γ(β)Γ(β)Γ(γ β) { ut( )} α { vt( + )} α dudv, where u, v, u + v ; Re(β) > and Re(γ β) >. (7..5) n Γ(γ + γ ) (u) γ ( u) γ (γ + γ ) n Γ(γ)Γ(γ ) α, β; F ut( α, β; ) F ( u)t( + ) du, γ ;

3 Chapter 7: A study of Bedient polynomials 63 where Re(γ) >, Re(γ ) >, t( ) < & t(+ ) <. (7..6) By taking γ γ α in (7..6), we get n Γ(α) u α [ ut( (α) n Γ(α)Γ(α) )] β ( u) α [ ( u)t( + )] β du, where Re(α) >, t( ) < & t( + ) <. (7..7) Similarly by taking γ γ β in (7..6) we get n Γ(β) u β [ ut( (β) n Γ(β)Γ(β) )] α ( u) β [ ( u)t( + )] α du, wherere(β) >, t( ) < & t( + ) <. (7..8) n (α) n Γ(α) u β [ ut( Γ(β)Γ(α β) )] α α, β; ( u) α β F ( u)t( + ) du, α β; where Re(α) >, t( ) < & t( + ) <. (7..9) Similarly n (β) n Γ(β) u α [ ut( Γ(α)Γ(β α) )] β α, β; ( u) β α F ( u)t( + ) du, β α; where Re(β) >, t( ) < & t(+ ) <. (7..)

4 Chapter 7: A study of Bedient polynomials 64 n Γ( γ)γ( γ )Γ(γ + γ ) (γ + γ ) n (πi) [:] p ( u) γ (u ) γ α, β; F ut( α, β; ) F ( u)t( + ) du, γ ; where ma{ ut( ), ( u)t( + ) } < along the contour. (7..) n (α + ) n G n (α, α + ; )tn (γ) n [ e u u α Ξ α, α + Γ(α) ; t( ), u 4 t( + ] ) du, ( Re t( + ) ) < ; Re(α) >. (7..) n (γ) n [ e u u β Ξ α, α, β; t( ), ut( + Γ(β) ) ] du, Re ( t( + ) ) < ; Re (β) >. (7..3) 7.3 Binomial and Trinomial Operator Representations In this section we have obtained certain results of binomial and trinomial operator representation type for Bedient polynomials R n (β, ) and G n (α, β; ) by using their Rodrigues formula. The results obtained are as follows: n, β, γ n; n, β, γ n; (D + D y ) n 3 F 3 F y γ + β n, ; γ + β n, ; n k ( ) n n n!( γ n) k ( γ + β n) k (γ) n (γ β) n (γ) k (γ β) k R n k (β, )R n (β, y). (7.3.)

5 Chapter 7: A study of Bedient polynomials 65 (D + D y + D z ) n 3 F n, β, γ n; 3 F γ + β n, ; n, β, γ n; y γ + β n, ; 3 F n, β, γ n; z γ + β n, ; ( )n n n! (γ) n (γ β) n n n r r s ( γ n) r+s (γ) r (γ) s (γ β) r ( γ + β n) r+s (γ β) s R n r s (β, )R r (β, y)r s (β, z). (7.3.) (D D y + D D z + D y D z ) n 3 F n, β, γ n; γ + β n, ; 3 F n, β, γ n; y 3 F γ + β n, ; n, β, γ n; z γ + β n, ; { n n! (γ) n (γ β) n } n n r r s ( n) r+s (r + s)!( γ n) r ( γ + β n) r (γ) r+s (γ β) r+s ( n) r ( n) s ( γ n) s( γ + β n) s R n s (β, )R n r (β, y)r r+s (β, z). (7.3.3) r! s! (D + D y ) n 3 F n, α, β;, α + β; 3 F n, α, β; y, α + β; n ( ) n n n! G n k (α, β; )G n (α, β; y). (7.3.4) k n, α, β; n, α, β; n, α, β; (D + D y + D z ) n 3 F 3 F y 3 F z, α + β;, α + β;, α + β; ( ) n n n! n n r r s G n r s (α, β; )G r (α, β; y)g s (α, β; z). (7.3.5)

