63, 4 (2011), 253–262 December 2011

63, 4 (2011), 253–262 December 2011

research paper

OPERATOR REPRESENTATIONS OF GENERALIZED

HYPERGEOMETRIC FUNCTIONS AND CERTAIN POLYNOMIALS Mumtaz Ahmad Khan and Mohd. Khalid Rafat Khan

Abstract. A new technique is evolved to give operator representations of hypergeometric functions and certain polynomials.

1. Introduction In 1731, Euler defined the derivative formula

Dⁿ_xx^λ= Γ(1 +λ)

Γ(1 +λ−n)x^λ−n, Dx≡ d dx wherenis a positive integer. Its general form is

D^µ_xx^λ= Γ(1 +λ)

Γ(1 +λ−µ)x^λ−µ, (1.1)

whereλandµare arbitrary complex numbers. Here (1.1) is given to facilitate the use of D⁻ⁿ, i.e. replacing µ by −n in (1.1), where n is a positive integer. Here series representations of different operators have been used to establish operational representation of hypergeometric functions and various known polynomials. The technique used and the results obtained are believed to be new.

2. Definitions and notation

In deriving the operator representations of hypergeometric functions and cer- tain polynomials use has been made of the fact of the following notations:

IfDx denotes a derivative operator, thenD_x⁻¹ is nothing but the inverse op- erator ofDx. Now we can write the following:

2010 AMS Subject Classification: 33C45, 33C20, 47F05.

Keywords and phrases: Operational representation; polynomials; generalized hypergeomet- ric functions.

253

D⁻¹_x = 1

Dx, D⁻ⁿ_x =D⁻¹_x D⁻¹_x . . .(ntimes) = (D_x⁻¹)ⁿ = µ 1

Dx

¶_n , (α)n x⁻ⁿ=x^α(−Dx)ⁿ x^−α, (2.1)

1

(α)n xⁿ=x^−α+1( 1

Dx)ⁿ x^α−1. (2.2)

Some results used in the proofs are e^x= P^∞

n=0

xⁿ

n!, (2.3)

(1−x)^−a = P^∞

n=0 (a)nxⁿ

n! . (2.4)

We also need the definitions of the following generalized hypergeometric func- tions and polynomials in terms of hypergeometric function and also their notations (see [4], [11], [12]).

Generalized hypergeometric function. The generalized hypergeometric function is defined as

pFq

·a1,· · · , ap; b1,· · · , bq;x

¸

= X∞ n=0

Q_p

i=1(ai)n xⁿ Q_q

j=1(bj)n n!. (2.5) Laguerre polynomial. It is denoted by the symbolL^(α)n (x) and is defined as

L^(α)_n (x) =(1 +α)n

n! ¹F1

· −n;

1 +α;x

¸ .

Legendre polynomial. It is denoted by the symbolPn(x) and is defined as Pn(x) =2F1

·−n, n+ 1 ; 1;

1−x 2

¸ .

Jacobi polynomial. It is denoted by the symbolPn^(α,β)(x) and is defined as P_n^(α,β)(x) =(1 +α)n

n! ²F1

·−n,1 +α+β+n; 1 +α;

1−x 2

¸ .

Ultraspherical polynomial. The special case ofβ =αof the Jacobi poly- nomial is called ultraspherical polynomial and is denoted by Pn^(α,α)(x). It is thus defined as

P_n^(α,α)(x) =(1 +α)n

n! ²F1

·−n,1 + 2α+n; 1 +α;

1−x 2

¸ .

Gegenbauer polynomial. The Gegenbauer polynomialC_n^ν(x) is the gener- alization of Legendre polynomial and is defined as

C_n^ν(x) =(2ν)n

n! ²F1

·−n, 2ν+n; ν+¹₂;

1−x 2

¸ .

Bessel polynomial. Simple Bessel polynomialyn(x) is defined as yn(x) =2F0

·−n, n+ 1 ;

−;

−x 2

¸ ,

and the generalized Bessel polynomialsyn(a, b, x) is defined as yn(a, b, x) =2F0

·−n, a−1 +n;

−;

−x b

¸ .

