IMPLICIT SUMMATION FORMULA FOR 2-VARIABLE LAGUERRE-BASED POLY-GENOCCHI POLYNOMIALS

The main object of this paper is to introduce a new class of Laguerre-based poly-Genocchi polynomials and investigate some properties for these polynomials and related to the Stirling numbers of the second kind. We derive summation formulae and general symmetry identities by using different analytical means and applying generating functions.

where C 0 (x) denotes the 0 th order Tricomi function.The n th order Tricomi functions C n (x) are defined as: with the following generating function: for t = 0 and for all finite x.
From (1.9) and (1.10), we get Thus, we have where L n (x) are the classical Laguerre polynomials (see [1]).Now, we recall here the following definition as follows: The Stirling number of the first kind is given by and the Stirling number of the second kind is defined by generating function: so that n (0, 0) are called the poly-Genocchi numbers.For k = 1 in (2.1), we have where L G n (x, y) is Laguerre-based Genocchi polynomials (see [13]).
Thus, we have On setting x = 0, (2.1) reduces to the known result of Kim et al. [14.,p.Eq.( 4)4776]: (2.4) Theorem 2.1.The following explicit summation formulae for Laguerre-based poly-Genocchi polynomials holds true: Proof.Using generating function for Laguerre-based poly-Genocchi polynomials (2.1), we have In particular k = 2, we have Replacing n by n − m in the r.h.s of above equation, we have On equating the coefficients of the like powers of t in both sides, we get (2.5).
Corollary 2.1.For n ≥ 0, we have Proof.From (2.1), we have Replacing n by n − m in above equation and comparing the coefficients of t n , we get From (2.8), we have Therefore by (2.9), we obtain the result (2.7). Remark (2.10) Theorem 2.3.For n ≥ 0, we have (2.11) Proof.By using (2.1), we can be written as From equations (2.12) and (2.13), we get Theorem 2.4.For n ≥ 1, we have Proof.By using definition (2.1), we have Replacing n by n − p in the above equation and comparing the coefficients of t n in both sides, we obtain the result (2.15).Corollary 2.4.For n ≥ 1, we have (2.16) Theorem 2.5.For d ∈ N with d ≡ 1(mod2), we have (2.17) Proof.From equation (2.1), we can be written as Replacing n by n − p in above equation and comparing the coefficient of t n in both sides, we get (2.17).

Summation formulae for Laguerre-based poly-Genocchi polynomials
In this section, we establish summation formula for Laguerre-based poly-Genocchi polynomials by using series techniques method.Proof.Replacing t by t + u and rewrite the generating function (2.1) as Replacing y by z in the above equation and equating the resulting equation to the above equation, we get On expanding exponential function (3.3) gives which on using formula [16, p.52(2)] in the left hand side becomes ∞ m,n=0 Now replacing l by l − m, p by p − n and using the lemma [16, p.100(1)] in the left hand side of (3.6), we get Finally on equating the coefficients of the like powers of t and u in the above equation, we get the required result.
Remark 3.2.Replacing z by z + y in (3.8), we obtain Theorem 3.2.The following summation formula for Laguerre-based poly-Genocchi polynomials H G holds true: Proof.Using (2.1), we can be written as Now replacing n by n − j and comparing the coefficients of t n in both sides, we obtain (3.10).holds true: Proof.From (2.1) and (1.10), we have holds true: n−m (x, y). (3.12) Proof.Using definition (2.1), we have Finally, equating the coefficients of the like powers of t n , we get (3.12).
Theorem 4.1.Let a, b > 0 and a = b, x, y ∈ R, n ≥ 0, then the following identity holds true: Since G(t) is symmetric in a and b and G(t) can written as Similarly, we can show that Comparing the coefficients of t n n! in (4.3) and (4.4), we arrive at the desired result.On comparing the coefficients of t n n! in (4.6) and (4.7), we arrive at the desired result (4.5).

Now replacing nTheorem 3 . 4 .
by n − m and comparing the coefficients of t n in both sides, we get (3.11).The following summation formula for Laguerre-based poly-Genocchi polynomials L G (k) n (x, y)