generating relations involving 4-variable modified

Journal of Indian Acad. Math. Vol. 33, No. 2 (2011) pp. 455-468

GENERATING RELATIONS INVOLVING 4-VARIABLE MODIFIED HERMITE POLYNOMIALS Subuhi Khan1, Mumtaz Ahmad Khan2 and Rehana Khan3

Abstract: In this paper, the 4-variable modified Hermite polynomials (4VMHP) are framed into the context of the representation theory of the harmonic oscillator Lie algebra G(0,1) . Generating relations involving 4VMHP are derived. Certain new and known results for polynomials related to 4VMHP are also obtained as special cases. Keywords: Modified Hermite polynomials, Hermite polynomials, generating relations, Harmonic oscillator Lie algebra . Mathematics Subject Classification: 33C45, 33C50, 33C80. 1. Introduction The 2-variable Hermite-Kampé de Fériet polynomials (2VHKdFP) H n ( x, y ) ([2];p.341(23)) are defined as: n [ ] 2

x n− 2 r y r H n ( x, y ) = n! ∑ . r =0 ( n − 2 r )! r!

(1.1)

In terms of classical Hermite polynomials H n (x ) or H en (x) [1], it is easily seen from the definition (1.1) that H n (2 x,−1) = H n ( x )

(1.2a)

1  H n  x,−  = H en ( x ). 2 

(1.2b)

and

Also, there exists the following close relationship ([2];p.341(21)):  ix H n ( x, y ) = ( −i) n y n/2 H n  2 y  3

Corresponding author Email address: [email protected]

 n    = i (2 y ) n/2 H en  x    i 2y    

(1.3)

with the classical Hermite polynomials. The usage of second variable (parameter) y in the 2VHKdFP H n ( x, y ) is found to be convenient from the viewpoint of their applications. Indeed, from an entirely different viewpoint and considerations, Hermite polynomials of several variables are introduced and investigated by Erdélyi et al.([7];p.283). Recently the 4-variable modified Hermite polynomials (4VMHP) H nα , β ,γ ,δ ( x, y, z , w) are

introduced [9] n [ ] 4

n! (δw) r H nα−,4βr,γ ( x, y, z ) . r! (n − 4r )! r =0

H nα , β ,γ ,δ ( x, y, z , w) = ∑

(1.4)

The generating function for these polynomials is given by

[

]

∞

exp αxt − (1 + β y )t 2 + γzt 3 + δwt 4 = ∑H nα , β ,γ ,δ ( x, y, z , w) n=0

tn . n!

(1.5)

The theory of special functions from the group-theoretic point of view is a well established topic, providing a unifying formalism to deal with the immense aggregate of the special functions and a collection of formulae such as the relevant differential equations, integral representations, recurrence formulae, composition theorems, etc., see for example [15,16]. The first significant advance in the direction of obtaining generating relations by Lie-theoretic method is made by Weisner [17-19] and Miller [13].Within the group-theoretic context, indeed a given class of special functions appears as a set of matrix elements of irreducible representations of a given Lie group. The algebraic properties of the group are then reflected in the functional and differential equations satisfied by a given family of special functions, whilst the geometry of the homogeneous space determines the nature of the integral representation associated with the family. Recently some contributions related to Lie-theoretical representations of generalized Laguerre and Hermite polynomials and Bessel functions of two variables have been given, see for example Khan [10], Khan and Pathan [11] and Khan et al. [12]. Motivated by these contributions, in this paper, we derive generating relations involving 4VMHP H nα , β ,γ ,δ ( x, y, z , w) by using representation ↑ω ,µ of Lie algebra G(0,1) . In Section 2, we consider the properties and special cases of H nα , β ,γ ,δ ( x, y, z , w) . In Section 3, we consider the problem of framing 4VMHP into the context of the representation ↑ω ,µ of four dimensional Lie algebra G(0,1)

and obtain generating relations involving H nα , β ,γ ,δ ( x, y, z , w) and associated Laguerre polynomials Lαn (x) . In Section 4, we consider some applications of the generating relations obtained in Section 3. 2. Properties and Special Cases of 4VMHP H nα , β ,γ ,δ ( x, y, z , w)