Lagrange polynomial. It is denoted by the symbolgn^(α,β)(x, y) and is defined by

g_n^(α,β)(x, y) =(α)n

n! ²F1

· −n, β;

1−α−n;

y x

¸ .

Sylvester polynomial. It is denoted by the symbolϕn(x) and is defined as ϕn(x) = xⁿ

n! ²F0

·−n, x;

−; x⁻¹

¸ .

Shively’s pseudo Laguerre polynomial. It is denoted by the symbol Rn(a, x) and is defined as

Rn(a, x) = (a)2n

n!(a)n1F1

· −n; a+n; x

¸ .

Hermite polynomial. It is denoted by the symbolHn(x) and is defined as Hn(x) = (2x)ⁿ 2F0

·−¹₂n,−¹₂n+¹₂ ;

−; − 1

x²

¸ .

Bateman’s Zn(x). Bateman’s polynomialZn(x) is defined as Zn(x) =2F2

·−n, n+ 1 ; 1, 1; x

¸ .

Bateman’s generalization of Zn(x). Bateman moved from Zn(x) to the more general polynomial

2F2

·−n, 2v+n; v+¹₂,1 +b; t

¸ .

It may be remarked here that the above polynomial is the Gegenbauer type gener- alization ofZn(x). We will therefore adopt the symbolZ_n^v(b, t). Thus we have

Z_n^v(b, t) =2F2

·−n,2v+n; v+¹₂, 1 +b;t

¸ .

A Jacobi type generalization ofZn(x) may be denoted by the symbolZn^(α,β)(b, x) and is defined as

Z_n^(α,β)(b, x) =2F2

· −n, 1 +α+β+n; 1 +α, 1 +b; x

¸ .

Tchebycheff polynomials. The Tchebycheff polynomials Tn(x) andUn(x) of the first and second kinds respectively are special ultraspherical polynomials. In details

Tn(x) = n!

(¹₂)n

Pn^{(−12,−12)}(x), Un(x) =(n+ 1)!

(³₂)n

Pn^(12,12)(x), and in terms of hypergeometric function their definition will be as follows

Tn(x) =2F1

·−n, n;

1 2;

1−x 2

¸

, Un(x) = (n+ 1)2F1

·−n, n+ 2;

3 2;

1−x 2

¸ .

Ces´aro polynomial. It is denoted by the symbolg^(s)n (x) and is defined as g_n^(s)(x) =

µs+n n

¶

2F1

·−n, 1 ;

−s−n;x

¸ .

Bedient’s polynomials. Bedient [2], in his study of some polynomials asso- ciated with Appell’sF2 andF3, introduced

Rn(β, γ, x) =(β)n(2x)ⁿ n! ³F2

·−ⁿ₂,−ⁿ₂ +¹₂ , γ−β;

γ,1−β−n; 1 x²

¸ , Gn(α, β;x) = (α)n(β)n(2x)ⁿ

n!(α+β)n 3F2

·−¹₂n,−¹₂n+¹₂, 1−α−β−n;

1−α−n,1−β−n; 1 x²

¸ . Rice polynomial. Rice polynomialHn(ξ, p, v) is defined as

Hn(ξ, p, v) = 3F2

·−n, n+ 1 , ξ;

1, p; v

¸ .

A Jacobi type generalization of Rice polynomial Hn(ξ, p, v) is due to Khandekar [10] who denoted his generalized polynomial by the symbol Hn^(α,β)(ξ, p, v) and is defined as

H_n^(α,β)(ξ, p, v) =(1 +α)_n n! ³F2

·−n, 1 +α+β+n , ξ;

1 +α, p; v

¸ .

Sister Celine’s polynomial. Sister M. Celine denoted her polynomial by the symbolfn

·a1,· · · , ap; b1,· · ·, bq;x

¸

and is defined as fn

·a1,· · ·, ap; b1,· · ·, bq;x

¸

=p+2Fq+2

·−n, n+ 1, a1,· · ·, ap; 1, ¹₂ , b1,· · ·, bq; x

¸ .

