italian journal of pure and applied mathematics – n.
31−2013 (15−20)
15
ON BINOMIAL OPERATOR REPRESENTATIONS OF SOME POLYNOMIALS
K.S. Nisar Department of Mathematics Salman bin Abdul aziz University Wadi Addawser Riyad region Kingdom of Saudi Arabia e-mail:
[email protected]
Mumtaz Ahmad Khan1 Department of Applied Mathematics Faculty of Engineering and Technology Aligarh Muslim University Aligarh – 202002, U.P. India e-mail: mumtaz ahmad khan
[email protected]
Abstract. Based on the technique used by M.A. Khan and A.K. Shukla [2] here finite series representations of bionomial partial differential operators have been used to establish operator representations of various polynomials not considered in the earlier mentioned paper. The results obtained are believed to be new. Keywords: Binomial operator, Operational represenations. AMS (MOS) Subject Classification (2000): Primary 47F05, Secondary 33C45.
1. Introduction Recently, in 2009, M.A. Khan and A.K. Shukla [2] evolved a new technique to give operator representations of certain polynomials. They give binomial and trinomial operator representation of certain polynomials. The aim of the present paper is to strengthen the technique evolved by obtaining binomial operator representations of some more polynomials not considered in the above mentioned paper. 1
Corresponding author
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k.s. nisar, mumtaz ahmad khan
2. The definitions, notations, and results used In deriving the operational representations of various polynomials use has been made of the fact that D µ xλ =
(2.1)
Γ(1 + λ) xλ−µ , Γ(1 + λ − µ)
D≡
d dx
where λ and µ, λ ≥ µ are arbitrary real numbers. In particular, use has been made of the following results: (2.2)
Dr e−x = (−1)r e−x
(2.3)
Dr x−α = (α)r (−1)r x−α−r , α is not an integer
(2.4)
Dr x−α−n = (α + n)r (−1)r x−α−n−r
(2.5)
Dn−r xα−1+n =
(2.6)
D
n−r
x
−α
(α)n α−1+r x (α)r
(α)n (−1)n = x−α−n+r , α is not an integer, (1 − α − n)r
where n and r are denoting the positive integers and (a)n = a (a + 1) · · · · · · (a + n − 1);
(a)0 = 1.
We also need the definitions of the following polynomials in terms of hypergeometric function and also their notations (see [1], [3], [4], [5]). Cesaro polynomials (s)
It is denoted by the symbol gn (x) and is defined as µ ¶ · ¸ s+n −n, 1 ; (s) (2.7) gn (x) = x 2 F1 n −s − n ; Miexner polynomials It is denoted by the symbol Mn (x; β, c) and is defined as · ¸ −n, − x ; −1 Mn (x; β, c) = 2 F1 1−c , β; (2.8) β>0, o < c < 1, x = 0, 1, 2, ..., N
on binomial operator representations of some polynomials Krawchouk polynomials It is denoted by the symbol Kn (x; P, N ) and is defined as · ¸ −n, − x ; Kn (x; P, N ) = 2 F1 P −1 , − N; (2.9) o < P < 1, x = 0, 1, 2, ..., N Hahn polynomials It is denoted by the symbol Qn (x; α, β, N ) and is defined as · ¸ −n, − x , α + β + n + 1; Qn (x; α, β, N ) = 3 F2 1 − N, α + 1; (2.10) α, β > −1, n, x = 0, 1, 2, ..., N Sylvester polynomials It is denoted by the symbol ϕn (x) and is defined as ¸ · xn −n, x; −1 x (2.11) ϕn (x) = 2 F0 −; n! Gottlieb polynomials It is denoted by the symbol ln (x; λ) and is defined as ¸ · −n, − x ; −nλ λ (2.12) ln (x; λ) = e 1−e 2 F1 1; Charlier polynomials It is denoted by the symbol Cna (x) and is defined as · ¸ −n, −x; 1 n a Cn (x) = (−a) 2 F0 −; a Mittag-Leffler polynomials It is denoted by the symbol gn (z, γ) and is defined as · (−γ)n −n, z ; (2.14) gn (z, γ) = 2 F1 − γ; n!
¸ 2
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k.s. nisar, mumtaz ahmad khan
Shively’s pseudo Laguerre polynomials It is denoted by the symbol Rn (a, x) and is defined as · ¸ (a)2n −n ; (2.15) Rn (a, x) = x 1 F1 a+n ; n!(a)n
3. Operational representations If Dx ≡
∂ ∂ and Dy ≡ , M.A. Khan and A.K. Shukla [2] wrote the binomial ∂x ∂y
expansion for (Dx + Dy )n as
n
(3.1)
(Dx + Dy ) ≡
n X
n
Cr Dxn−r Dyr
r=0
where
n
Cr =
n! . r!(n − r)!
By writing the finite series on the right of (3.1) M.A. Khan and A.K. Shukla [2] wrote (3.1) also as n
(3.2)
(Dx + Dy ) ≡
n X
n
Cr Dxr Dyn−r
r=0
If F(x,y) is a function of x and y, they obtained the following from (3.1) and (3.2) (3.3)
n X (−n)r (−1)r n−r r (Dx + Dy ) F (x, y) ≡ Dx Dy F (x, y) r! r=0
(3.4)
n X (−n)r (−1)r r n−r (Dx + Dy ) F (x, y) ≡ Dx Dy F (x, y) r! r=0
n
n
In particular, if F (x, y) = f (x)g(y) then (3.3) and (3.4) in the form n X (−n)r (−1)r n−r Dx f (x)Dyr g(y) r! r=0
(3.5)
(Dx + Dy )n f (x)g(y) ≡
(3.6)
n X (−n)r (−1)r r (Dx + Dy ) f (x)g(y) ≡ Dx f (x)Dyn−r g(y) r! r=0 n
Now, by taking special values of f (x) and g(y) in (3.5), we obtain the following partial differential operator representations of the polynomials given above:
on binomial operator representations of some polynomials
ν+1
© ª y(Dx + Dy ) xν−1 y −1 = (−1)n n! gn(ν) n
(3.7)
x
(3.8)
n o −1 (Dw + Dy Dz )n wβ−1+n y x e(1−c )z
(3.9)
= (β)n wβ−1 y x e(1−c )z Mn (x; β, c(w/y)) n o n −N −1+n x p−1 z (Dw + Dy Dz ) w y e
µ ¶ x y
−1
−1
(3.10)
(3.11) (3.12) (3.13) (3.14)
= (−N )n w−N −1 y x e−p z Kn (x; P, N ), if w/y = 1 © ª (Dv Dw − Dy Dz )n v −N −1+n wα+n y x z −1−α−β−n µ ¶ vw −N −1 α x −1−α−β−n = (−N )n (1 + α)n v w y z Qn x; α, β, N yz n o −1 xyez/x (1 + Dy Dz )n y −x e x z = n!ϕn (y/x) o n n n x (1−eλ )z = n!y x e(1−λ)z ln (x; λ), when w/y = 1 (Dw + Dy Dz ) w y e 1 n o eaz x a1 z (1 + Dy Dz ) y e = Cn(a) (x) for y = 1 n (−a) © ª (1 − Dy )n y a+2n−1 = n!y a+2n−1 Rn (a, y)
n
To validate the above mentioned equations, here we illustrate some proofs: Proof of (3.7). (Dx + Dy )n {x−s−1 y −1 } n X (−n)r (−1)r (1 + s)n (−1)n −s−1−n+r = x (1)r (−1)r y −1−r r! (−s − n) r r=0 µ ¶ x = xν+1 y(−1)n n! gn(ν) y Proof of (3.8). n o −1 (Dw + Dy Dz )n wβ−1+n y x e(1−c )z n X (−n)r (−1)r n−r β−1+n r x r (1−c−1 )z Dy y Dz e Dw w = r! r=0 n X (−n)r (−x)r (1 − c−1 )r wr = (β)n w y e r! (β)r y r r=0 · ¸ −n, − x ; β−1 x (1−c−1 )z −1 = (β)n w y e (1 − c) (w/y) . 2 F1 β; β−1 x (1−c)−1 )z
−1 )z
= (β)n wβ−1 y x e(1−c)
Mn (x; β, c(w/y))
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Proof of (3.13). n
(Dw + Dy Dz ) {w
−γ−1+n −z 2z
y e }=
n X (−n)r (−1)r
r!
r=0
= (−γ)n w
Dwn−r w−γ−1+n Dyr y −z Dzr e2z ·
−γ−1 −z 2z
y e
2 F1
¸ −n, z ; 2(w/y) −γ ;
Proof of (3.14). (1 − Dy )n {y a−1+2n } =
n X (−n)r (−1)n−r (a + n)n r=0
= n!
(a)2 n (a)n
r! xa−1+n
y a+n−1+r
(a + n)r · ¸ −n ; y . 1 F1 a + n;
by (3.6)
= n!y a−1+n Rn (a, y) Reference [1] Chihar, T.S., An introdution to Orthogonal Polynomials, Gordon and Breach, New York, 1978. [2] Khan, M.A., Shukla, A.K., On Binomial and Trinomial operator representations of certain Polynomials, Italian Journal of Pure and Applied Mathematics, 25 (2009), 103-110. [3] Mumtaz Ahmad Khan, Nisar, K.S., Operational representations of bilinear and bilateral generating formulae for polynomials, International J. of Math. Sci. and Engg. Appls. (IJMSEA), vol. IV (4) (2010), 1-19. [4] Rainville, E.D., Special Functions, MacMillan, New York, 1960, Reprinted by Chelsea Publishing Company, Bronx, New York, 1971. [5] Srivastava, H.M., Manocha, H.L., A Treatise on Generating Functions, Ellis Horwood Limited, Chichester, 1984. Accepted: 11.11.2009