The 4VMHP satisfy the following differential and pure recurrence relations ∂ α ,β ,γ ,δ Hn ( x, y, z, w) = αnH nα−,1β ,γ ,δ ( x, y, z, w), ∂x ∂ α ,β ,γ ,δ Hn ( x, y , z, w) = − βn(n − 1) H nα−,2β ,γ ,δ ( x, y , z, w), ∂y ∂ α ,β ,γ ,δ Hn ( x, y , z, w) = γn(n − 1)(n − 2) H nα−,3β ,γ ,δ ( x, y, z, w), ∂z ∂ α ,β ,γ ,δ Hn ( x, y , z, w) = δn(n − 1)(n − 2)(n − 3) H nα−,β4 ,γ ,δ ( x, y , z, w) ∂w

(2.1)

and H nα+,1β ,γ ,δ ( x, y , z, w) = αxH nα ,β ,γ ,δ ( x, y, z, w) − 2(1 + βy )nH nα−,1β ,γ ,δ ( x, y , z, w)

+ 3γzn(n − 1) H nα−,2β ,γ ,δ ( x, y, z, w) + 4δwn (n − 1)(n − 2)H nα−,3β ,γ ,δ ( x, y , z, w).

(2.2)

We note the following special cases of 4VMHP H nα , β ,γ ,δ ( x, y, z , w) : (1)

H nα , β ,γ ,δ ( x, y, z,0) = H nα ,β ,γ ( x, y, z ),

(2.3)

where H nα , β ,γ ( x, y , z ) denotes 3-variable modified Hermite polynomials (3VMHP) defined by the generating function [9]. ∞

exp (αxt − (1 + β y )t 2 + γzt 2 ) = ∑ n=0

(2)

H nα ,β ,γ ( x, y , z )t n n!

H n2,1,1, δ ( x, y → ( y − 1), z,0) = H n ( x, y, z ),

(2.4)

(2.5)

where H n ( x, y , z ) denotes 3-variable Hermite polynomials (3VHP) defined by the generating function ([4]; p.511(17)) ∞

exp(2 xt − yt 2 + zt 3 ) = ∑H n ( x, y, z ) n =0

tn . n!

(2.6)

(3)

H n1,1,1, δ ( x, y → (1 − y ), z,0) = H n ( x, y, z ), (2.7)

where H n ( x, y , z ) denotes another form of 3-variable Hermite polynomials defined by the generating function ([3]; p.114(22)) ∞

exp( xt + yt 2 + zt 3 ) = ∑H n ( x, y , z ) n=0

(4)

tn . n!

H nα , β ,γ ,δ ( x, y,0,0) = H nα , β ( x, y ),

(2.8) (2.9)

where H nα , β ( x, y ) denotes 2 variable modified Hermite polynomials (2VMHP) defined by the generating function [9]. ∞

exp(α xt − (1 + β y )t 2 ) = ∑ n =0

(5)

H nα , β ( x, y )t n n!

H n2,1,γ ,δ ( x, y → ( y − 1),0,0) = H n ( x, y ),

(2.10)

(2.11)

where H n ( x, y ) denotes 2-variable Hermite polynomials (2VHP) defined by the generating function ([4]; p.510(8)) ∞

tn . n! n= 0 H n1,1, γ ,δ ( x, y → (1 − y ),0,0) = H n ( x, y ), exp(2 xt − yt 2 ) = ∑H n ( x, y )

(6)

(2.12) (2.13)

where H n ( x, y ) denotes 2VHKdFP defined by the generating function ([3]; p.112(8)) ∞

exp ( xt + yt 2 ) = ∑H n ( x, y ) n=0

(7)

tn . n!

H n2,1,γ ,δ ( x, y,0,0) = H n ( x, y ),

(2.14) (2.15)

where H n ( x, y ) denotes 2-variable Hermite polynomials (2VHP) defined by the generating function [8]. ∞

H n ( x, y )t n . n! n =0

(8)

exp(2 xt − ( y + 1)t 2 ) = ∑

(2.16)

H nα , β ,γ ,δ ( x,0,0,0) = H nα ( x),

(2.17)

where H nα (x) denotes modified Hermite polynomials (MHP) defined by the generating function [9].

H nα ( x)t n . n! H n2,1,γ ,δ ( x,0,0,0) = H n ( x ), exp (αxt − t 2 ) =

(9)

(2.18) (2.19)

where H n (x ) denotes ordinary Hermite polynomials [1].

