A class of proper and improper partial bilateral generating Functions for some special Polynomials

In this paper, the group theoretic method is used to derive some classes of proper and improper partial bilateral generating functions for certain special polynomial. Some new and known results are obtained as special cases of the main results.


Introduction and Preliminaries:
Generating functions play a large role in the study of special functions.Generating functions which are available in the literature are almost bilateral in nature.There is a death of trilateral generating functions in the field of special functions.Group-theoretic method of obtaining generating functions for various special functions has been receiving much attention in recent years.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 grouptheoretic significance of generating functions for Hypergeometric, Hermite, and Bessel functions [10, 11 and 12] respectively.This technique was used by Khan et.al [2,3,4] for obtaining generating functions for various special functions.Miller, McBride, Srivastava and Manocha [6, 5 and 9], respectively , reported Group theoretic method for obtaining generating relations in their books.
In this paper, we have used the group-theoretic method to obtain new classes of generating functions from a given class of generating function.In this section, we give the brief introduction to Konhauser, Gegenbauer and Laguerre polynomials.Also, we define the recently used terms proper and improper partial bilateral generating functions.

 − t v vzt
Next, we define the recently used terms proper partial bilateral generating relation and improper partial bilateral generating relation for two classical polynomials introduced by Sarkar (cf.[8]).

Definition 1.1 Proper partial bilateral generating function for two classical polynomials is the relation
where the coefficients , n a s are quite arbitrary, where the coefficients In the following section, we have obtained new classes of improper bilateral generating functions for some orthogonal polynomials.

Improper partial bilateral generating functions:
We derive the main result in the form of the following theorem: Theorem (1) : If there exist the following class of improper partial bilateral generating functions for the Konhauser and Gegenbauer polynomials by means of the relation where n a is arbitrary , then the following generating function holds

Proof of the Theorem 1
on multiplying both sides of above equation by s t y  , which gives us Finally, when putting 1 = = = h t y in the above equation (2.6), we arrive at the result (2.2).
It may be of special interest to point out that, for s = m, the above theorem has become a nice class of generating functions forms proper partial bilateral generating functions.We state these results in the form of following corollary: Corollary If there exist the following class of (proper) partial bilateral generating function for the Konhauser and Gegenbauer polynomials by means of the relation where n a is arbitrary , then the following general class of generating function is hold respectively.In the following section, we have obtained new and known classes of proper partial bilateral generating functions as particular cases of the above main results.
Particular Cases: I. On putting 1 = k in Theorem 1, we get the following known result due to Sarkar [8]: where For m s = in (2.10) gives the following class of (proper) partial bilateral generating function for the Laguerre and Gegenbauer polynomials due to Sarkar [8]: the parameters  and  , respectively.where   and are complex numbers , n is a positive integer or zero and m is integer number.Definition 1.2 Improper partial bilateral generating relation for two classical polynomials, is the relation

(
1) and using relation (1.7), we get the following improper partial bilateral generating functions for Konhauser and Tchebycheff polynomials: