ON MODELS OF q–REPRESENTATIONS OF sl(2, C) AND q –CONFLUENT HYPERGEOMETRIC FUNCTIONS

ON MODELS OF q–REPRESENTATIONS OF sl(2, C) AND q –CONFLUENT HYPERGEOMETRIC FUNCTIONS

Vivek Sahai and Shalini Srivastava

(Received October 2001)

Abstract. We construct new two variable models of irreducible q–representations of the Lie algebra sl(2, C). The q–Mellin transformaton is defined and used to transform the models in the form of q–difference dilation models, involving ba- sic hypergeometric function

φ

as basis functions. All the models culminate in many special function identities and recurrence relations.

1. Introduction

Manocha [6] introduced the idea of irreducible q–representations of the spe- cial complex Lie algebra sl (2, C) and developed theory centering around it. A q–analogue of fractional calculus technique [1, 7] was used to construct models of irreducible q–representations of sl (2, C) followed by the Weisner’s method [18] to obtain identities involving q–special functions of one and several variables. These models were in terms of q–differintegral operators acting on q–hypergeometric func- tions 2 φ 1 . Further, in [15], q–differintegral operator models of sl (2, C) with basis functions involving k

+1 φ k

functions were constructed and a good number of iden- tities were derived from them.

The theory of q–representation of Lie algebras together with the theory of q–

integral transformations has also been a rich source of results in q–special function theory. In Sahai [12, 13, 14], the q–Euler integral transformation, which is moti- vated by q–integral representation of beta function, is utilized to obtain models in terms of q–difference dilation operators of sl (2, C) along with identities involving q–special functions. Both these models, viz. q–differintegral operator models and q–difference dilation operator models, are of q–representations D q (u, α) and ↑ ^q (u) of sl (2, C) in the particular case u = 0.

In this paper, we construct new models of q–representations D q (u, α) and ↑ q (u) of sl (2, C) corresponding to u ̸= 0 acting on q–confluent hypergeometric functions

1 φ 1 . We then define q–Mellin transformation, which is motivated by q–integral representation of gamma function. The q–Mellin transformation is the q–analogue of Mellin integral transformation used in [5, 10]. We utilize this transformation to transform these models of sl (2, C) to new models involving q–difference dilation operators with the q–hypergeometric functions 2 φ 1 as basis functions. The discus- sion culminates in interesting identities and recurrence relations involving 1 φ 1 and

2 φ 1 functions.

Section–wise treatment is as follows.

2000 AMS Mathematics Subject Classification: 33D80, 17B37.

In Section 2, we give definitions involving hypergeometric and basic hyperge- ometric series, needed for our discussion. In Section 3, Lie algebra sl (2, C) is presented along with its q–deformed version. In Section 4, we construct model of D q (−γ/2, α) and ↑ q (−γ/2) in two variables acting on 1 φ 1 and obtain identities and recurrence relations based on it. In Section 5, we introduce q–Mellin transformation and then utilize it to transform the models given in Section 4 to q–difference dila- tion models acting on basis functions 2 φ 1 . The transformed models are further used in obtaining identities and recurrence relations involving 2 φ 1 . Finally, in Section 6, we give an application of these techniques to q–Laguerre polynomials.

2. Preliminaries

The generalized basic or q–hypergeometric series r φ s is defined as [3]

r φ s

! a 1 , . . . , a r

b 1, . . . , b s ; q, x "

= # ^∞

n=0

(a 1 ; q) _n · · · (a ^r ; q) _n (b 1 ; q) _n · · · (b ^s ; q) _n (q; q) _n

$ (−1) ⁿ q(

)% ^1+s−r x ⁿ ,

(2.1) where q–shifted factorial (a; q) _n is defined by

(a; q) _n = (a; q) _∞

(aq ⁿ ; q) _∞ , (a; q) _∞ = & ^∞

r=0

(1 − q ^r a) . (2.2) In other words,

(a; q) _n =

⎧ ⎨

⎩

(1 − a) (1 − aq) · · · *

1 − aq ^{n −1} + , n = 1, 2, . . .

1; n = 0

,* 1 − aq ⁻¹ + *

1 − aq ⁻² +

· · · (1 − aq ⁻ⁿ ) - −1 ; n = −1, −2, . . . .

(2.3)

The series r φ s terminates if one of the numerator parameter is of the form q ^−m , m = 0, 1, 2, . . . , and q ̸= 0. When 0 < |q| < 1, the series ^r φ s converges absolutely for all x if r ≤ s; and for |x| < 1 if r = s + 1. If |q| > 1 and |x| < _|a ^|b

^···b _···a

^| _| , then also

r φ s converges absolutely. It diverges for x ̸= 0 if 0 < |q| < 1 and r > s + 1, and if

|q| > 1 and |x| > _|a ^|b

^···b _···a

^| _| , unless it terminates.

