A new class of Hermite-Fubini polynomials and its properties

In this paper, we introduce a new class of Hermite-Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we derive symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions.

The Hermite polynomial H n (x,y) (see ([9, 10]) is defined by means of the following generating function as follows: Geometric polynomials (also known as Fubini polynomials) are defined as follows (see [2]): where is the Stirling number of the second kind (see [5]).
For x = 1 in (1.3), we get n th Fubini number (ordered Bell number or geometric number) F n [2,4,5,6,8,12] is defined by The exponential generating functions of geometric polynomials is given by (see [2]): and related to the geometric series (see [2]): Let us give a short list of these polynomials and numbers as follows: and Geometric and exponential polynomials are connected by the relation (see [2]): Recently, Pathan and Khan [9] introduced two variable Hermite-Bernoulli polynomials is defined by means of the following generating function: ( t e t − 1 On setting α = 1 in (1.7), the result reduces to known result of Dattoli et al. [3].
The manuscript of this paper as follows: In section 2, we consider generating functions for Hermite-Fubini numbers and polynomials and give some properties of these numbers and polynomials.In section 3, we derive summation formulas of Hermite-Fubini numbers and polynomials.In Section 4, we construct a symmetric identities of Hermite-Fubini numbers and polynomials by using generating functions.

A new class of Hermite-Fubini numbers and polynomials
In this section, we define three-variable Hermite-Fubini polynomials and obtain some basic properties which gives us new formula for H F n (x, y, z).Moreover, we shall consider the sum of products of two Hermite-Fubini polynomials.The sum of products of various polynomials and numbers with or without binomial coefficients have been studied by (see [2,4,5,6,8]): We introduce 3-variable Hermite-Fubini polynomials by means of the following generating function: It is easily seen from definition (2.1), we have For y = 0 in (2.1), we obtain 2-variable Fubini polynomials which is defined by Kargin [8].
When investigating the connection between Hermite polynomials H n (x, y) and Fubini polynomials F n (z) of importance is the following theorem.
Theorem 2.1.The following summation formula for Hermite-Fubini polynomials holds true: 3) where Proof.On separating the power series on r.h.s. of (2.1) in to their even and odd terms by using the elementary identity and then replacing t by it where i 2 = −1 and equating the real and imaginary parts in the resulting equation, we get the summation formulae (2.2) and (2.3).
Corollary 2.1.The following summation formula for Hermite-Fubini polynomials holds true: 2 Theorem 2.2.For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: Proof.Using definition (2.1), we have Comparing the coefficients of (2.8) Proof.We begin with the definition (2.1) and write Then using the definition of Kampé de Fériet generalization of the Hermite polynomials H n (x, y) and (2.1), we have Finally, comparing the coefficients of t n n! , we get (2.8).
Theorem 2.3.For n ≥ 0 and z 1 ̸ = z 2 , the following formula for Hermite-Fubini polynomials holds true: Proof.The products of (2.1) can be written as By equating the coefficients of t n n! on both sides, we get (2.9).
Theorem 2.4.For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: Comparing the coefficients of t n n! on both sides, we obtain (2.10).
Remark 2.3.On setting x = y = 0 and x = −1 in Theorem 2.4, we find and (2.12) Theorem 2.5.For n ≥ 0, p, q ∈ R, the following formula for Hermite-Fubini polynomials holds true: (2.13) Proof.Rewrite the generating function (2.1), we have Replacing k by k − 2j in above equation, we have Finally, equating the coefficients of t n on both sides, we acquire the result (2.13).
Theorem 2.6.For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: Proof.From (2.1), we have Replacing n by n − l in above equation, we get Comparing the coefficients of t n n! in both sides, we get (2.14).
Theorem 2.7.For n ≥ 0, the following formula for Hermite-Fubini polynomials holds true: (2.15) Proof.Replacing x by x + r in (2.1), we have Replacing n by n − l in above equation, we get Comparing the coefficients of t n n! in both sides, we get (2.15).

Summation Formulae for Hermite-Fubini polynomials
First, we prove the following result involving the Hermite-Fubini polynomials H F n (x, y; z) by using series rearrangement techniques and considered its special case: Theorem 3.1.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Proof.Replacing t by t + u in (2.1) and then using the formula [11,p.52(2)]: in the resultant equation, we find the following generating function for the Hermite-Fubini polynomials H F n (x, y; z): H F q+l (w, y; z) On expanding exponential function (3.4) gives H F q+l (w, y; z) which on using formula (3.2) in the first summation on the left hand side becomes H F q+l (w, y; z) Now replacing q by q − n, l by l − p and using the lemma ([11, p.100(1)]): in the l.h.s. of (3.6), we find Remark 3.2.Replacing w by w + x in (3.9), we obtain Theorem 3.2.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Proof.Consider the product of the Hermite-Fubini polynomials, we can be written as generating function (2.1) in the following form: T m m! .
(3.12) Replacing x by w, y by u, X by W and Y by U in (3.12) and equating the resultant to itself, which on using the generating function (3.7) in the exponential on the r.h.s., becomes Corollary 3.2.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Theorem 3.3.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Proof.We replace x by x + w and y by y + u in (2.1), use (1.2) and rewrite the generating function as: Equating the coefficients of the like powers of t on both sides, we get (3.19).
Theorem 3.6.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true: Proof.Using the generating function (2.1), we have Finally, equating the coefficients of the like powers of t on both sides, we get (3.21).

Symmetric identities
Recently, Khan [7], Pathan and Khan [9,10] have been introduced symmetric identities.In this section, we establish general symmetry identities for the generalized Hermite-Fubini polynomials H F n (x, y; z) by applying the generating function (2.1) and (2.2).Theorem 4.1.Let x, y, z ∈ R and n ≥ 0, then the following identity holds true:

Proof. Start with
Then the expression for A(t) is symmetric in a and b and we can expand A(t) into series in two ways to obtain: Similarly, we can show that (4.6)By comparing the coefficients of t n on the right hand sides of the last two equations, we arrive at the desired result.

. 8 ) 1 . 3 . 1 .Corollary 3 . 1 .
Finally, on equating the coefficients of the like powers of t and u in the above equation, we get the assertion (3.1) of Theorem 3.Remark Taking l = 0 in assertion (3.1) of Theorem 3.1, we deduce the following consequence of Theorem 3.1.The following summation formula for Hermite-Fubini polynomials H F n (x, y; z) holds true:

( 3 . 2 . 3 . 3 .
13) Finally, replacing n by n − r and m by m − k and using equation (3.7) in the r.h.s. of the above equation and then equating the coefficients of like powers of t and T , we get assertion (3.11) of Theorem 3.Remark Replacing u by y and U by Y in assertion (3.11) of Theorem 3.2, we deduce the the following consequence of Theorem 3.2.

30 April 2019 doi:10.20944/preprints201904.0333.v1
Replacing x by w in the above equation and equating the resultant equation to the above equation, we find exp .3) Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted:

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0333.v1
Now replacing n by n − s in l.h.s. and comparing the coefficients of t n on both sides, we get the result (3.15).Now replacing n by n − 2s in r.h.s. and comparing the coefficients of t on both sides, we arrive at the desired result(3.16).