The study of Lie theory and its connection with special functions have been a rich source of interesting results in mathematical analysis. For example, authors such as Miller [4], Manocha [2], Manocha and Sahai [3], Sahai [6], Sahai and Yadav [7, 8], etc. have studied the Lie theory of several special functions and deformation of Lie algebras and special functions. The theory of special functions and group representations has been well discussed by the authors such as Manocha and Srivastava [10], Vilenkin [11], Wawrzynczyk [12] etc. Also, tools of the underlying theory have been studied in [1, 5, 9].

In the present paper, we study the connection between the 5-dimensional complex Lie algebra and certain special functions. We construct new two variable models of the irreducible representations R(ω, m₀, μ) and ↑_ω,μ of corresponding to μ ≠ 0, where basis function appears in the form of ₁F₀. We also use an Euler type integral transformation to obtain the transformed models of the irreducible representations R(ω, m₀, μ) and ↑_ω,μ of . In these models, the basis functions appear in terms of ₂F₁. As an application, these models result in several generating functions and recurrence relations. Section-wise treatment is as follows:

In Section 2, we present some prelimnaries. We define [4] the Lie algebra and give its one variable model in the representations R(ω, m₀, μ) and ↑_ω,μ. We also take an Euler type integral transformation to be used later on. In Section 3, we frame new two variable models of the irreducible representations of in which the basis functions appear as ₁F₀(−n;−; x)yⁿ and ₁F₀(n;−; x)yⁿ. To make our discussion more fruitful, we exponentiate these models into the local multiplier representations of the corresponding Lie group K₅. In Section 4, we obtain the transformed models by using the Euler type integral transformation, defined in Section 2. The basis functions for these models are given in terms of ₂F₁. In Section 5, we derive several generating functions for ₁F₁ and recurrence relations for ₂F₁ that are believed to be new.

2.1. Lie Algebra

The complex Lie algebra as defined in [4] is the 5-dimensional complex Lie algebra with basis , ℰ and commutation relations:

Let g ∈ K₅ i.e. g=[1ceτbe-τ2a-bcτ0eτ2qe-τb-2qc000e-τ-c00001000001], where a, b, c, q, τ ∈ ℂ.

The elements of are of the type i=[0α4α3α2α50α52α1α3000-α5-α400001000001], where α1=dqdt,α2=dadt,α3=dbdt,α4=dcdt and α5=dτdt in the neighbourhood of identity. The basis elements of the algebra are:

Let us consider an irreducible representation ρ of on the vector space V and let

then the operators J^±, J³, Q, E obey the commutation relations same as (2.1). Define the set of all eigenvalues of J³ to be the spectrum S of J³. Further, let the irreducible representation ρ satisfy the conditions:

• Each eigenvalue of J³ has multiplicity equal to one.

• There exists a denumerable basis for V consisting of all the eigenvalues of J³.

This guarantees that S is denumerable and that there exists a basis for V consisting of vectors f_m such that J³f_m = mf_m. From Miller [4], a one variable model of the irreducible representations is given by:

Representation R(ω, m₀, μ)

where m∈S = {m₀ + n : n is an integer}.

Representation ↑_ω,μ

where m∈S ={−ω+ n : n is nonnegative integer}.

For these representations there is a basis of V consisting of vectors f_m, defined for each m∈S, such that

2.2. Euler Integral Transformation

Let us consider the complex vector space V of all functions f(x), representable as a power series about x = 0. We use

Then the isomorphic image of V under the transformation I:f(x) → h(β, γ) is W=IV. Now according to our requirement, we obtain transform of some expressions under the transformation I in terms of difference operators defined as

The following are the transforms under I:

We give below the two-variable models of the Lie algebra . Also, given along with are the multiplier representations of the local Lie group K₅ induced by the operators on ℱ, the space of all analytic functions in a neighbourhood of (x₀, y₀).

Model IA

Representation R(ω, m₀, μ):

where m ∈ S = {m₀ + n : n is an integer}.

The multiplier representation T₁(g)f of the Lie group K₅ induced by the operators on ℱ is

where |c/y(1 − x)| < 1 and lies in a small enough neighbourhood of e.

Model IIA

Representation R(ω, m₀, μ):

where m ∈ S = {m₀ + n : n is an integer}.

The multiplier representation T₂(g)f of the Lie group K₅ induced by the operators on ℱ is

where |c(1 − x)/y| < 1 and lies in a small enough neighbourhood of e.

Model IIIA

Representation ↑_ω,μ:

where m ∈ S = {−ω + n : n is nonnegative integer}.

The multiplier representation T₃(g)f of the Lie group K₅ induced by the operators on ℱ is

where lies in a small enough neighbourhood of e.

Model IVA

Representation ↑_ω,μ:

where m ∈ S = {−ω + n : n is nonnegative integer}.

The multiplier representation T₄(g)f of the Lie group K₅ induced by the operators on ℱ is

where lies in a sufficiently small neighbourhood of e.

To get the models, with the basis function appearing as hypergeometric functions, we reproduce a theorem as in [6]:

Theorem 4.1

Let the basis operators {J^±, J³, Q, E} on a representation space V with basis functions {f_m : m∈S} gives the irreducible representation ρ of the Lie algebra. Then the transformation I induces another irreducible representation σ ofon the representation space W = IV having basis functions {h_m : m∈S} in terms of operators {K^±, K³, Q₁, E₁}, where

i.e. ρ and σ are isomorphic.

We give below the transforms of models IA, IIA, IIIA and IVA introduced in section 3.

We will use models IA, IIA, IIIA and IVA for obtaining generating functions and the transformed models IB, IIB, IIIB and IVB for deriving recurrence relations. To obtain generating functions, we follow the method given in Manocha and Sahai [3]. We omit details and present the results only as follows:

5.1. Generating Functions

When we put q=0 in equation (3.2), we get

and when we put q = 0 in equation (3.4), we get

Also, when we put q=0 in equation (3.6), we get

and q=0 in equation (3.8) gives

5.2. Recurrence Relations

By using the Model IB, we get

Similarly, the more recurrence relations which we get from Models IB, IIB, IIIB and IVB are as follows:

The financial assistance provided to the second author in the form of a CSIR fellowship (File No.:09/1050(0006)/2016-EMR-I) from CSIR, HRDG, New Delhi, India is gratefully acknowledged.