The generalized hypergeometric series r F s is defined by [9]

r F s

. α 1 , . . . , α s

β 1, . . . , β s

; z /

=

# ∞ n=0

(α 1 ) _n · · · (α ^r ) _n

(β 1 ) _n · · · (β ^s ) _n n! z ⁿ , (2.4) where (α) _n is Pochhammer’s symbol defined by

(α) _n =

0 α (α + 1) · · · (α + n − 1) = ^Γ(α+n) Γ(α) ; n = 1, 2, . . .

1; n = 0 . (2.5)

The q–analogue of the binomial function is

1 φ 0

! a

− ; x "

= (ax; q) _∞

(x; q) _∞ , |q| < 1, |x| < 1. (2.6) The q–analogues of the exponential functions are

e q (x) = # ^∞

n=0

x ⁿ

(q; q) _n = 1

(x; q) _∞ , |q| < 1 (2.7)

and

E q (x) =

# ∞ n=0

q(

)x ⁿ

(q; q) _n = (−x; q) _∞ , |q| < 1. (2.8) We shall also make use of the function

Γ q (α) = e q (q ^α )

e q (q) (1 − q) ^{1 −α} (2.9)

defined for α ̸= 0, −1, −2, . . . . This is a q–analogue of the gamma function and satisfies the functional equation

Γ q (α + 1) = 1 − q ^α

1 − q Γ q (α) . (2.10)

We need the q–analogue of Kamp´e de F´eriet function [16]

F _C:E;E ^A:B;B

! (α):(β); ( ^β

)

(γ):(δ);(δ

) ; x, y "

= # ^∞

m,n=0

Π ^A _j=1 (α j ) _m+n Π ^B _j=1 (β j ) _m Π ^B _j=1

* β _j ^′ +

n x ^m y ⁿ Π ^C _j=1 (γ j ) _m+n Π ^E _j=1 (δ j ) _m Π ^E _j=1

*

δ ^′ _j +

n m!n!

(2.11) in the form [6]

φ ^A:B;B _C:E;E

! (a) : (b) ; (b ^′ )

(c) : (e) ; (e ^′ ) ; x, y q ^r , q ^s

"

=

# ∞ m,n=0

Π ^A _j=1 (a j ; q) _m+n Π ^B _j=1 (b j ; q) _m Π ^B _j=1

* b ^′ _j ; q +

n q ^r (

)q ^s (

)x ^m y ⁿ Π ^C _j=1 (c j ; q) _m+n Π ^E _j=1 (e j ; q) _m Π ^E _j=1

*

e ^′ _j ; q +

n (q; q) _m (q; q) _n . (2.12) For r = s = 0, we denote the l.h.s. of (2.12) simply by

φ ^A:B;B _C:E;E

! (a) : (b) ; (b ^′ ) (c) : (e) ; (e ^′ ) ; x, y

"

. The q–derivative operator is defined by

∆ x (f (x)) = f (x) − f (qx)

(1 − q) x = (1 − T x )

(1 − q) x f (x) , (2.13) where the q–dilation operator T x is given by

T x (f (x)) = f (qx) . (2.14)

The q–integral [3] is defined by 1 ∞

0

f (t) d q t = (1 − q)

# ∞ n=−∞

f (q ⁿ ) q ⁿ . (2.15)

3. Irreducible q–representations of sl( 2, C)

The complex Lie algebra sl (2, C) = L {SL (2, C)}, the Lie algebra of complex Lie group

SL (2, C) =

23 a b c d

4

: a, b, c, d ∈ C, ad − bc = 1 5

, (3.1)

consists of all 2 × 2 matrices with trace zero, that is, sl (2, C) = 23

e f

g −e 4

: e, f, g ∈ C 5

. (3.2)

It has a basis J ⁺ = 3

0 −1 0 0 4

, J ⁻ = 3 0 0

−1 0 4

, J ⁰ = 3 ₁

2 0

0 − ¹ ₂ 4

(3.3) satisfying the commutation relations

, J ⁰ , J ^± -

= ±J ^± , ,

J ⁺ , J ⁻ -

= 2J ⁰ . (3.4)

Let V q be a complex vector space consisting of q–special functions with a basis {φ ^λ | λ ∈ S} such that the functions {f ^λ = lim _q→1 φ λ | λ ∈ S} form a basis of a vector space, say V . Let A(V q ) be the associative algebra of all linear operators on V q over the complex field.

A q–representation of sl (2, C) on V q is a mapping ρ q : sl (2, C) → A (V q ) satisfying the following conditions:

(i) ρ q (ax + by) = aρ q (x) + bρ q (y)

(ii) There exists a Lie algebra representation ρ of sl(2) on V such that

q lim →1 ρ q (x) φ λ = ρ (x) f λ , for all x, y ∈ sl(2, C) and a, b ∈ C .

The representation ρ q of sl(2, C) is said to be irreducible if there is no proper subspace W q of V q which is invariant under ρ q .