3. Representation ↑ω , µ of G(0,1) and Generating Relations The irreducible representation ↑ω , µ of G(0,1) is defined for each ω , µ ∈ C such that µ ≠ 0 . The spectrum S of this representation is the set {−ω + k : k a nonnegative integer} and there is a basis ( f m ) m∈S for the representation space V , with the properties J 3 f m = mf m , Ef m = µf m , J + f m = µf m +1 , J − f m = (m + ω ) f m−1 , C0,1 f m = ( J + J − − EJ 3 ) = µωf m , µ ≠ 0.

(3.1)

(Here f −ω −1 ≡ 0 , so J − f −ω = 0 ). The commutation relations satisfied by the operators J 3 , J ± , E are [ J 3 , J ± ] = ± J ± , [ J + , J − ] = − E , [ E , J ± ] = [ E, J 3 ] = 0.

(3.2)

In order to obtain a realization of the representation ↑ω , µ of G(0,1) on a space of functions of two complex variables x and y , Miller ([13]; p.104) has taken the functions f m ( x, y ) = Z m ( x)e my , such that relations (3.1) are satisfied for all m ∈ S , where the differential operators J 3 , J + , J − , E are given by

∂ , ∂y ∂ 1  J + = e y  − µ x ,  ∂x 2   ∂ 1  J − = e − y  − − µx   ∂x 2  E = µ.

J3

=

(3.3)

In particular, we look for the functions

f m ( x, y, z, w, t ) = Z mα ,β ,γ ,δ ( x, y, z, w)t m , such that relation (3.1) are satisfied. for all m ∈ S .

(3.4)

We take that the set of linear differential operators K ± , K 3 , I as follows: K+

= αxt −

K−

=

K3

2(1 + βy )t ∂ 3γzt ∂ 4δwt ∂ − + , ∂x α β ∂y γ ∂z

1 ∂ , αt ∂x

= t

(3.5)

∂ , ∂t

I = 1 and note that these operators satisfy the commutation relations identical to (3.2). There is no loss of generality for special function theory if we set ω = 0 and µ = 1 . For this choice of ω and µ and in terms of the function Z m (x ) , relation (3.1) reduce to ([13]; p.104(4.72)) d 1   − x  Z m ( x)  dx 2 

= Z m +1 ( x),

 d 1   − − x  Z m ( x) = mZ m−1 ( x ),  dx 2   d 2 x2 1   − 2 + − − m  Z m ( x ) = 0. 4 2  dx 

(3.6)

Further, for the same choice of ω and µ and using operators (3.5), relations (3.1) in terms of the functions Z mα , β ,γ ,δ ( x, y, z, w) take the form  2(1 + βy ) ∂ 3γz ∂ 4δw ∂  α , β ,γ ,δ − + Zm ( x, y, z, w) = Z mα +, β1 ,γ ,δ ( x, y , z, w), αx −  α ∂x β ∂y γ ∂z    1 ∂  α , β ,γ ,δ ( x, y, z, w) = mZ mα −, β1 ,γ ,δ ( x, y , z, w),  α ∂x  Z m    ∂ 2(1 + β y ) ∂ 2 3γz ∂ 2  4δw ∂ 2 x − − + − m  Z mα ,β ,γ ,δ ( x, y, z , w) = 0.  2 2 α ∂x αβ ∂x∂y αγ ∂x∂z  ∂x 

(3.7)

Miller [13] have used relations (3.6) to compute the functions Z m (x ) , which are easily expressed in terms of parabolic cylinder functions or Hermite polynomials H n (x ) . In fact, Z m ( x) = (−1) m Dm ( x) = (−1) m exp (−

x 2 −m/2 x )2 H m ( ), m = 0,1,2, K 4 2

(3.8)

Similarly, we observe that, for all m ∈ S , the choice for Z mα , β ,γ ,δ ( x, y , z, w) = H mα , β ,γ ,δ ( x, y , z, w) satisfy Eqs.

(3.7).

It

follows

from

the

above

discussion

that

the

functions

f m ( x, y, z, w, t ) = H mα , β ,γ ,δ ( x, y , z, w)t m , m ∈ S form a basis for a realization of the representation ↑ 0,1 of G(0,1) . By using ([13]; p.18(Theorem (1.10))), this representation of G(0,1) can be extended to a local multiplier representation T (g ) , g ∈ G (0,1) defined on F , the space of all functions analytic in a neighbourhood of the point ( x 0 , y 0 , z 0 , w 0 , t 0 ) = (1,1,1,1,1) . Using operators (3.5), the local multiplier representation takes the form

