A Note on the Generalized and Universal Associated Legendre Equations∗

Keegan L.A.Kirk, Kyle R.Bryenton, and Nasser Saad

School of Mathematical&Computational Sciences,University of Prince Edward Island,550 University Avenue,Charlottetown,PEI C1A 4P3,Canada

1 Introduction

Recently,the Universal Associated Legendre Polynomials

has been the subject of many interesting studies.[1−7]These polynomials are solutions to the differential equation(see Ref.[2]and the references therein):

in which b=0,m′=Through partial-fraction decomposition of the rational coefficient of the Pm′ℓ′(r)term,Eq.(2)is shown to be a slight modification of the well-known Generalized Associated Legendre Equation[8−19]

The differential equation(3)was introduced first by Bateman in his analysis of the harmonic equations(Ref.[8],page 389).Following Bateman’s work,this was later intensively studied in a series of research articles by Kuipers and Meulenbeld for complex-valued parameters k,m,and n.[9−17]The recent book of Virchenko and Fedotova[19]was devoted to the subsequent development of the theory of the Generalized Associated Legendre Functions and their applications.

As we shall prove in the present work,both differential equations(2)and(3)are members of a more general class of differential equations characterized by

where aj,bj,j=1,2,3,c3are real parameters and ξ1

The exact solutions of the differential equation(4)are given,along with their relations to the published solutions of the Generalized and Universal Associated Legendre Differential Equations(2)and(3).New solvable classes of differential equations useful for the analysis of quantum systems are obtained.[20−22]

2 Exact Solutions

The differential equation(4)has three regular singular points,r ∈ {ξ1,ξ2,∞}with exponents µ1, µ2,andµ∞,respectively,determined as the roots of the indicial(quadratic)equations:

According to the classical theory of ordinary differential equations,[23]Eq.(4)is reducible to the hypergeometric equation.To this end,the general solutions of Eq.(4)take the form

where the exponents µ1and µ2are evaluated using Eqs.(5)and(6)respectively.The substitution of Eq.(8)into Eq.(4)yields the following hypergeometric-type equation for the function f≡f(r):

Employing the Möbius transformation,z=(αr+ β)/(γr+ δ),for αδ− βγ0,yields

Thus,if γ=0,the change of variables r→z≡z(r)transforms Eq.(10)into an equation of the same type as that of Eq.(9).This in turn implies

To express the solutions of this equation in terms of hypergeometric functions,one must impose either of the following necessary conditions on α, β,and δ:

Should we impose the conditions as given(i),Eq.(11)reduces to(denoting f→)

(z)=z1−2µ2+(b1+a1ξ2)/(ξ2−ξ1)

More precisely,the analytic solutions of the differential equation(4),for arbitrary constants A1and A2,are

where

Equivalently,the exact solutions of the differential equation(4)may also be expressed as(denoting f→)

As a result,we are able to obtain Eq.(18)from Eq.(21)and vice versa.This relationship may be confirmed using the linear transformation(Ref.[24],Eq.(15.8.4)):

After some simplification,it follows that

For the second solution(17),we may again apply Eq.(24)to obtain

To obtain the expression in terms ofthe Pfaff transformation(Ref.[24],Eq.(15.8.1)),

must be used for the hypergeometric functions on the right-hand side of Eq.(26).It is sufficient to focus on either one of the solution sets.

3 Connection with the Generalized Associated Legendre Equation

This section serves to demonstrate the relationship between the differential equations(3)and(4).First denote

then,using partial-fraction decomposition,Eq.(4)reads

Using Eq.(28),the indicial equations(5)and(6)may be expressed as

Thus,Eq.(29)reduces to

with the exact solutions,determined via Eq.(21),given by:

as found earlier by Kuipers et al.forµ1=n/2 andµ2=−m/2.In this case equation(30)reads

For ξ2= −ξ1=1,Eq.(33)reduces to the Generalized Associated Legendre Differential Equation.Different choices of ξ1and ξ2give rise to other interesting classes of differential equations,(e.g. ξ1=0, ξ2=1).The mathematical properties of these other classes(such as the weight-function,the recurrence relation,the orthogonality conditions,etc.)will be the focus of future work.

4 Connection with the Universal Associated Legendre Equation

To establish the connection between the solutions of the differential equation(4)and that of the Universal Associated Legendre Equation(2)as given by Eq.(1),we express Eq.(1)in terms of the hypergeometric function.Using the

Legendre Duplication Formula,Γ(2z)=22z−1Γ(z)Γ(z+1/2)/Eq.(1)can be written as

Further,by means of the Pochhammer identity Γ(z − ν)=(−1)νΓ(z)/(1 − z)ν,we obtain

With the assumption that −(1/2)(ℓ′− m′)or(1/2)(1 − (ℓ′− m′))=0,−1,−2,...,this equation may now be written in terms of the hypergeometric equation

with the understanding that the limit of the right-hand side is well-defined as r→0.Since

it easily follows that

The identity(Ref.[24],Eq.(15.8.6))

implies

With a1=2(µ1+µ2−1),the differential equation(4)reads after an application of partial-fraction decomposition,

Using the identity(Ref.[24],Eq.(15.8.20)):

the solution of Eq.(39),namely,

can be written as

Using the identity(Ref.[24],Eq.(15.8.7)),

and assuming

it follows that c=1/2.Finally,the solution(41)now reads

which reduces,up to a multiplicative constant,to the solution(38)for b1=0, ξ1= −1 and ξ2=1.

5 Conclusion

The classical Generalized and the recent Universal Associated Legendre Equations are members of the more broad class of differential equations given by Eq.(4).We established the hypergeometric solutions of this class of equations and demonstrated that they lead to the Generalized and Universal Associated Legendre hypergeometric solutions.These new solutions open the door for further compelling studies,including the examination of their mathematical properties and the investigation of their applicability to problems in mathematical physics.

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