Define

J _q ⁺ = ρ q (J ⁺ ), J _q ⁻ = ρ q (J ⁻ ), J _q ⁰ = ρ q (J ⁰ ), (3.5) where J _q ⁺ , J _q ⁻ , J _q ⁰ ∈ A (V ^q ).

Manocha [6] has defined the following commutator rules for J q –operators:

J _q ⁰ J _q ⁺ − qJ q ⁺ J _q ⁰ = J _q ⁺ , qJ _q ⁰ J _q ⁻ − J q ⁻ J _q ⁰ = −J q ⁻ ,

qJ _q ⁺ J _q ⁻ − J q ⁻ J _q ⁺ = 2q ^2u J _q ⁰ − (1 − q)q ^2u J _q ⁰ J _q ⁰ , u ∈ C. (3.6) These commutator rules were later generalized by Sahai [11] for the 4–dimensional complex Lie algebra G (a, b), a, b ∈ C. Further, the commutation relations (3.6) are also equivalent to those introduced by Jimbo [4]. For details, see [6, 14, 15].

If we define an operator C q on V q by

C q = qJ _q ⁺ J _q ⁻ + q ^2u J _q ⁰ J _q ⁰ − q ^2u J _q ⁰ (3.7)

then it is easy to check that

qJ _q ⁺ C q = C q J _q ⁺ , J _q ⁻ C q = qC q J _q ⁻ ,

J _q ⁰ C q = C q J _q ⁰ (3.8)

As q → 1, the operators J q ⁺ , J _q ⁻ , J _q ⁰ reduce to J ⁺ , J ⁻ , J ⁰ which satisfies the com- mutation relations obeyed by sl (2, C) and the operator C q reduces to the Casimir operator C. Hence, as q → 1, ρ ^q reduces to a Lie algebra representation ρ of sl (2, C). For more details, see [6, 8].

4. Models of Irreducible q–representations

We give below new two variable models for each of the q–representations D q (u, α) and ↑ q (u) corresponding to u = −γ/2, in which case they will be denoted by D q (−γ/2, α) and ↑ q (−γ/2), respectively.

4.1. Representation D _q (−γ/2, α).

Consider an irreducible representation D q (−γ/2, α) of sl (2, C), α, γ ∈ C and γ ̸=

0, −1, −2, . . . on the representation space V q with basis {f λ | λ = α + n, n ∈ Z}, such that action of sl (2, C) on V q is given by

J _q ⁺ f λ = q ^−γ − q ^λ−γ 1 − q f λ+1 ,

J _q ⁻ f λ = − 1 − q ^{λ −γ} 1 − q f λ −1 ,

J _q ⁰ f λ = 1 − q ^{λ −γ/2} 1 − q f λ ,

* qJ _q ⁺ J _q ⁻ + q ^−γ J _q ⁰ J _q ⁰ − q ^−γ J _q ⁰ + f λ =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ−γ f λ . (4.1) To find a realization of (4.1) in terms of q–dilation operators, we choose

J _q ⁺ = t 1 − q

* q ^−γ − q ^−γ T z T t + ,

J _q ⁻ = t ⁻¹ 1 − q T _z ⁻¹ *

q ^−γ T t − T ^z − q ^−γ zT z T t + ,

J _q ⁰ = 1 − q ^−γ/2 T t

1 − q ,

C q = qJ _q ⁺ J _q ⁻ + q ^−γ J _q ⁰ J _q ⁰ − q ^−γ J _q ⁰ , (4.2) with basis functions

f λ = 1 φ 1

! q ^λ

q ^γ ; q, −z "

t ^λ , λ = α, α ± 1, α ± 2, . . . .

Model (4.2) obeys (4.1) and the following J _q ⁰ J _q ⁺ − qJ q ⁺ J _q ⁰ = J _q ⁺ , qJ _q ⁰ J _q ⁻ − J q ⁻ J _q ⁰ = −J q ⁻ ,

qJ _q ⁺ J _q ⁻ − J q ⁻ J _q ⁺ = 2q ^−γ J _q ⁰ − (1 − q)q ^−γ J _q ⁰ J _q ⁰ , γ ∈ C (4.3) as well as

qJ _q ⁺ C q = C q J _q ⁺ , J _q ⁻ C q = qC q J _q ⁻ ,

J _q ⁰ C q = C q J _q ⁰ . (4.4)