6 Chapter 7: A study of Bedient polynomials 66 n, α, β; n, α, β; (D D y + D D z + D y D z ) n 3 F 3 F y, α + β;, α + β; n, α, β; 3 F z, α + β; n n! n n r r s (n r)!(n s)!(r + s)! G n s (α, β; ) r! s! (n r s)! 7.4 Hypergeometric Representations Following are the Hypergeometric representations of G n (α, β; ) { t( + (γ) )} α n n F :; :; γ β : α, β; α; γ : γ β; ; G n r (α, β; y)g r+s (α, β; z). (7.3.6) t( t( + ), ) t( +. (7.4.) ) { t( + (γ) )} β n n F :; :; γ α : α, β; β; t( t( + ), ) γ : γ α; ; t( +. (7.4.) ) { t( + (γ) )} γ α β n n F :; :; γ α, γ β : α, β; ; γ : γ α, γ β; ; t{( ) t}, t( + ). (7.4.3) G n (α, β; )t n { t( + )} α n α : α, β; α; F :; :; t( t( + ), ) α + β : α; ; t( +. (7.4.4) )

7 Chapter 7: A study of Bedient polynomials 67 n (α) n { t( + )} β F :; :; α : α, β; β; α : α; ; t( t( + ), ) t( +. (7.4.5) ) 7.5 Relationship with Generalized Rice Polynomials Bedient polynomials are related to the generalized Rice polynomials by the following relations R n (β, ) (β) nγ( β n)( ) n Γ( β) ( β n, β γ n) H n G n (α, β; ) (α) n(β) n Γ( α n)( ) n (α + β)nγ( α) ( 7.6 Fractional Integrals ( α n, α+β ) H n ( β, + ). (7.5.) α, β; + ). (7.5.) In this section we find some fractional integrals of R n (β, ) and G n (α, β; ). These are as given below: n (γ + γ ) n Γ(γ + γ ) (u) µ ( u) γ+γ µ Γ(γ + γ µ)γ(µ) α, β; F ut( α, β; ) F ( u)t( + ) du, µ; γ + γ µ; where Re(γ) >, Re(γ ) >, t( ) < & t( + ) <. (7.6.)

8 Chapter 7: A study of Bedient polynomials 68 [ )] I µ n β R n (β, n (β) n Γ( β n) β µ+n Γ( β + µ n) n, n +, γ β; 3 F γ, β + µ n; I µ [ n α G n (α, β; )] n (α) n (β) n Γ( α n) n!(α + β) n α µ+n Γ( α + µ n) n, n +, α β n; 3 F α + µ n, β n;. (7.6.). (7.6.3) Riemann-Liouville left sided fractional integral of order α ai α {R n (β, ( a))} (β) n n ( a) n+α Γ(n + α + ) n α, n α+, γ β; 3 F γ, β n; ( a). (7.6.4) ai λ {G n (α, β; ( a))} (α) n(β) n n ( a) n+λ (α + β) n Γ(n + λ + ) 3 F n λ, n λ+, α β n; α n, β n; ( a). (7.6.5) Riemann-Liouville right sided fractional integral of order α Ib α {R n (β, (b ))} n (β) n (b ) n+α Γ(n + α + ) n α, n α+, γ β; 3 F γ, β n; (b ). (7.6.6) I λ b {G n (α, β; (b ))} (α) n(β) n n (b ) n+λ (α + β) n Γ(n + λ + ) 3 F n λ, n λ+, α β n; α n, β n; (b ). (7.6.7)

9 Chapter 7: A study of Bedient polynomials 69 The Weyl integral of f() of order α, denoted by W α, is given by W α {R n (β, )} ( )n (β) n n+α ( + α) n n! 3F n α, n α+, γ β; γ, β n;. (7.6.8) W λ {G n (α, β; )} (α) n(β) n ( ) λ () n n+λ ( + n) λ (α + β) n n! 3 F n λ, n λ+, α β n; α n, β n;. (7.6.9) Erdelyi-Kober operator of first kind I[α, η; R n (β, )] (β) n () n ( + n + η) α Γ( + n) 5 F 4 (, n), (, n α η), γ β; γ, β n, (, n η);. (7.6.) I[λ, η; G n (α, β; )] (α) n (β) n () n (α + β) n ( + n + η) λ Γ( + n) (, n), (, n λ η), α β n; 5 F 4 (, n η), α n, β n;. (7.6.) Erdelyi-Kober operator of second kind I[α, η; R n (β, )] (β) n () n ( n + η) α Γ( + n) (, n), (, n + η), γ β; 5 F 4 γ, β n, (, n + α + η);. (7.6.) I[λ, η; G n (α, β; )] (α) n (β) n () n (α + β) n ( n + η) λ Γ( + n) (, n), (, n + η), α β n; 5 F 4 α n, β n, (, n + λ + η);. (7.6.3)

