On Some Mixed Trilateral Generating Functions of Modified Jacobi Polynomials by Group Theoretic Method

In this note, we have obtained some novel results on mixed trilateral generating functions involving P (α+n, β) n (x), a modification of Jacobi polynomials by group-theoretic method. We have introduced a linear partial differential operator and found the corresponding extended form of the group. Finally, we obtained a novel generating function with the help of which, our desired result has been established.


Introduction
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 dearth 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.
In fact, the idea of group-theoretic method in the study of generating functions of various special functions started in the middle of the last century by L. Weisner [1] while investigating generating functions of Hypergeometric functions. From seventies and onwards (i.e. just after the publication of the book "Obtaining generating functions" by E. B. McBride [2]) of the last century, Weisner's method has been extensively utilized by researchers in the derivation of generating functions of various special functions. In the present article, we have adopted group theoretic method to obtain where P (α, β) n (x) is defined by [3]: The main result of our investigation is stated in the form of the following theorem. For previous works on trilateral generating functions of Jacobi / modified Jacobi polynomials by group theoretic method, one can see the works [4][5][6].
Theorem-I: If there exists a bilateral generating relation of the form: where g n (u) is an arbitrary polynomial of degree n, then To prove the theorem, we introduce a linear partial differential operator and the corresponding extended form of the group in the next section. Finally, as an application of the operator, we shall obtain a novel generating relation of the special function under consideration with the help of which Theorem-1 will be proved.
2 Derivation of the operator, its extended form of the group and generating function.

Derivation of the operator
At first we seek the following first order linear partial differential operator: where A i (i =0,1,2,3) are functions of x, y, z but independent of n, α and a n is a function of n, α, β but independent of x, y, z.
Now using the following differential recurrence relation [3]: we obtain,

Extended form of the group generated by R
Now we find the extended form of the group generated by R i.e., we shall find where f (x, y, z) is arbitrary function and a is an arbitrary constant, real or complex. Let ϕ(x, y, z) be a function such that R ϕ = 0. Then on solving R ϕ = 0, we get a solution as i.e, Now let X, Y , Z be a set of new variables for which so that E reduces to D = ∂ ∂X . Now solving (2.7), we get a set of solutions as follows: from which we get .
Then e aR f (x, y, z) On calculation, we have ] . (2.13)

Application of the operator R
From (2.13), we notice that ) . (2.14) But by using (2.5), we get (2.15) Equating (2.14) and (2.15) and then replacing −2y 2 zw by t, we get the above generating function is worthy of notice.

Proof of the Theorem
We shall now prove the Theorem-1 stated above by using the generating relation (2.16).

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On Some Mixed Trilateral Generating Functions of Modified Jacobi Polynomials by Group Theoretic Method Now the right hand side of (1.3) which is Theorem-1.
Here we would like to point it out that Theorem-1 can be proved as follows by the direct application of the operator R by using the method as discussed in [7].
Let us now assume that where g n (u) is an arbitrary polynomial of degree n.
Replacing t by tz in (3.2) and then multiplying both sides of the same by y α , we get Operating exp(wR) on both sides of (3.3), we get .
The left hand side of (3.4), with the help of (2.13), becomes . (3.5) The right hand side of (3.4), with the help of (2.5), becomes On Some Mixed Trilateral Generating Functions of Modified Jacobi Polynomials by Group Theoretic Method Equating (3.5) and (3.6) we get (3.7) Putting −2w y 2 = 1 in (3.7), we get This completes the proof of Theorem-1.
Finally, if we use the following symmetry relation we shall get the following result.

Conclusion
The importance of the Theorem-1 lies in the fact that whenever one knows a bilateral gener-