4.1.1. Identities based on model (4.2).

As

f λ (z, t) = 1 φ 1

! q ^λ

q ^γ ; q, −z "

t ^λ satisfies

C q f λ (z, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^{λ −γ} f λ (z, t) . (4.5) It immediately follows that

u (z, t) = φ ^1:0;2 _0:1;1

! a : −; b ^′ , b ^′′

− : q ^γ ; c ^′ , z, t q, 1

"

t ^α

= # ^∞

n=0

(a; q) _n (b ^′ ; q) _n (b ^′′ ; q) _n (c ^′ ; q) _n (q; q) _n ¹ φ 1

! q ^λ

q ^γ ; q, −z "

t ^λ , (4.6) where q ^α = a, also satisfies

C q u (z, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ−γ u (z, t) . (4.7) Using the fact that

e q * qsJ _q ⁺ +

C q = C q e q * sJ _q ⁺ +

, (4.8)

which follows from (4.4), we have

C q , e q *

sJ _q ⁺ +

u - (z, t) =

6 ast q

(1 −q) ; q 7 6 ∞

st q

(1 −q) ; q 7

∞

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 + (1 − q) ² q ^−γ

× φ ^1:0;2 1:1;1

! a : −; b ^′ , b ^′′

ast

q

(1 −q) : q ^γ ; c ^′ ; z, qt q, 1

"

(tq) ^α where

, e q * sJ _q ⁺ +

u - (z, t) =

6 ast q

(1 −q) ; q 7 6 ∞

st q

(1−q) ; q 7

∞

φ ^1:0;2 _1:1;1

! a : −; b ^′ , b ^′′

ast

q

(1−q) : q ^γ ; c ^′ ; z, t q, 1

"

t ^α . (4.9)

Using the expansion , e q *

sJ _q ⁺ +

u - (z, t) = # ^∞

n=0

i n f α+n (z, t) , (4.10) where i n are obtained by putting z = 0, we get the following identity, after suitable rescaling:

6 at q

; q 7 6 ∞

t q

; q 7

∞

φ ^1:0;2 _1:1;1

! a : −; b ^′ , b ^′′

at

q

: q ^γ ; c ^′ ; z, −t/w q, 1

"

=

# ∞ n=0

(a; q) _n (q; q) _n

* q ^−γ t + n 3 φ 1

! q ⁻ⁿ , b ^′ , b ^′′

c ^′ ; q, − q ^n+γ w

"

1 φ 1

! aq ⁿ q ^γ ; q, −z

"

, (4.11) where w = − _{1 −q} ^s .

4.1.2. Recurrence relations from model (4.2).

The recurrence relations originating from the action of J _q ⁺ , J _q ⁻ on f λ are given by

* 1 − q ^λ +

1 φ 1 (λ + 1; z) + q ^λ 1 φ 1 (λ; qz) = 1 φ 1 (λ; z) , (4.12) and

zq ^λ−γ 1 φ 1 (λ; qz) + 1 φ 1 (λ; qz) − q ^λ−γ 1 φ 1 (λ; z) = *

1 − q ^λ−γ +

1 φ 1 (λ − 1; qz) (4.13)

respectively, where 1 φ 1 (λ; z) stands for 1 φ 1

! q ^λ q ^γ ; q, −z

"

.

4.2. Representation ↑ q (−γ/2).

We look for an irreducible representation ↑ q (−γ/2) of sl (2, C), γ ∈ C − {0, −1, −2, . . . }, on the representation space V ^q with basis functions {f ^λ | λ = 0, 1, 2, . . . } such that the action of sl (2, C) on V ^q is given by

J _q ⁺ f λ = q ^−γ − q ^λ 1 − q f λ+1 ,

J _q ⁻ f λ = − 1 − q ^λ 1 − q f λ −1 ,

J _q ⁰ f λ = 1 − q ^λ+γ/2 1 − q f λ ,

* qJ _q ⁺ J _q ⁻ + q ^−γ J _q ⁰ J _q ⁰ − q ^−γ J _q ⁰ + f λ =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ f λ . (4.14)

To meet this requirement, we choose J _q ⁺ = t

1 − q

* q ^−γ − T ^z T t − zT ^z T t + ,

J _q ⁻ = t ⁻¹

1 − q T _z ⁻¹ (T t − T ^z ) ,

J _q ⁰ = 1 − q ^γ/2 T t

1 − q ,

C q = qJ _q ⁺ J _q ⁻ + q ^−γ J _q ⁰ J _q ⁰ − q ^−γ J _q ⁰ , (4.15) with basis functions

f λ = 1 φ 1

! q ^−λ

q ^γ ; q, −q ^λ+γ z

"

t ^λ , λ = 0, 1, 2, . . . . Model (4.15) satisfies (4.3) as well as (4.4).

4.2.1. Identities based on model (4.15).

As

f λ = 1 φ 1

! q ^−λ

q ^γ ; q, −q ^λ+γ z

"

t ^λ , satisfies

C q f λ (z, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ f λ (z, t) , (4.16) it immediately follows that

u (z, t) = φ ^1:0:0 _1:1;0

! a : −; −

c ^′ : q ^γ ; − ; − q ^γ zt, t q ² , 1

"

=

# ∞ n=0

(a; q) _n

(c ^′ ; q) _n (q; q) _n ¹ φ 1

! q ⁻ⁿ

q ^γ ; q, −q ^n+γ z

"

t ⁿ , (4.17) satisfies

C q u (z, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ⁿ u (z, t) . (4.18) Using the fact that

