Multiplier Representations and Generating Functions

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Multiplier Representations and Generating Functions

S. K. Raizada and Rajeev Kumar Department of Mathematics & Statistics,

Dr. R. M. L. Avadh University, Faizabad, U., P., INDIA.

e-mail: [email protected], e-mail: [email protected] (Received on: February 5, 2015)

ABSTRACT

Group theoretic method has been applied and some Generating functions have been obtained for proper hypergeometric function _
  , ;   ; 

by introducing Lie-operators effecting the one numerator and one denominator parameter in it simultaneously.

Keywords: Hypergeometric function / Lie-operators / Multiplier representation / Group theoretic method of obtaining generating functions.

INTRODUCTION

The Lie-group theoretic method suggested by Weisner¹and Miller²has been successfully used by Manocha³ to find certain generating functions involving double hypergeometric series for _
^s.Then Jain and Agarwal⁴ extended this technique to proper hypergeometric function
_, ; ;  and obtained some new generating functions for it by variation of the numerator parameter  only. In the present note, the authors have introduced Lie-operators affecting the numerator parameter  and denominator parameter  simultaneously and thus have deduced some newgenerating relations for proper hypergeometric functions.

FORMULATION OF THE PROBLEM We consider an ordinary differential equation

 ,_^ ,    0 , (1)

with a solution_,and thereby show that the corresponding partial differential equation

 ,   ,      ,   0 (2)

remains invariant by the transformation induced by the multiplier representation

  , of a Lie-group, which is isomorphic to some Lie-group!2, #.

Thus (2) can yield a solution  ,  serving as the generating function for solutions_ of (1).

DIFFERENTIAL OPERATORS AND MULTIPLIER REPRESENTATION We know that,

$ %
_{ }  , ;   ;  , (3)

is a solution⁵ of 1 ' ^_⁽⁾₍ *   '      1+^)_'    $  0 (4)

Now we replace the parameterby the operator    and construct the partial differential equation

,  % ,1 ' _-^-(₍ 1 ' _--.^-(  * '    1+_-^- ' _-.^- '  / , . (5) Clearly,   ^$is a solution of (5) iff$ is a solution of (4).

Next we introduce first order partial differential operators,

0¹ y_-.^- ^_   ' ─1 (6)

0³ 1 ' _-^- ' ^{ -}_-.'  (7)

0⁴ 1 ' ^{4 -}_-_-.^-   ' 1 ' ^4 (8) It can easily be seen that these operators0¹, 0³&0⁴satisfy the Commutator relations:

0¹, 0⁵  5 0⁵ , 0³, 0⁴  20¹ (9)

and hence generate a Lie-algebra of the Lie-group²!2, #.

The Casimir operator, 6  0³0⁴ 0¹0¹' 0¹

 1 '  71 '  ^

^ 1 '  ^

   *γ ' α  β  1z+ 

 ' βy ' αβ<  1

4 *1   '  ^+   ' 

 1 '  ^_>*1   '  ^+   ' , (10) commutes with each of the0- operators.

In terms of the Casimir operator 6 of (10),eq. (5) can be written as:

91 6,   ,^_>*1   '  ^+   ' / , . (11) Now, to determine the multiplier representation induced bythe0- operators, we need to compute the expression for?^@AB, ?^CAD& ?^)AE.

Using Miller²technique, it is seen that:

?^@AB,   1 ^@_.^F41 ^@_.^4G ^@3._.3@ , H  ; I^@_.I  JK1, ||^4 (12)

?^CAD,   1  M^4N ^3.C_3.C,_3.C^. ; |M| O 1 (13)

?^)AE,   ?^P()Q3R4G4, ?⁾. (14) Therefore, it may easily be deduced that:

  ,   ?^@AB?^CAD?^)AE, 

 ?^P()Q3 R4 G41 ^@_.^F41 ^@_.^4G1  H  M^4N  @3.3.3@C

.3@3@3.C,_3@3.C^@3.ST , (15) where the complex parametersμ,M,$are related to V !2, # , such that:

% W XY Z;WZ ' XY  1, (16)

respectively by: μ  ─ Y W⁄ , M  ─WX &?^T( W.

Therefore, for in a sufficiently small neighborhood of the identity element

1 00 1 V !2, # ,

  ,   W^F4\41 '_^.^]^F4Z ' X^4N ^.4]4_.

^.4]4_.,_{4_.}^{^.4]}; (17)

|W` W| O a,|W` Z| O a , I^_._I O 1 , I_^.^]I O JK 1, ||^4.

