GENERATING FUNCTIONS ON EXTENDED JACOBI POLYNOMIALS FROM LIE GROUP VIEW POINT

Generating functions play a large role in the study of special functions. The present paper deals with the derivation of some novel generating functions of extended Jacobi polynomials by the application of group-theoretic method introduced by Louis Weisner. In fact, by suitably interpreting the index (n) and the parameter (β) of the polynomial under consideration we define four linear partial differential operators and on showing that they generate a Lie-algebra, we obtain a new generating relation (3.3) as the main result of our investigation. Furthermore, some generating functions of Laguerre, Hermite, Bessel and Jacobi polynomials are obtained as the special cases of our main result. Some applications of our results are also pointed out.


Introduction
The extended Jacobi polynomials as defined by I. Fujiwara [1] are as follows: (1.1) They satisfy the following ordinary differential equation: Recently, some attempts [2], [3] have been made by researchers for deriving generating functions of the polynomials under consideration from the Lie group view point.The aim at presenting this article is to apply L. Weisner's group-theoretic method [4] with the simultaneous suitable interpretations of the index (n) and the parameter (β) in the study of extended Jacobi polynomials.It may be mentioned that in the course of constructing a four dimensional Lie algebra we have obtained two operators such that, when operated on the polynomials under consideration, simultaneously raise (lower) and lower (raise) the index and the parameter by one unit.Such type of operators do not seem to have appeared earlier in the study of extended Jacobi polynomials.The main results of this paper are the formulas (3.3) to (3.6) given in Section 3.

Group Theoretic Method
Replacing d dx by ∂ ∂x , n by y ∂ ∂y , β by z ∂ ∂z and u by v(x, y, z) in (1.2), we get the following partial differential equation: Let us now introduce a set of linear partial differential operators, A i , i = 1, 2, 3, 4, defined as follows: (2.2)

Then (2.3)
On generating functions of extended Jacobi polynomials 5 We now proceed to find the commutator relations satisfied by A i (i = 1, 2, 3, 4).Using the notation So from the above commutator relations we can easily state the following: Theorem.The set of operators {1, A i (i = 1, 2, 3, 4)} where 1 stands for the identity operator, generates a Lie-algebra L.

Now the partial differential operator L given by
. We can easily verify that each of A i (i = 1, 2, 3, 4) commutes with L. In other words, Now the extended forms of the groups generated by A i (= 1, 2, 3, 4) are given as follows: (2.7) e a1A1 f (x, y, z) = f (x, e a1 y, z), Thus we have e a4A4 e a3A3 e a2A2 e a1A1 f (x, y, z

Generating Functions
From (2.1) it is seen that F n (α, β; x)y n z β is a solution of the system: From (2.5) we observe that where S = e a4A4 e a3A3 e a2A2 e a1A1 .
Therefore, the transformation S(F n (α, β; x)y n z β ) is annihilated by (x − a)L.Now putting α 1 = α 2 = 0 and replacing f (x, y, z) by F n (α, β; x)y n z β we get Equating (3.1) and (3.2) we get our main result: Before discussing the particular cases of (3.3) we would like to point out that the operators A 3 , A 4 being non commutative, the relation (3.3) will change if we change the order of the Lie element e a4A4 e a3A3 .This is done in Section 4.

Now we discuss several cases:
Case 1. Putting a 4 = 0, a 3 = 1 and − z y = t in (3.3) we get Case 2. Putting a 3 = 0, a 4 = 1 and y z = t in (3.3) we get We now proceed to find some particular cases of interest of results (3.4) and (3.5).

Variants of the Result (3.3)
Since [A 3 , A 4 ] = 0, we can well aply the operator e a3A3 × e a4A4 on the function F n (α, β; x)y n z β .Now Again On generating functions of extended Jacobi polynomials 11 Equating (4.1) and (4.2) we get the following result

Application
Relations (3.4) and (3.5) may be applied in deriving bilateral generating functions involving the special function under consideration.We shall give an application by using the relation (3.5) in deriving the following theorem as bilateral generating relation.

Theorem. If
where Proof: The importance of this theorem lies in the fact that one can get a large number of bilateral generating relations from (ii) by attributing different suitable values to a n in (i).