E q

! 1 q sJ _q ⁻

"

C q = C q E q * sJ _q ⁻ +

, (4.19)

which follows from (4.4), we have

C q , E q *

sJ _q ⁻ +

u - (z, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 + (1 − q) ²

× φ ^1:0;1 1:1;0

! a : −; (1 −q)t ^s

c ^′ : q ^γ ; − ; − q ^γ+1 zt, qt q ² , 1

"

, (4.20)

where

, E q * sJ _q ⁻ +

u - (z, t) = φ ^1:0;1 _1:1;0 ! a : −; (1 −q)t ^s

c ^′ : q ^γ ; − ; − q ^γ zt, t q ² , 1

"

. (4.21) The expansion

, E q * sJ _q ⁻ +

u - (z, t) = # ^∞

n=0

A n f n (z, t) , (4.22) leads to the following identity

φ ^1:0;1 _1:1;0

! a : −; −w/t

c ^′ : q ^γ ; − ; − q ^γ zt, t q ² , 1

"

=

# ∞ n=0

(a; q) _n

(q; q) _n (c ^′ ; q) _n ¹ φ 1

! aq ⁿ

c ^′ q ⁿ ; q, −w "

1 φ 1

! q ⁻ⁿ

q ^γ ; q, −q ^n+γ z

"

t ⁿ . (4.23)

4.2.2. Recurrence relations.

The recurrence relations arising from model (4.15) are given by

1 φ 1 (λ; z) − q ^{λ+γ 1} φ 1 (λ; qz) − q ^λ+γ z 1 φ 1 (λ; qz) = *

1 − q ^λ+γ +

1 φ 1 (λ + 1; z) , (4.24)

1 φ 1 (λ; qz) − q ^λ 1 φ 1 (λ; z) = * 1 − q ^λ +

1 φ 1 (λ − 1; qz) , (4.25) where 1 φ 1 (λ; z) stands for 1 φ 1

! q ^−λ

q ^γ ; q, −q ^λ+γ z

"

. 5. Transformed Models of sl (2, C)

We introduce the q–Mellin transformation I q . Define

h (β, x) = I q [f (ux)] = q

Γ q (β)

1 ∞

0

e q (−u) u ^{β −1} f (ux) d q u, (5.1) which is motivated from the q–integral representation of gamma function, as in [17],

Γ q (γ) = q

1 ∞

0

e q (−u) u ^{γ −1} d q u. (5.2) Using the substitution z = ux, which gives z∆ z = u∆ u = x∆ x and T z f (z) = f (qz) = T u f (ux), we have the following transforms of certain operator expressions which are needed for the discussion.

I q [uf (ux)] = q ^−β − 1

1 − q E β h (β, x) ,

I q [u∆ u f (ux)] = x∆ x h (β, x) , (5.3)

where E β h (β, x) = h (β + 1, x).

To obtain transformed models of D q (−γ/2, α) and ↑ q (−γ/2), we put z = ux in models (4.2) and (4.15). Note that if ρ q is an irreducible representation of sl (2, C) on the representation space V q having the basis vectors {f λ | λ ∈ S} in terms of 8

J _q ⁺ , J _q ⁻ , J _q ⁰ 9 , then the transformation I q induces another irreducible representation σ q of sl (2, C) on the representation space W q = I q V q in terms of 8 K _q ⁺ = I q J _q ⁺ I _q ⁻¹ , K _q ⁻ = I q J _q ⁻ I _q ⁻¹ , K _q ⁰ = I q J _q ⁰ I _q ⁻¹ 9 with basis vectors {h ^λ = I q f λ | λ ∈ S}.

5.1. Transformed Model of D q (−γ/2, α).

K _q ⁺ = tq ^−γ

1 − q (1 − T x T t ) ,

K _q ⁻ = t ⁻¹ 1 − q T _x ⁻¹

!

q ^−γ T t − T ^x − q ^−β − 1

1 − q E β q ^−γ xT x T t

"

,

K _q ⁰ = 1 − q ^−γ/2 T t

1 − q , (5.4)

with basis functions as

h λ (x, t) = 2 φ 1

! q ^λ , q ^β

q ^γ ; q, x (1 − q) q ^β

"

t ^λ . Indeed,

K _q ⁰ K _q ⁺ − qK q ⁺ K _q ⁰ = K _q ⁺ , qK _q ⁰ K _q ⁻ − K q ⁻ K _q ⁰ = −K q ⁻ ,

qK _q ⁺ K _q ⁻ − K q ⁻ K _q ⁺ = 2q ^−γ K _q ⁰ − (1 − q) q ^−γ K _q ⁰ K _q ⁰ (5.5) and

K _q ⁺ h λ = q ^−γ 1 − q ^λ 1 − q h λ+1 ,

K _q ⁻ h λ = − 1 − q ^λ−γ 1 − q h _λ−1 ,

K _q ⁰ h λ = 1 − q ^{λ −γ/2} 1 − q h λ , C _q ^′ h λ = *

qK _q ⁺ K _q ⁻ + q ^−γ K _q ⁰ K _q ⁰ − q ^−γ K _q ⁰ + h λ

=

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ−γ h λ . (5.6)

