Kyungpook Mathematical Journal 2018; 58(4): 615-625
Certain Models of the Lie Algebra and Their Connection with Special Functions
Department of Mathematics, NIT, Kurukshetra-136119, India
e-mail : sarasvati1201@nitkkr.ac.in and geetarani.5121992@gmail.com
*Corresponding Author.
Received: July 15, 2018; Revised: October 28, 2018; Accepted: November 6, 2018; Published online: December 23, 2018.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In this paper, we discuss the connection between the 5-dimensional complex Lie algebra and Special functions. We construct certain two variable models of the irreducible representations of . We also use an Euler type integral transformation to obtain the new transformed models, in which the basis function appears as 2F1. Further, we utilize these models to get some generating functions and recurrence relations.

Keywords: Lie algebra, hypergeometric function.
1. Introduction

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 , Manocha , Manocha and Sahai , Sahai , 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 , Vilenkin , Wawrzynczyk  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  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. Prelimnaries

2.1. Lie Algebra The complex Lie algebra as defined in  is the 5-dimensional complex Lie algebra with basis , ℰ and commutation relations:

$[J3,J±]=±J±, [J3,Q]=2Q,[J-,J+]=E, [J-,Q]=2J+, [J+,Q]=0,[J+,E]=[J-,E]=[J3,E]=[Q,E]=0.$

Let gK5 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:

$Q=, E=, J+=J-=[0100000000000-100000000000], J3=[000010100000-1000000000000].$

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

$J±=ρ(J±), J3=ρ(J3), Q=ρ(Q), E=ρ(E),$

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 fm such that J3fm = mfm. From Miller , a one variable model of the irreducible representations is given by:

Representation R(ω, m0, μ)

$J+=μz,J-=(m0+ω)z-1+ddz,J3=m0+zddz,Q=μz2,E=μ,fm(z)=zn,$

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

Representationω,μ

$J+=μz,J-=ddz,J3=zddz-ω,Q=μz2,E=μ,fm(z)=zn,$

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

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

$J3fm=mfm,J+fm=μfm+1,J-fm=(m+ω)fm-1,Efm=μfm,Qfm=μfm+2.$

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

$h(β,γ)=I[f(x)]=Γ(γ)Γ(β)Γ(γ-β)∫01xβ-1(1-x)γ-β-1f(x)dx,Re γ>Re β>0.$

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

$Eβh(β,γ)=h(β+1,γ),Lβh(β,γ)=h(β-1,γ),Δβh(β,γ)=(Eβ-1)h(β,γ).$

The following are the transforms under I:

$I[xf]=βγEγβ.hI[x∂xf]=βΔβ.hI[(1-x)f]=(γ-β)γEγ.hI[(1-x)-1f]=(γ-1)(γ-β-1)Lγ.h.$
3. Two Variable Models

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):

$J-=(m0+ω)y-1(1-x)-1+∂∂y-y-1x∂∂x,J3=m0+y∂∂y,J+=μy(1-x),Q=μy2(1-x)2,E=μ,fm(x,y)=F10(-n;-;x)yn,$

where mS = {m0 + n : n is an integer}.

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

$[T1(g)f](x,y)=exp(μ(y2(1-x)2q+a+y(1-x)b)+m0τ)×(1+cy(1-x))m0+ωf(xyy+c,(y+c)eτ),$

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

Model IIA

Representation R(ω, m0, μ):

$J-=(m0+ω)y-1(1-x)+∂∂y-y-1x(1-x)∂∂x,J3=m0+y∂∂y,J+=μy(1-x)-1,Q=μy2(1-x)-2,E=μ,fm(x,y)=F10(n;-;x)yn,$

where mS = {m0 + n : n is an integer}.