3. Operational representation If Dxi ≡ _∂x^∂

i and Dyi ≡ _∂y^∂

i, where i = 1,2,3, . . . , by using these partial differential operators, we define the following function:

]q p

[b1, b2. . . , bp;c1, c2, . . . , cq;x1, x2, . . . , xq :y1, y2, . . . , yp]

= YP j=1

y_j^bj Yq

i=1

x^−c_i ⁱ⁺¹exp

Ã(−1)^pQ_P

j=1Dyj

Q_q

i=1Dxi

! _P Y

j=1

y_j^−bj Yq

i=1

x^c_iⁱ⁻¹, (3.1)

and we can show that the result (3.1) is equivalent to the following:

]q

p

[b1, b2. . . , bp;c1, c2, . . . , cq;x1, x2, . . . , xq :y1, y2, . . . , yp]

= pFq

·

b1, b2, . . . , bp;c1, c2, . . . , cq;x1x2. . . xq

y1y2..yp

¸

. (3.2) Again we define another function, as follows:

]q p+1

[a, b1, b2. . . , bp;c1, c2, . . . , cq;x1, x2, . . . , xq :y1, y2, . . . , yp]

= Yp

j=1

y^b_j^j Yq

i=1

x^−c_i ⁱ⁺¹ Ã

1−(−1)^pQ_p

j=1Dyj

Q_q

i=1Dx_i

!−a PY

j=1

y_j^−bj Yq

i=1

x^c_iⁱ⁻¹. (3.3) Further, we can show that the result (3.3) is equivalent to the following

]q p+1

[b1, b2. . . , bp;c1, c2, . . . , cq;x1, x2, . . . , xq:y1, y2, . . . , yp]

= p+1Fq

·

a, b1, b2, . . . , bp;c1, c2, . . . , cq;x1x2. . . xq

y1y2..yp

¸

. (3.4) Proof of(3.2). Taking the R.H.S of (3.1) and applying the result (2.3), we get Yp

j=1

y^b_j^j Yq

i=1

x^−c_i ⁱ⁺¹ X∞

r=0

(−1)^prQ_p

j=1(Dyj)^r r! Q_q

i=1(Dxi)^r Yp

j=1

y^−b_j ^j Yq

i=1

x^c_iⁱ⁻¹

= X∞ r=0

1 r!

Yp

j=1

n

y_j^bj(−Dyj)^ry_j^−bj o Y^q

i=1

½ x^−c_i ⁱ⁺¹

µ 1 Dxi

¶_r x^c_iⁱ⁻¹

¾ .

Now applying the results (2.1) and (2.2), we get

= X∞ r=0

1 r!

Yp

j=1

©(bj)r y^−r_j ª Y^q

i=1

½ 1 (ci)r x^r_i

¾ . From (2.5), we get the result (3.2).

Proof of(3.4). Taking the R.H.S of (3.3) and applying the results (2.4), we get Yp

j=1

y^b_j^j Yq

i=1

x^−c_i ⁱ⁺¹ X∞ r=0

(a)r (−1)^prQ_p

j=1(Dyj)^r r! Q_q

i=1(Dxi)^r

Yp

j=1

y_j^−bj Yq

i=1

x^c_iⁱ⁻¹

= X∞ r=0

(a)r

r!

Yp

j=1

n

y_j^bj(−Dy_j)^ry_j^−bjo Y^q

i=1

½ x^−c_i ⁱ⁺¹

µ 1 Dxi

¶_r x^c_iⁱ⁻¹

¾ ,

and now applying the results (2.1) and (2.2), we get the following:

= X∞ r=0

(a)r

r!

Yp

j=1

©(bj)r y_j^−rª Y^q

i=1

½ 1 (ci)r x^r_i

¾ .

Again from (2.5), we get the result (3.4).

4. Operational representations of hypergeometric functions By taking different values of pand qin (3.1)–(3.4), we get operational repre- sentations of hypergeometric functions. If we take p= 1 and q = 0 in (3.2), we get

]0

1

[a;−;−:x] = 1F0

· a;−;1

x

¸ , and from (3.1)

1F0

· a;−;1

x

¸

=x^a e^−Dx x^−a. Similarly, we can define the following

1F1

· a;b;x

y

¸

=y^a x^−b+1e^−DyDx y^−ax^b−1

1F1[a;b;x] =x^−b+1 µ

1− 1 Dx

¶_−a x^b−1

2F0

· a, b;1

x

¸

=x^a(1 +Dx)^−bx^−a

2F0

· a, b;1

x

¸

=x^b(1 +Dx)^−ax^−b

2F0

· a, b; 1

xy

¸

=x^ay^b e^DxDyx^−ay^−b

2F1

·

a, b;c;x y

¸

=x^−c+1y^b µ

1 +Dy

Dx

¶_−a

x^c−1y^−b

2F1

·

a, b;c;x y

¸

=x^−c+1y^a µ

1 + Dy

Dx

¶_−b

x^c−1y^−a

2F1

·

a, b;c; x yz

¸

=x^−c+1y^az^b exp

µDyDz

Dx

¶

x^c−1y^−az^−b

2F2

h

a, b;c, d;xy z

i

=x^−c+1y^−d+1z^b µ

1 + Dz

DxDy

¶_−a

x^c−1y^d−1z^−b

2F2

h

a, b;c, d;xy z

i

=x^−c+1y^−d+1z^a µ

1 + Dz

DxDy

¶_−b

x^c−1y^d−1z^−a

2F2

h

a, b;c, d;xy z

i

=x^−d+1y^−c+1z^b µ

1 + Dz

DxDy

¶_−a

x^d−1y^c−1z^−b

2F2

h

a, b;c, d;xy zw i

=x^−c+1y^−d+1z^a w^b exp

µDzDw

DxDy

¶

x^c−1y^d−1z^−a w^−b

2F2

h

a, b;c, d;xy zw i

=x^−d+1y^−c+1z^b w^a exp

µDzDw

DxDy

¶

x^d−1y^c−1z^−b w^−a

3F0

·

a, b, c;−; 1 xy

¸

=x^by^c(1−DxDy)^−ax^−by^−c

3F0

·

a, b, c;−; 1 xy

¸

=x^ay^c(1−DxDy)^−bx^−ay^−c

3F0

·

a, b, c;−; 1 xy

¸

=x^ay^b(1−DxDy)^−cx^−ay^−b

3F0

·

a, b, c;−; 1 xyz

¸

=x^ay^bz^c exp (DxDyDz)x^−ay^−bz^−c

3F1

·

a, b, c;d; x yz

¸

=x^−d+1y^az^b µ

1−DyDz

Dx

¶_−c

x^d−1y^−az^−b

3F1

·

a, b, c;d; x yz

¸

=x^−d+1y^az^c µ

1−DyDz

Dx

¶_−b

x^d−1y^−az^−c

3F1

·

a, b, c;d; x yz

¸

=x^−d+1y^bz^c µ

1−DyDz

Dx

¶_−a

x^d−1y^−bz^−c

3F1

·

a, b, c;d; x yzw

¸

=x^−d+1y^bz^cw^a exp

µ−DzDyDw

Dx

¶

w^−ax^d−1y^−bz^−c.

5. Operational representations of certain polynomials The operational representations of several polynomials are given as follows

L^(α)_n (x) = x^−α

n! (Dx−1)ⁿx^n+α L^(α)_n (x) =(1 +α)n

n! x^−α µ

1− 1 Dx

¶_n x^α L^(α)_n (x

y) =(1 +α)n

n! x^−αy⁻ⁿe^−DyDx x^αyⁿ Pn

µ 1−2x

y

¶

=y⁻ⁿ n! Dⁿ_x

µ 1 + Dy

Dx

¶_−n−1 xⁿyⁿ Pn

µ 1−2x

y

¶

=y⁻ⁿ µ

1 + Dy

Dx

¶_−n−1 x⁰yⁿ Pn

µ 1−2x

y

¶

=yⁿ⁺¹ µ

1 + Dy

Dx

¶_n

x⁰y⁻ⁿ⁻¹ P_n^(α,β)

µ 1−2x

y

¶

=x^−αy⁻ⁿ n! D_xⁿ

µ 1 + Dy

Dx

¶_{−n−α−β−1}

x^n+αyⁿ

P_n^(α,β) µ

1−2x y

¶

= (1 +α)nx^−αy^1+α+β+n n!