Moreover,

qK _q ⁺ C _q ^′ = C _q ^′ K _q ⁺ , K _q ⁻ C _q ^′ = qC _q ^′ K _q ⁻ ,

K _q ⁰ C _q ^′ = C _q ^′ K _q ⁰ . (5.7)

5.1.1. Identities based on transformed model D q (−γ/2, α).

As

h λ (x, t) = 2 φ 1

! q ^λ , q ^β

q ^γ ; q, x (1 − q) q ^β

"

t ^λ is a solution of

C _q ^′ h λ (x, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ−γ h λ (x, t) , (5.8) the function

u (x, t) = φ ^1:1;2 _0:1;1 !

a : q ^β ; b ^′ , b ^′′

− : q ^γ ; c ^′ , q ^−β x 1 − q , t

"

t ^α

= # ^∞

n=0

(a; q) _n (b ^′ ; q) _n (b ^′′ ; q) _n (c ^′ ; q) _n (q; q) _n ² φ 1

! q ^λ , q ^β

q ^γ ; q, x (1 − q) q ^β

"

t ^λ (5.9) is also a solution of

C _q ^′ u (x, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^{λ −γ} u (x, t) . (5.10) Using qK _q ⁺ C _q ^′ = C _q ^′ K _q ⁺ , we have

C _q ^′ , e q *

sK _q ⁺ +

u - (x, t) =

6 ast q

(1 −q) ; q 7 6 ∞

st q

(1 −q) ; q 7

∞

q ^−γ *

1 − q ^−γ/2 + *

1 − q ^−γ/2+1 + (1 − q) ²

× φ ^1:1;2 1:1;1

! a : q ^β ; b ^′ , b ^′′

ast

q

(1 −q) : q ^γ ; c ^′ ; q ^−β x 1 − q , qt

"

(qt) ^α , (5.11) where

, e q * sK _q ⁺ +

u - (x, t) =

6 ast q

(1 −q) ; q 7 6 ∞

st q

(1 −q) ; q 7

∞

φ ^1:1;2 _1:1;1

! a : q ^β ; b ^′ , b ^′′

ast

q

: q ^γ ; c ^′ ; q ^−β x 1 − q , t

"

t ^α . (5.12) From the expansion

, e q * sK _q ⁺ +

u - (x, t) = # ^∞

n=0

A n h α+n (x, t) ,

where A n is found by putting x = 0, we arrive at the following identity after suitable rescaling:

6 at q

; q 7 6 ∞

t q

; q 7

∞

φ ^1:1;2 _1:1;1

! a : q ^β ; b ^′ , b ^′′

at

q

: q ^γ ; c ^′ ; q ^−β x 1 − q , −t/w

"

=

# ∞ n=0

(a; q) _n (q; q) _n

* q ^−γ t + n 3 φ 1

! q ⁻ⁿ , b ^′ , b ^′′

c ^′ ; q, − q ^n+γ w

"

2 φ 1

! aq ⁿ , q ^β

q ^γ ; q, x (1 − q) q ^β

"

.

(5.13)

5.1.2. Recurrence relations from model (5.4).

The recurrence relations originating from the action of K _q ⁺ and K _q ⁻ on h λ are given by

2 φ 1 (λ; β; x) − q ^λ 2 φ 1 (λ; β; qx) = * 1 − q ^λ +

2 φ 1 (λ + 1; β; x) , (5.14)

xq ^λ−γ−β 1 − q ^β

1 − q ² φ 1 (λ; β + 1; qx) + 2 φ 1 (λ; β; qx) − q ^λ−γ 2 φ 1 (λ; β; x)

= *

1 − q ^{λ −γ} +

2 φ 1 (λ − 1; β; qx) , (5.15) where 2 φ 1 (λ; β; x) stands for 2 φ 1

! q ^λ , q ^β

q ^γ ; q, _{(1 −q)q} ^x

"

.

5.2. Transformed Model of ↑ q (−γ/2).

K _q ⁺ = t 1 − q

!

q ^−γ − T ^x T t − q ^−β − 1

1 − q E β xT x T t

"

, (5.16)

K _q ⁻ = t ⁻¹

1 − q T _x ⁻¹ (x∆ x − t∆ ^t ) ,

K _q ⁰ = 1 − q ^γ/2 T t

1 − q , with basis functions as

h λ (x, t) = 2 φ 1

! q ^−λ , q ^β

q ^γ ; q, q ^{λ+γ −β} 1 − q x

"

t ^λ . Model (5.16) satisfies (5.5) and (5.7) and

K _q ⁺ h λ = q ^−γ − q ^λ 1 − q h λ+1 ,

K _q ⁻ h λ = − 1 − q ^λ 1 − q h λ −1 ,

K _q ⁰ h λ = 1 − q ^λ+γ/2 1 − q h λ , C _q ^′ h λ = *

qK _q ⁺ K _q ⁻ + q ^−γ K _q ⁰ K _q ⁰ − q ^−γ K _q ⁰ + h λ

=

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ h λ . (5.17)