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

$[T2(g)f] (x,y)=exp(μ(y2(1-x)-2q+a+y(1-x)-1b)+m0τ)×(y+c(1-x)y)m0+ωf(xyy+c(1-x),(y+c)eτ),$

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

Model IIIA

Representation ↑ω,μ:

$J-=∂∂y-y-1x∂∂x,J3=y∂∂y-ω,J+=μy(1-x),Q=μy2(1-x)2,E=μ,fm(x,y)=F10(-n;-;x)yn,$

where mS = {−ω + n : n is nonnegative integer}.

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

$[T3(g)f](x,y)=exp(μ(y2(1-x)2q+a+y(1-x)b)-ωτ)f(xyy+c,(y+c)eτ),$

where lies in a small enough neighbourhood of e.

Model IVA

Representation ↑ω,μ:

$J-=∂∂y-y-1x(1-x)∂∂x,J3=y∂∂y-ω,J+=μy(1-x)-1,Q=μy2(1-x)-2,E=μ,fm(x,y)=F10(n;-;x)yn,$

where mS = {−ω + n : n is nonnegative integer}.

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

$[T4(g)f](x,y)=exp(μ(y2(1-x)-2q+a+y(1-x)-1b)-ωτ)×f(xyy+c(1-x),(y+c)eτ),$

where lies in a sufficiently small neighbourhood of e.

4. Transformed Models Of Lie Algebra To get the models, with the basis function appearing as hypergeometric functions, we reproduce a theorem as in :

Theorem 4.1

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

$K+=IJ+I-, K-=IJ-I-, K3=IJ3I-Q1=IQI-, E1=IEI-hm=Ifm, m∈S,$

i.e. ρ and σ are isomorphic.

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

Model IB$K-=(m0+ω)y-1(γ-1γ-β-1)Lγ+ddy-y-1βΔβ,K+=μy(γ-βγ)Eγ,K3=m0+yddy,Q1=μy2(γ-β) (γ-β+1)γ(γ+1)Eγ2,E1=μ,hm(β,γ,y)=F21(-n,β;γ;1)yn,$

where mS = {m0 + n : n is an integer}.

Model IIB$K-=(m0+ω)y-1(γ-β)γEγ+ddy-y-1β(γ-β)γΔβEγ,K+=μy(γ-1)(γ-β-1)Lγ,K3=m0+yddy,Q1=μy2(γ-1) (γ-2)(γ-β-1) (γ-β-2)Lγ2,E1=μ,hm(β,γ,y)=F21(n,β;γ;1)yn,$

where mS = {m0 + n : n is an integer}.

Model IIIB$K-=ddy-y-1βΔβ,K+=μy(γ-βγ) Eγ,K3=yddy-ω,Q1=μy2(γ-β) (γ-β+1)γ(γ+1)Eγ2,E1=μ,hm(β,γ,y)=F21(-n,β;γ;1)yn,$

where mS = {−ω + n : n is nonnegative integer}.

Model IVB$K-=ddy-y-1β(γ-β)γΔβEγ,K+=μy(γ-1)(γ-β-1)Lγ,K3=yddy-ω,Q1=μy2(γ-1) (γ-2)(γ-β-1) (γ-β-2)Lγ2,E1=μ,hm(β,γ,y)=F21(n,β;γ;1)yn,$

where mS = {−ω + n : n is nonnegative integer}.

The models given above satisfy the following:

$[K3,K±]=K±, [K3,Q1]=2Q1,[K-,K+]=E1, [K-,Q1]=2K+, [K+,Q1]=0,[K+,E1]=[K-,E1]=[K3,E1]=[Q1,E1]=0,$

and thus represent a representation of .

Also

$K3hm=mhm,K+hm=μhm+1,K-hm=(m+ω)hm-1,E1hm=μhm,Q1hm=μhm+2.$
5. Recurrence Relations and Generating Functions

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 . 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

$exp(μ(a+y(1-x)b)+m0τ) (1+cy(1-x))m0+ωf(xyy+c,(y+c)eτ)=∑l=-∞∞exp(μa+(m0+k)τ)ck-lΓ(k+ρ+1)Γ(k-l+1)Γ(ρ+l+1)×F11(-ρ-l;k-l+1;-μbc) [y(1-x)]l,$

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

$exp(μ(a+y(1-x)-1b)+m0τ) (1+cy(1-x)-1)m0+ωf(xyy+c(1-x),(y+c)eτ)=∑l=-∞∞exp(μa+(m0+k)τ)ck-lΓ(k+ρ+1)Γ(k-l+1)Γ(ρ+l+1)×F11(-ρ-l;k-l+1;-μbc) [y(1-x)-1]l.$

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

$exp(μ(a+y(1-x)b)-ωτ)f(xyy+c,(y+c)eτ)=∑l=-∞∞exp(μa+(k-ω)τ)ck-lΓ(k+1)Γ(k-l+1)Γ(l+1)×F11(-l;k-l+1;-μbc) [y(1-x)]l,$

and q=0 in equation (3.8) gives

$exp(μ(a+y(1-x)-1b)-ωτ)f(xyy+c(1-x),(y+c)eτ)=∑l=-∞∞exp(μa+(k-ω)τ)ck-lΓ(k+1)Γ(k-l+1)Γ(l+1)×F11(-l;k-l+1;-μbc) [y(1-x)-1]l.$

5.2. Recurrence Relations

By using the Model IB, we get

$(n+β)F21(-n,β;γ;1)+((m0+ω) (γ-1)γ-β-1) F21(-n,β;γ-1;1)-βF21(-n,β+1;γ;1)=(m+ω)F21(-n+1,β;γ;1).$

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

$(γ-β) (γ-β+1)γ(γ+1)F21(-n,β;γ+2;1)=F21(-n-2,β;γ;1),$$(γ-β)γF21(-n,β;γ+1;1)=F21(-n-1,β;γ;1),$$(m0+ω) (γ-β)γF21(n,β;γ+1;1)-β(γ-β)γ[F21(n,β+1;γ+1;1)-F21(n,β;γ+1;1)]+nF21(n,β;γ;1)=(m+ω)F21(n-1,β;γ;1),$$(γ-1)(γ-β-1)F21(n,β;γ-1;1)=F21(n+1,β;γ;1),$$(γ-1) (γ-2)(γ-β-1) (γ-β-2)F21(n,β;γ-2;1)=F21(n+2,β;γ;1).$
Acknowledgements

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.

References
1. Gasper, G, and Rahman, M (2004). Basic hypergeometric series: Cambridge University Press
2. Manocha, HL (1989). Lie algebras of difference-differential operators and Appell functions F1. J Math Anal Appl. 138, 491-510.
3. Manocha, HL, and Sahai, V (1990). On models of irreducible unitary representations of compact Lie group SU (2) involving special functions. Indian J pure Appl Math. 21, 1059-1071.
4. Miller, W (1968). Lie theory and special functions. Mathematics in Science and Engineering. 43.
5. Rainville, ED (1960). Special functions. New York: Macmillan
6. Sahai, V (1995). On certain models of irreducible representations of sl (2,ℂ) and Euler integral transformation. Indian J pure appl Math. 26, 1177-1189.
7. Sahai, V, and Yadav, S (2007). Representations of two parameter quantum algebras and p, q-special functions. J Math Anal Appl. 335, 268-279.
8. Sahai, V, and Yadav, S (2008). On models of certain p, q-algebra representations: The p, q-oscillator algebra. J Math Phys. 49, 053504.
9. Sneddon, IN (1961). Special functions of mathematical physics and chemistry. Edinburgh: Oliver and Boyd, Ltd
10. Srivastava, HM, and Manocha, HL (1984). A treatise on generating functions. New York: Ellis Horwood, Chichester, Halsted Press
11. JaVilenkin, N (1968). Special functions and the theory of group representations: American Mathematical Society
12. Wawrzynczyk, A (1984). Group representations and special functions: Examples and Problems prepared by Aleksander Strasburger. Warszawa: Polish Scientific Publishers 