µ 1 + Dy

Dx

¶_n

x^αy^{−1−α−β−n} P_n^(α,β)

µ 1−2x

y

¶

=(1 +α)nx^−αy⁻ⁿ n!

µ 1 +Dy

Dx

¶_{−n−α−β−1} x^αyⁿ P_n^(α,α)

µ 1−2x

y

¶

= x^−αy⁻ⁿ n! Dⁿ_x

µ 1 +Dy

Dx

¶_{−n−2α−1}

x^n+αyⁿ P_n^(α,α)

µ 1−2x

y

¶

= (1 +α)nx^−αy⁻ⁿ n!

µ 1 + Dy

Dx

¶_{−n−2α−1} x^αyⁿ P_n^(α,α)

µ 1−2x

y

¶

= (1 +α)nx^−αy^1+2α+n n!

µ 1 + Dy

Dx

¶_n

x^αy^{−1−2α−n} C_n^(ν)

µ 1−2x

y

¶

= (2ν)nx^−ν+12y⁻ⁿ n!

µ 1 + Dy

Dx

¶_−n−2ν

x^ν−12yⁿ C_n^(ν)

µ 1−2x

y

¶

= (2ν)nx^−ν+12y^2ν+n n!

µ 1 +Dy

Dx

¶_n

x^ν−12y^−2ν−n yn

µ1 x

¶

=x⁻ⁿ µ

1−Dx

2

¶_−n−1 xⁿ yn

µ1 x

¶

=xⁿ⁺¹ µ

1−Dx

2

¶_n x⁻ⁿ⁻¹ yn

µ a, b,1

x

¶

=x⁻ⁿ µ

1−Dx

b

¶_−n−a+1 xⁿ yn

µ a, b,1

x

¶

=x^n+a−1 µ

1−Dx

b

¶_−n

x^−n−a+1 g^(α,β)_n (x, y) =(α)n

n! y^n+αx⁻ⁿ(1 + Dx

Dy

)^−βy^−n−αxⁿ g_n^(α,β)

µ1 x,1

y

¶

= (α)n

n! x^n+αy^β µ

1 + Dy

Dx

¶_n

x^−n−αy^β Φn(x) = 1

n!(1 +Dx)^−xxⁿ Rn(a, x) = (a)2n

(a)nn!x^−n−a+1 µ

1 + 1 Dx

¶_n

x^n+a−1 Rn

µ a,x

y

¶

= (a)2n

(a)nn!x^−n−a+1y⁻ⁿexp µ−Dy

Dx

¶

x^n+a−1yⁿ Hn

³x 2

´

=e^−D2xxⁿ Hn(√

x) = 2ⁿ(1−Dx)ⁿ²⁻¹²xⁿ² Hn(√

x) = 2ⁿx¹²(1−Dx)ⁿ² xⁿ²⁻¹²

Zn(x) = 1 n!Dⁿ_x

½ xⁿ

µ 1− 1

Dx

¶_n x⁰

¾

Zn(x) = 1

(n!)²D_xⁿ{xⁿ(Dx−1)ⁿxⁿ} Z_n^ν

³ b,xy

z

´

=z^2ν+ny^−bx^−ν+12 µ

1 + Dz

DxDy

¶_n

z^−2ν−ny^bx^ν−12 Z_n^ν

³ b,xy

z

´

=x^−by^−ν+12z⁻ⁿ µ

1 + Dz

DxDy

¶_−2ν−n

x^by^ν−12zⁿ Z_n^ν

³ b,xy

zw

´

=z⁻ⁿy^−bx^−ν+12w^2ν+nexp

µDzDw

DxDy

¶

zⁿy^bx^ν−12w^−2ν−n Z_n^(α,β)