5.2.1. Identities based on transformed model ↑ q (−γ/2).

As

h λ (x, t) = 2 φ 1

! q ^−λ , q ^β

q ^γ ; q, q ^λ+γ−β 1 − q x

"

t ^λ , λ = 0, 1, 2, . . . satisfies

C _q ^′ h λ (x, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ^λ h λ (x, t) . (5.18) This, in turn, gives

u (x, t) = φ ^1:1;0 _1:1;0

. a : q ^β ; − c ^′ : q ^γ ; − ; − ^q

γ−β

xt 1 −q , t

q, 1 /

= # ^∞

n=0

(a; q) _n

(c ^′ ; q) _n (q; q) _n ² φ 1

! q ⁻ⁿ , q ^β

q ^γ ; q, q ^{n+γ −β} 1 − q x

"

t ⁿ , (5.19) satisfies

C _q ^′ u (x, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 +

(1 − q) ² q ⁿ u (x, t) . (5.20) Using K _q ⁻ C _q ^′ = qC _q ^′ K _q ⁻ , we have

C _q ^′ , E q *

sK _q ⁻ +

u - (x, t) =

* 1 − q ^−γ/2 + *

1 − q ^−γ/2+1 + (1 − q) ²

× φ ^1:1;1 1:1;0

. a : q ^β ; _{q(1 −q)t} ^s c ^′ : q ^γ ; − ; − ^q

γ−β+1

xt 1 −q , qt

q, 1 /

, (5.21) where

, E q * sK _q ⁻ +

u -

(x, t) = φ ^1:1;1 _1:1;0

. a : q ^β ; _{(1 −q)t} ^s c ^′ : q ^γ ; − ; − ^q

γ−β

xt 1 −q , t

q, 1 /

. (5.22)

The expansion

, E q * sK _q ⁻ +

u - (x, t) = # ^∞

n=0

C n h n (x, t) , (5.23) gives rise to the following identity:

φ ^1:1;1 _1:1;0

. a : q ^β ; − ^w t

c ^′ : q ^γ ; − ; − ^q

γ−β

xt 1 −q , t

q, 1 /

= # ^∞

n=0

(a; q) _n

(q; q) _n (c ^′ ; q) _n ¹ φ 1

! aq ⁿ

c ^′ q ⁿ ; q, −w "

2 φ 1

! q ⁻ⁿ , q ^β

q ^γ ; q, q ^{n+γ −β} 1 − q x

"

t ⁿ .

(5.24)

5.2.2. Recurrence relations.

The recurrence relations that come from (5.17) are given by

2 φ 1 (λ; β; x) − q ^λ+γ 2 φ 1 (λ; β; qx) − xq ^{λ+γ −β} 1 − q ^β

1 − q ² φ 1 (λ; β + 1; qx)

= * 1 − q ^λ +

2 φ 1 (λ + 1; β; x) , (5.25)

2 φ 1 (λ; β; qx) − q ^λ 2 φ 1 (λ; β; x) = * 1 − q ^λ +

2 φ 1 (λ − 1; β; qx) , (5.26) where 2 φ 1 (λ; β; x) stands for 2 φ 1

! q ^−λ , q ^β

q ^γ ; q, ^q

_{1 −q}

x

"

. 6. Conclusion

In this section, we give an application of our models to q–Laguerre polynomials.

Let α > −1 and 0 < q < 1. Then q–Laguerre polynomials are defined by [3]

L ^α _n (z; q) =

* q ^α+1 ; q +

n

(q; q) _n ¹ φ 1

! q ⁻ⁿ

q ^α+1 ; q, −zq ^n+α+1

"

. (6.1)

If we take z = (1 − q) x then it can be easily seen that lim q →1

L ^α _n (z; q) = L ^α _n (x), where L ^α _n (x) are classical Laguerre polynomials. Consider the model (4.15) in the particular case γ = 1 + α (that is, a model of representation ↑ q (− (1 + α) /2)) with basis functions involving q–Laguerre polynomials, given as

J _q ⁺ = t 1 − q

6 q ^−(1+α) − T ^z T t − zT ^z T t

7 ,

J _q ⁻ = t ⁻¹

1 − q T _z ⁻¹ (T t − T ^z ) ,

J _q ⁰ = 1 − q ^(1+α)/2 T t

1 − q ,

f n (z, t) = L ^α _n (z; q) t ⁿ . (6.2)

The action of these J q –operators on f n (z, t) will take the following form:

J _q ⁺ f n = q ^−(1+α) 1 − q ⁿ⁺¹ 1 − q f n+1 ,

J _q ⁻ f n = − 1 − q ^α+n 1 − q f _n−1 ,

J _q ⁰ f n = 1 − q

⁺ⁿ

1 − q f n . (6.3)

We can obtain identities involving q–Laguerre polynomials using the method ex- plained above. For example, taking

u (z, t) =

# ∞ n=0

* q ^α+1 ; q +

n

(c

; q) _n (q; q) _n ¹ φ 1

! q ⁻ⁿ

q ^α+1 ; q, −zq ^n+α+1 "

t ⁿ (6.4)

and proceeding as in Section 4.2.1, we arrive at the following identity:

φ ^1:0;1 _1:1;0

! q ^α+1 : −; − ^w _t

c ^′ : q ^1+α ; − ; − q ^α+1 zt, t q ² , 1

"

=

# ∞ n=0

t ⁿ

(c

; q) _n L ^α _n (z; q) 1 φ 1

! q ^α+1+n ,

c ^′ q ⁿ ; q, −w "

, (6.5) which is a special case of (4.23).

The recurrence relations involving q–Laguerre polynomials can also be obtained by using first two equations of (6.3). These will shape as

L ^α _n (z; q) − q ^1+α+n L ^α _n (qz; q) − zq ^n+α+1 L ^α _n (qz; q) = *

1 − q ⁿ⁺¹ +

L ^α _n+1 (z; q) (6.6) and

q ⁿ L ^α _n (z; q) − L ^α n (qz; q) = − *

1 − q ^α+n +

L ^α _n−1 (qz; q) , (6.7) respectively. Eliminating L ^α _n (z; q) from (6.6) and (6.7) gives rise to a three term recurrence relation as follows

* 1 − q ^α+n +

L ^α _{n −1} (qz; q) + *

zq ^1+α+2n − *

1 − q ^1+α+2n ++

L ^α _n (qz; q) + q ⁿ *

1 − q ⁿ⁺¹ +

L ^α _n+1 (z; q) = 0, (6.8) n = 1, 2, . . . .

Note that (6.8) is a q–analogue of the three term recurrence relation for classical Laguerre polynomials [2],

(α + n) L ^α _n−1 (x) + (x − 1 − α − 2n) L ^α n (x) + (n + 1) L ^α _n+1 (x) = 0. (6.9) Acknowledgements. The authors thank the referee for his valuable suggestions which led to a better presentation of the paper.

References

1. M.A. Al-Bassam and H.L. Manocha, Lie theory of solutions of certain differ- integral operators, SIAM J. Math. Anal. 18 (1987), 1087–1100.

2. G.E. Andrews, R.Askey and R. Roy, Special Functions, Cambridge Univ. Press, 1999.

3. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ.

Press, 1990.

4. M. Jimbo , A q–difference analogue of U (g) and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.

5. H.L. Manocha, Lie algebras of difference differential operators and Appell func- tion F 1 , J. Math. Anal. Appl. 138 (1989), 491–510.

6. H.L. Manocha, On models of irreducible q–representations of sl(2, C), Appl.

Anal. 37 (1990), 19–47.

7. H.L. Manocha, and V. Sahai, On models of irreducible unitary presentations of compact Lie group SU(2) involving special functions, Indian J. Pure Appl.

Math. 21 (1990), 1059–1071.

8. W. Miller, Lie Theory and Special Functions, Academic Press, 1968.

9. E.D. Rainville, Special Functions, Macmillan, 1960.

10. V. Sahai, Lie theory of polynomials of the form F D via Mellin transformation, Indian J. Pure Appl. Math. 22 (1991), 927–941.

11. V. Sahai, q–representations of the Lie algebras G(a, b), J. Indian Math. Soc. 63 (1997), 83–98.

12. V. Sahai, On models of U q

* sl(2) +

and q–Euler integral transformation, Indian J. Pure Appl. Math. 29 (1998), 819–827.

13. V. Sahai, On models of U q * sl(2) + and m–fold q–Euler integral transformation, J. Indian Math. Soc. 66 (2000), 123–132.

14. V. Sahai, On models of U q * sl(2) + and q–Appell functions using a q–integral transformation, Proc. Amer. Math. Soc. 127 (1999), 3201–3213.

15. V. Sahai, On models of irreducible q–representations of sl(2, C) by means of fractional q–calculus, New Zealand J. Math., 28(1999), 237-249.

16. H.M. Srivastava and H.L. Manocha, A treatise on generating functions, Ellis–

Horwood, 1984.

17. N. Ja Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions, Vol. 3, Kluwer, Dordretch, 1992.

18. L. Weisner, Group theoretic origin of certain generating functions, Pacific J.

Math. 5 (1955), 1033–1039.

Vivek Sahai

Department of Mathematics and Astronomy Lucknow University

Lucknow 226007 INDIA

[email protected]

Shalini Srivastava

Department of Mathematics and Astronomy Lucknow University

Lucknow 226007

INDIA