Generating Functions for Legendre Polynomials by using ………………….Gamal Ali Qashash Generating Functions for Legendre Polynomials by using Group Theoretic Method

In this paper we obtain generating functions for the Legendre polynomials ( ) x P n in its modified form by using Weisner's group theoretic method. Whereas, we deployed it to determine the new generating relations between the generalized Legendre polynomials and with easy way. The ideas in consent with proofs are originated from the book of McBride [8] and is used to determine new generating relations which involve modified Legendre polynomials.


Introduction
The unification of generating functions has a great importance in connection with ideas and principles of special functions. Group Theoretic Method proposed by Louis Weisner in 1955, who employed this method to find generating relations for a large class of special functions. Weisner discussed the group-theoretic significance of generating functions for Hypergeometric, Hermite, and Bessel functions in his papers [13,14] and [15] respectively. McBride [8] deployed Weisner's method to determine the new generating relations that involve Hermite, Bessel, generalized Laguerre, Gegenbauer polynomials. In this directions, some important steps has been made by researchers, namely Singhal and Srivastava [10], Chaterjea [1,2] and Chongdar [3]. In their study, Desale and Qashash [5] have obtained a new general class of generating functions for the generalized modified Laguerre polynomials ( ) x L n ) ( by group theoretic method. Also, they have introduced the bilateral generating function for the generalized modified Laguerre and Jacobi polynomials, with the help of two linear partial differential operators. Further, continuing their study [6,7], they used the group theoretic method to obtain proper and improper partial bilateral as well as trilateral generating functions. Now, continuing the work in connection with class of generating functions, we extend our ideas to obtain new generating relations that involve between the generalized Legendre polynomials. Legendre Polynomials ( ) x P n for n = 0, 1, 2, . . . are defined as [9]; . ; 1 We see that these polynomials are particular solutions of Legendre differential equation One may consult Rainville [9] for more details about Legendre Polynomials. Note that subscripts in the following relations are nonnegative integers: where D is the differential operator

Linear Differential Operators
In this section, we define some linear partial differential operators in two independent variables x and y . We will investigate their commutative properties while operating on Legendre Polynomials. If we use (1.2) and (1.3) repeatedly, then we see that So, we can rewrite the above equation in the form differential operator L as; Hence, we define a linear partial differential operator L as Next, we seek to define the linear differential operators A, B, and C, which will commute with L with L x) (  . Note that  is some function of x to be determined. Let us define these operators as follows: As in previous paragraph, we define linear partial differential operators. It is interesting to see that how does these operators act on Legendre polynomials Following are the actions of these operators on respective functions: Let us consider an arbitrary 2 l function u = u(x, y) in two independent variables. Hence, if we operate the operator L on u = u(x, y), we get Since u is an arbitrary function, we conclude that

Extended form of the group of Operators
In this section, we extend the operators B and C which defined in the previous section to the exponential form. Consider an arbitrary function ) , ( y x f f = in two independent variables. Also, we consider the arbitrary constants b and c. exp (bB) and exp (cC) are called the extended form of the transformation groups generated by B and C. For doing this business, we will follow the method suggested by Weisner [13]. One may refers to McBride [8] for similar kind of analysis.
In order to find the extended form of the group generated by operators B and C, we will change the form of differential operator and finally by the In We will transform E into -D. A simple use of this substitution, and applying Taylor's theorem in the form On the other hand, we can expand left hand side of (4.1) in series form and then repeated application of (2.8) on the same side of (4.1), we get which is the new generating function that involves modified Legendre polynomial and Gegenbauer polynomial. (4.6) Separately, we consider the left hand side of (4.6) and we write exponential operators in series form so that we have the following relation: in the above equation and simplifying, we obtains the following generating function between modified Legendre polynomials, which we believe to be new generating function.

Conclusion
In this section, we conclude the findings of this paper. Basically, we adopt the Weisner method to determine the new generating functions for modified Legendre polynomial. In the beginning of this paper, we defined the linear operators A, B, C, and L. In section 2, we discussed the commutative properties of these operators. We have extended these operators in exponential them to determine the new generating functions such as (4.2), (4.4), (4.5) and (4.8). We also believe that the operators 1, A, B and C form a Lie group.