[T (exp aE) f ]( x, y, z, w, t ) = exp(a) f ( x, y, z, w, t ), [T (exp bJ + ) f ]( x, y , z, w, t ) = exp (αxtb − (1 + βy )t 2b 2 + γzt 3b 3 + δwt 4b 4 )

  2(1 + βy )tb 3γzt 2b 2 4δwt 3b 3   3γztb 6δwt 2b 2  , y 1 − × f  x1 − + + − ,  αx αx αx   βy βy      4δwtb  , w,t , z 1 + γz   

  c   [T (exp cJ − ) f ]( x, y, z, w, t ) = f  x1 + , y , z, w, t ,   αxt   [T (exp τJ 3 ) f ]( x, y , z, w, t ) = f ( x, y , z, w, teτ ),

(3.9)

for f ∈ F , where J + , J − , J 3 , E are the basis element for Lie algebra G(0,1) ([13];p.9(1.31)). If

g ∈ G (0,1) has coordinates (a, b, c,τ ) then we have g = (exp bJ + )(exp cJ − )(exp τJ 3 )(exp aE ). Thus the operator T (g ) acting on f ∈ F is given by [T ( g ) f ]( x, y , z, w, t ) = [T (exp bJ + )T (exp cJ − )T (exp τJ 3 )T (exp aE ) f ]( x, y , z, w, t ). and hence we obtain [T ( g ) f ]( x, y, z, w, t ) = exp(a + η ) f ( xθ , yφ , zψ , w, teτ ), where

η := αxtb − (1 + βy )t 2b 2 + γzt 3b3 + δwt 4b 4 ,

(3.10)

θ := 1 −

c 2(1 + βy )tb 3γzt 2b 2 4δwt 3b 3 + + + , αx αx αx αxt

3γztb 6δwt 2b 2 − , φ := 1 − βy βy

ψ := 1 +

4δwtb . γz

(3.11)

The matrix elements of T (g ) with respect to the analytic basis f m ( x, y, z, w, t ) = H mα , β ,γ ,δ ( x, y , z, w)t m , are the functions Alk (g ) , uniquely determined by ↑ 0,1 of G(0,1) and we obtain relations ∞

[T ( g ) f k ]( x, y , z, w, t ) = ∑Alk ( g ) f l ( x, y, z, w, t ), k = 0,1,2,K ,

(3.12)

l =0

which on using (3.10) yields exp(a + η + kτ ) H kα ,β ,γ ,δ ( xθ , yφ , zψ , w) ∞

= ∑Alk ( g ) H lα , β ,γ ,δ ( x, y, z, w)t l − k , k = 0,1,2,K

(3.13)

l =0

where η , θ , φ , ψ are given in (3.11) The matrix element Alk (g ) are given by ([13]; p.87(4.26)) Alk ( g ) = exp(a + kτ )c k −l Lkl −l (−bc), k , l ≥ 0.

(3.14)

Substituting (3.14) into (3.13) and simplifying, we obtain the generating relation exp(η ) H kα ,β ,γ ,δ ( xθ , yφ , zψ , w) ∞

= ∑c k −l Lkl −l (−bc ) H lα , β ,γ ,δ ( x, y, z, w)t l − k , k = 0,1,2,L,

(3.15)

l =0

where η , θ , φ , ψ are given in (3.11).

4. Applications We consider the following applications of generating relation (3.15). 1. Taking b → 0 in generating relation (3.15) and using the limit ([13]; p.88(4.29))

n n l

c L (bc) |b=0

 n + l  n c  =  n  0 

if n ≥ 0,

(4.1)

if n < 0,

we get c   k H k  x + , y , z, w  = ∑k αt   l =0

c k −l H lα ,β ,γ ,δ ( x, y , z, w)t l −k .

(4.2)

k −l

Again, taking c → 0 in generating relation (3.15) and using the limit ([13]; p.88(4.29))

n n l

c L (bc) |c=0

if n > 0, 0  −n − b ( ) = , if n ≤ 0,  (−n)!

(4.3)

we get   2(1 + βy )tb 3γzt 2b 2 exp (αxtb − (1 + βy )t 2b 2 + γzt 3b 3 + δwt 4b 4 ) H kα , β ,γ ,δ  x1 − + α x αx  

+

4δwt 3b 3   3γztb 6δwt 2b 2   4δwtb   , y 1 − , z1 + , w − αx   βy βy   γz   ∞

=∑ l =0

b l −k H lα , β ,γ ,δ ( x, y , z , w)t l −k (l − k )!