DEDUCTION OF GENERATING FUNCTIONS

First we choose ,  to be a common eigen function of the operator6and  ' 10¹0¹ 1 ' 0¹0⁴

2 b

2    ' ' 2 ^c' 3 '    ' ' 3e 0⁴ f ' g ' ^c' 1   '   1hi0¹ (18) i.e. ,  satisfy either of the following simultaneous equations:

6,   ,^_>*1  α ' β  γ^+   ' / , , (19)  ' 10¹0¹ 1 ' 0¹0⁴

2 b

2    ' ' 2 ^c' 3 '    ' ' 3e 0⁴

* '  ' ^c' 1   '   1+0¹ ,^._   ' ' 2 ^c' 1 b^_   '  ' 1   '   1i '^_>   ' ' 1  '   1/ , , (20) which can be re-written respectively as:

,1 ' _-^-(₍ 1 ' _{- -.}^-(  *γ ' α  β  1z+_-^- ' βy_-.^- ' αβ/ ,   0 (21) ,1 ' _-.^-(₍ 1 ' _{- -.}^-(  *γ ' α  β  1y+_-.^- ' βz_-^- ' αβ/ ,   0 (22) These equations have a solution⁶:

,   F, , ^c;  ;  ,  ; (23) where,

F_, , ^c;  ;  ,   ∑ ^NlDm_F ^{\l \n m l.m}

lDm o! q!

ro,qst .

Therefore,

  ,   W^F4\41 '_^.^]^F41 '_^.^]^4\Z ' X^4N
_,, , ^c; ;^.4]4_.

^.4]4_.,_{4_.} ^{^.4]}/(24) Now   ,  satisfies:

6  ,     6 , 

   uv1

4 1   '  ^   ' w x , 

 b^_>1  α ' β  γ^   ' e   ,  (25) which shows that (5) is invariant with respect to transformation induced by the multiplier representation of !2, #.

Therefore  ,  has an expansion of the form

  ,   ∑^rs4rz  _
  , ;   ;  . (26) To compute the coefficientsz_  , we put   0 on both sides to get,

z_   W^F4\4X^Z4N4 Ґ Q 3 |

ҐQ  ∑r ^{Q 3 }~! Ґ33~^{} 4F}}

~st _^^_]^~F_' ' `, , <sup>c</sup>;  ' `; 1, 'W X (27) Thus the generating relation becomes:

1 '_^.^]^R41 '_^.^]^4G1 '^_._^4QF_,, , ^c; ;^.4]4_.

^.4]4_.,_{4_.} ^{^.4]}/

93

  Ґα  n

Ґ α X

Z ‚

r 

s4r

_  , ;   ; 

 α  ~ 1 ' ~

`! Ґ1    `

r

~st

XY WZ‚

~ƒ

F_,' ' `, , <sup>c</sup>;  ' `; 1, 'W X / ; (28) I_^.^]I O JK 1, ||^4) ,I^_._I O 1, WZ ' XY  1

where the terms corresponding to   '1, '2 , ' ' ' ' ' ' ' ' ' 'are well defined because of the relation;

@‡4qlim α  H~ 1 ' ~

`! Ґ 1  H  `

r

~st

XY WZ‚

~
,'H ' `, , <sup>c</sup>;  ' `; 1, 'W X /

 ^{Q4 q}_q! ^{m 4Fm}_^^_]^q∑ ^Q_{~! 3q}^{} 4F3q}}

}

r~st _^^_]^~F,' `, , <sup>c</sup>;  ' ` ' ˆ; 1, 'W X / (29)

Equation (28) is the required result.

SPECIAL CASE

As a special case, result (28) reduces to the following generating relation

1 'M

‚

R41 'M

 ‚

4G
_u, , ^c; ; ' M

 ' M ,  ' Mx

 ∑ ҐQ 3 |Ґg\ⁿ 3 |hҐ R

ҐQҐ\ⁿ ҐR3| Ґ|3 

rs4r _
  , ;   ;  _
α  n, ^c ; 1  n; 'w ^; (30)

ŠM

Š O JK 1, ||^4

Similarly choosing another common eigen function, we can obtain other generating function.

REFERENCES

1. Weisner L., Group theoretic origin of certain generating functions. Pacific J. Math 5:1033-1039 (1955).

2. Miller W., Lie theory and special functions. Academic press (1966).

3. Manocha H L., Lie theoretic generating function. Pub De I’ Institut. Mathematique (1976).

4. Jain R & Agarwal B M., Multiplier representation and generating functions. Comment Math. Univ. 30 (2):105-111 (1981).

5. Rainville ED. Special functions. Mc Millan (1960).

6. Srivastava H M & Manocha H L., A treatise on generating functions. Ellis Harwood Series (1984).