³ b,xy

z

´

=x^−αy^−bz⁻ⁿ µ

1 + Dz

DxDy

¶_{−n−α−β−1} x^αy^bzⁿ Z_n^(α,β)

³ b,xy

z

´

=x^−αy^−bz^n+α+β+1 µ

1 + Dz

DxDy

¶_n

x^αy^bz^{−n−α−β−1} Z_n^(α,β)

³ b,xy

zw

´

=x^−αy^−bz^n+α+β+1w⁻ⁿexp

µDzDw

DxDy

¶

x^αy^bz^{−n−α−β−1}wⁿ Tn

µ 1−2x

y

¶

=x¹²y⁻ⁿ µ

1 +Dy

Dx

¶_−n x⁻¹²yⁿ Un

µ 1−2x

y

¶

= (n+ 1)x⁻¹²y⁻ⁿ µ

1 + Dy

Dx

¶_−n−2 x¹²yⁿ g^(s)_n

µx y

¶

=

µs+n n

¶

x^s+n+1y⁻ⁿ µ

1 +Dy

Dx

¶−1

x^−s−n−1yⁿ Hn

³ ξ, p,y

x

´

=x^ξ n!Dⁿ_y

µ y^n−p+1

µ 1 +D_x

Dy

¶n

x^−ξy^−p−1

¶

H_n^(α,β)³ ξ, p,xy

zw

´

=x^−αy^−p+1z⁻ⁿw^ξ µ

1−DzDw

DxDy

¶_{−α−β−n−1}

x^αy^p−1zⁿw^−ξ Gn

µ

α, β; x 2√

yz

¶

= (α)n(β)n

n!(α+β)nz^α+n2y^β+n2 µ

1− D²_x DyDz

¶β+α+n−1

xⁿy^−β−nz^−α−n Rn

µ

β, γ; x 2√

yz

¶

=(β)n

n! z^−γ+1−n2y^β+n2 µ

1− D²_x DyDz

¶β−γ

xⁿy^−β−nz^γ−1.

REFERENCES

[1] W.A. Al-Salam, Operator representations for the Laguerre and other polynomials, Duke Math. J.31(1964), 127–142.

[2] P.E. Bedient,Polynomials related to Appell functions of two variables, Michigan, Thesis, 1958.

[3] L. Carlitz,A note on the Laguerre polynomials, Michigan Math. J.7(1960), 219–223.

[4] T.S. Chihara,An Introduction to Orthogonal Polynomials, Godon and Breach, New York, 1978.

[5] H.W. Gould, A.T. Hopper,Operational formulas connected with two generalizations of Her- mite polynomials, Duke Math. J.29(1962), 51–54.

[6] M.A. Khan,On a calculus for theTk,q,x-operator, Math. Balkanica6(1992), 84–90.

[7] M.A. Khan,On some operational generating formula, Acta Ciencia IndicaXIXM(1993), 35–38.

[8] M.A. Khan, M.K.R. Khan,Expansion and summation formulae for Laguerre polynomials of mvariable, submitted.

[9] M.A. Khan, A.K. Shukla, On binomial and trinomial operator representations of certain polynomials, PromathematicaXXII/No 43-44(2008), 89–101.

[10] P.R. Khandekar,On generalization of Rice’s polynomials, I. Proc. Nat. Acad. Sci. India Sect.

A34(1964), 157–162.

[11] E.D. Rainville,Special Functions, MacMillan, New York, 1960; Reprinted by Chelsea Pub- lishing Company, Bronx, New York, 1971.

[12] H.M. Srivastava, H.L. Manocha,A Treatise on Generating Functions, Ellis Horwood Limited, Chichester, 1984.

(received 03.07.2010, revised 10.11.2010)

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh - 202002, U.P., India.

E-mail:mumtaz ahmad khan [email protected], [email protected]