10 Chapter 7: A study of Bedient polynomials 7 Saigo integral operator of first kind I α,β,η + {R n (λ, µ; )} (λ) n n n β Γ( + n + η β) Γ( + n β)γ( + n + α + η) (, n + β), (, n α η), µ λ; 5 F 4 µ, λ n, (, n + β η); I λ,µ,η + {G n (α, β; )} (α) n (β) n n n µ Γ( + n µ + η) (α + β) n Γ( + n µ)γ( + n + λ + η) (, n + µ), (, n λ η), α β n; 5 F 4 α n, β n, (, n + µ η);. (7.6.4). (7.6.5) Saigo integral operator of second kind I α,β,η {R n (λ, µ; )} n (λ) n n Γ( n)γ( n β + η) Γ( + n)γ( n β)γ( n + α + η) (, n), (, n + ), (, n β + η + ), µ λ; 7 F 6 µ, λ n, (, n β + ), (, n + α + η + );. (7.6.6) I λ,µ,η {G n (α, β; )} (α) n (β) n n n Γ( n)γ( n µ + η) (α + β) n Γ( + n)γ( n µ)γ( n + λ + η) (, n), (, n + ), (, n λ + η + ), α β n; 7 F 6 α n, β n, (, n µ + ), (, n + λ + η + );, (7.6.7) where we have used t ρ (t ) c F a, b; c; d t Γ(c)Γ(ρ)Γ(ρ + c b a) Γ(ρ + c a)γ(ρ + c b) tρ+c, where ρ, a, b, c C, Re(ρ) >, Re(c) >, Re(ρ + c a b) > and a, b; ρ ( t) c F t Γ(c)Γ( ρ c)γ( ρ a b) d Γ( ρ a)γ( ρ b) t c; t ρ+c, where ρ, a, b, c C, Re(c) >, Re(ρ + c) < Re(ρ + a + b) <.

11 Chapter 7: A study of Bedient polynomials Fractional Derivatives A number of fractional derivatives of R n (β, ) and G n (α, β; ) are as follows: D β n, n + ; n Γ(γ β)n! γ β F β n; n (β) n Γ(γ) +γ R n (β, ). (7.7.) D α α β n F n, n + ; α n; (α) n(β) n n Γ( α β n) (α + β) n Γ( β n) n! β 3n Gn (α, β; ). (7.7.) The left sided Riemann-Liouville fractional derivative of order α ad α {R n (β, ( a))} (β) n n ( a) n α Γ(n α + ) 3 F n+α, n+α+, γ β; γ, β n; ( a). (7.7.3) ad λ {G n (α, β; ( a))} (α) n(β) n n ( a) n λ (α + β) n Γ(n λ + ) 3 F n+λ, n+λ+, α β n; α n, β n; ( a). (7.7.4) The right sided Riemann-Liouville fractional derivative of order α D α b {R n (β, (b ))} (β) n n (b ) n α Γ(n α + ) 3 F n+α, n+α+, γ β; γ, β n; (b ). (7.7.5)

12 Chapter 7: A study of Bedient polynomials 7 Db λ {G n (α, β; (b ))} (α) n(β) n n (b ) n λ (α + β) n Γ(n λ + ) n+λ, n+λ+, α β n; 3 F α n, β n; (b ). (7.7.6) The Weyl fractional derivative of f() of order α, denoted by D α, is given by D α {R n (β, )} ( )α () n (β) n n α Γ(n + α) 3F n+α, n+α+, γ β; γ, β n; (7.7.7) D {G λ n (β, )} (α) n(β) n ( ) λ () n n λ (α + β) n Γ(n + α) n+λ, n+λ+, α β n; 3 F α n, β n; D λ µ Y Y λ ( Y ) α 3F α, λ; 7.8 Laplace Transforms µ; X Y Γ(λ) Γ(µ) Y µ n. (7.7.8) (α) n R n (λ, µ; )t n (µ) n, where X ( )t and Y ( + )t. (7.7.9) The Laplace transform of R n (β, ) and G n (α, β; ) is as given below γ β; {R n (β, ) : s} n (β) n s F s n+. (7.8.) 4 γ, β n; {G n (α, β; ) : s} n (α) n (β) n (α + β) n s n+ F α β n; α n, β n; s 4. (7.8.) β; tα F ( β; )t F ( + )t : s Γ(α) s α (α) n R n (β, )t n. (7.8.3) n (γ) n s n

13 Chapter 7: A study of Bedient polynomials Mellin Transform Following are the Mellin transforms of R n (β, ) and G n (α, β; ) M{e R n (β, ) : s} n (β) n Γ(s + n + ) n! (; n), γ β; 3 F 4 γ, β n, (; n s + );. (7.9.) 4 M{e G n (α, β; ) : s} n (α) n (β) n Γ(s + n) Γ(n + )(α + β) n (; n), α β n; 3 F 4 α n, β n, (; n s + );. (7.9.) 4 The contents of this chapter accepted/published/communicated in the form of a paper as mentioned below: (i) A study of Bedient polynomials, communicated for publication.