(4.4)

1. Taking w = 0 in generating relation (3.15) and using Eq. (2.3), we get α , β ,γ

exp(αxtb − (1 + βy )t b + γzt b ) H k 2 2

3 3

  2(1 + βy )tb 3γzt 2b 2 c   x1 − , + +   α x α x α xt   

 3γztb   ∞ k −l k −l , z  = ∑c Ll (−bc) H lα ,β ,γ ( x, y , z )t l −k , y 1 − β y   l =0 

where H nα , β ,γ ( x, y , z ) denotes 3VMHP given by Eq. (2.4). Further, taking z = 0 in Eq. (4.5) and using Eq. (2.9), we get   2(1 + βy )tb c   exp (αxtb − (1 + βy )t 2b 2 ) H kα , β  x1 − + , y  αx αxt    

(4.5)

∞

= ∑c k −l Lkl −l (−bc) H lα ,β ( x, y )t l − k ,

(4.6)

l =0

where H nα , β ( x, y ) denotes 2VMHP given by Eq. (2.10). Furthermore, taking y = 0 in Eq. (4.6) and using Eq. (2.17), we get   2tb c   ∞ k −l k −l α l −k exp(αxtb − t 2b 2 ) H kα  x1 − +   = ∑c Ll (−bc) H l ( x )t , αx αxt   l =0  

(4.7)

where H nα (x) denotes MHP given by Eq. (2.18). 1. Taking α = 2 , β = γ = 1 , w = 0 and replacing y by ( y − 1) in generating relation (3.15) and using Eq. (2.5), we obtain ([14]; p.63(16)) (for τ = 0 )   ytb 3 zt 2b 2 c   3 ztb   , y1 − , z  exp(2 xtb − yt 2b 2 + zt 3b 2 ) H k  x1 − + + x 2 x 2 xt y       ∞

= ∑c k −l Lkl −l (−bc ) H l ( x, y, z)t l −k ,

(4.8)

l =0

where H n ( x, y , z ) denotes 3VHP given by Eq. (2.6). Further, taking z = 0 in Eq. (4.8) and using Eq. (2.11), we get ([14]; p.63(17)) (for τ = 0 )   ytb c   exp(2 xtb + yt 2b 2 ) H k  x1 − + , y  x αxt     ∞

= ∑c k −l Lkl −l (−bc ) H l ( x, y )t l −k ,

(4.9)

l =0

where H n ( x, y ) denotes 2VHP given by Eq. (2.12). 1. Taking α = β = γ = 1 , w = 0 and replacing y by (1 − y) in generating relation (3.15) and using Eq. (2.7), we get   2 ytb 3zt 2b 2 c   3ztb   , z  exp( xtb + yt 2b 2 + zt 3b 3 ) H k  x1 + + + , y 1 − x x xt y       ∞

= ∑c k −l Lkl −l (−bc ) H l ( x, y, z)t l −k , l =0

(4.10)

where H l ( x, y , z ) denotes 3VHP given by Eq. (2.8). Further, taking z = 0 in Eq. (4.10) and using Eq. (2.13), we get   2 ytb c   ∞ k −l k −l exp( xtb + yt 2b 2 ) H k  x1 + + , y  = ∑c Ll (−bc ) H l ( x, y )t l −k , (4.11) x xt   l =0  

where H l ( x, y ) denotes 2VHP given by Eq. (2.14). 1. Taking α = 2 , β = 1 , z = w = 0 in generating relation (3.15) and using Eq. (2.15), we get   (1 + y )tb c   exp(2 xtb − (1 + y )t 2b 2 )H k  x1 − + , y  x 2 xt     ∞

= ∑c k −l Lkl −l (−bc)H l ( x, y ),

(4.12)

l =0

where H n ( x, y ) denotes 2VHP given by Eq. (2.16). Further taking y = 0 in Eq. (4.12) and using Eq. (2.19), we get ([13]; p.106(4.76)) (for t = −t )   tb c   ∞ k −l k − l exp(2 xtb − t 2b 2 ) H k  x1 − +   = ∑c Ll (−bc) H l ( x), x 2 xt   l =0   where H n (x ) denotes Hermite polynomials.

(4.13)

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1. Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India E-mail address: [email protected]

(Received, Dec.7, 2010)

2. Department of Applied Mathematics Faculty of Engineering, Aligarh Muslim University, Aligarh--202002, India E-mail address: [email protected] 3. Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India E-mail address: [email protected]