A note on degenerate Hermite-Fubini numbers and polynomials

In this paper, we introduce a new class of degenerate 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 degenerate Hermite-Fubini numbers and polynomials by using generating functions.

The Hermite polynomial H n (x,y) (see ( [12,13]) is defined by means of the following generating function as follows: Recently, Khan [7] introduced degenerate Hermite polynomials by means of the following generating function as follows: (1 + λt)
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 (1.8), we get n th Fubini number (ordered Bell number or geometric number) F n [2,5,6,15] 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]): (1.11)In (2016), Khan [7] introduced two variable degenerate Hermite-poly-Bernoulli polynomials is defined by means of the following generating function: so that The object of this paper, we consider generating functions for degenerate Hermite-Fubini numbers and polynomials and give some properties of these numbers and polynomials.We derive summation formulas of degenerate Hermite-Fubini numbers and polynomials and we construct a symmetric identities of degenerate Hermite-Fubini numbers and polynomials by using generating functions.

Degenerate Hermite-Fubini numbers and polynomials
In this section, we define three-variable degenerate Hermite-Fubini polynomials and obtain some basic properties which gives us new formula for H F n,λ (x, y; z) as follows: We introduce 3-variable degenerate Hermite-Fubini polynomials by means of the following generating function: When x = y = 0. z = 1 in (2.1), we have On setting y = 0 in (2.1), we obtain 2-variable Fubini polynomials which is defined by Kim et al. [9].
Theorem 2.1.For n ≥ 0, we have Proof.Using definition (2.1), we have Proof.Using definition (2.1), we have Equating the coefficients of t n n! in both sides, we get (2.4).
Theorem 2.3.For n ≥ 0, the following formula for degenerate Hermite-Fubini polynomials holds true: (2.5) Proof.We begin with the definition (2.1) and write Now, we observe that, by (2.6), we get Then, we have Comparing the coefficients of t n n! in equation (2.7) and (2.8), we get (2.5).
Theorem 2.4.For n ≥ 0, the following formula for degenerate Hermite-Fubini polynomials holds true: (2.9) Proof.We begin with the definition (2.1) and write Then using the definition of Kampé de Fériet generalization of the degenerate Hermite polynomials H n,λ (x, y) (1.3) and (2.1), we have Finally, comparing the coefficients of t n n! , we get (2.9).
Theorem 2.5.For n ≥ 0 and z 1 ̸ = z 2 , the following formula for degenerate 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 (2.10).Theorem 2.6.For n ≥ 0, the following formula for degenerate Hermite-Fubini polynomials holds true:

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Proof.From (2.1), we have Comparing the coefficients of t n n! on both sides, we obtain (2.11).
Remark 2.3.On setting x = y = 0 and x = −1 in Theorem 2.6, we find and Theorem 2.7.For n ≥ 0, p, q ∈ R, the following formula for degenerate Hermite-Fubini polynomials holds true: (2.14) Proof.Rewrite the generating function (2.1), we have Replacing k by k − 2j in above equation, we have Again replacing n by n − k in above equation, we have Finally, equating the coefficients of t n on both sides, we acquire the result (2.14). (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 degenerate Hermite-Fubini polynomials
First, we prove the following result involving the degenerate Hermite-Fubini polynomials H F n,λ (x, y; z) by using series rearrangement techniques and considered its special case: Theorem 3.1.The following summation formula for degenerate Hermite-Fubini polynomials H F n (x, y; z) holds true: Consider the product of the degenerate Hermite-Fubini polynomials, we can be written as generating function (2.1) in the following form: Replacing x by u, y by v, X by U and Y by V in (3.2) and equating the resultant to itself, which on using the generating function [14] in the r.h.s., becomes (3.3) Finally, replacing n by n − r and m by m − k and using the lemma [14] in the r.h.s. of the above equation and then equating the coefficients of like powers of t and T , we get assertion (3.1) of Theorem 3.1.
Theorem 3.2.The following summation formula for degenerate 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 (1.3) and rewrite the generating function as: On replacing n by n − r in above equation, we get Equating the coefficients of the like powers of t on both sides, we get (3.5).
Theorem 3.4.The following summation formula for degenerate Hermite-Fubini polynomials H F n,λ (x, y; z) holds true: Finally, equating the coefficients of the like powers of t on both sides, we get (3.6).

Symmetric identities for degenerate Hermite-Fubini polynomials
In this section, we establish general symmetry identities for the degenerate Hermite-Fubini polynomials H F n,λ (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.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 April 2019 doi:10.20944/preprints201904.0334.v1 Theorem 2.8.
For n ≥ 0, the following formula for degenerate Hermite-Fubini polynomials holds true: Now replacing n by n − s in l.h.s. and comparing the coefficients of t n on both sides, we get the result (3.4).