Abstract. In the paper, the authors consider the **generating** **functions** of the **Hermite** polyno- mials and their squares, present explicit formulas for higher order derivatives of the **generating** **functions** of the **Hermite** **polynomials** and their squares, which can be viewed as ordinary differen- tial equations or derivative **polynomials**, find differential equations that the **generating** **functions** of the **Hermite** **polynomials** and their squares satisfy, and derive explicit formulas and recurrence relations for the **Hermite** **polynomials** and their squares.

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Special **polynomials** and numbers possess a lot of importances in many …elds of mathematics, physics, engineering and other related disciplines including the topics such as di¤erential equations, mathematical analysis, functional analysis, mathematical physics, quantum mechanics and so on. One of the most consid- erable **polynomials** in the theory of special **polynomials** is the **Hermite**-Kampé de Fériet (or Gould-Hopper) **polynomials** (see [1]) and one other is Bernoulli **polynomials** (see [10], [16]). Nowadays, these type polyno- mials and their several generalizations have been studied and used by many mathematicians and physicsics, see [1-17] and references therein. Araci et al. [2] introduced a new concept of the Apostol **Hermite**-Genocchi **polynomials** by using the modi…ed Milne-Thomson’s **polynomials** and also derived several implicit sum- mation formulae and general symmetric identities arising from di¤erent analytical means and **generating** **functions** method. Bretti et al. [4] de…ned multidimensional extensions of the Bernoulli and Appell poly- nomials by using the **Hermite**-Kampé de Fériet **polynomials** and gave the di¤erential equations, satis…ng by the corresponding 2D **polynomials**, derived from exploiting the factorization method. Bayad et al. [3] considered poly-Bernoulli **polynomials** and numbers and proved a collection of extremely important and fundamental identities satis…ed by the poly-Bernoulli **polynomials** and numbers. Cenkci et al. [5] considered poly-Bernoulli numbers and **polynomials** with a q parameter and developed some aritmetical and number theoretical properties. Dattoli et al. [6] applied the method of **generating** function to introduce new forms of Bernoulli numbers and **polynomials**, which were exploited to derive further classes of partial sums involving generalized many index many variable **polynomials**. Khan et al. [7] introduce the **Hermite** poly-Bernoulli **polynomials** and numbers of the second kind and examined some of their applications in combinatorics, num- ber theory and other …elds of mathematics. Kurt et al. [8] studied on the **Hermite**-Kampé de Fériet based second kind Genocchi **polynomials** and presented some relatieonships of them. Ozarslan [11] introduced an uni…ed family of **Hermite**-based Apostol-Bernoulli, Euler and Genocchi **polynomials** and then, acquired some symmetry identities between these **polynomials** and the generalized sum of integer powers. Ozarslan also gave explicit closed-form formulae for this uni…ed family and proved a …nite series relation between this uni…cation and 3d-**Hermite** **polynomials**. Pathan [12] de…ned a new class of generalized **Hermite**-Bernoulli

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Different methods of derivation of **generating** **functions** are presented in the monographs [2] [26]. The prob- lem of determination of the basic **generating** function for simple Laguerre 2D and **Hermite** 2D **polynomials** was solved in [9]-[12] [18]-[20]. A more difficult problem is the determination of **generating** **functions** for products of two Laguerre 2D **polynomials** or of a Laguerre 2D and a **Hermite** 2D polynomial. In [12], we derived some special **generating** **functions** for products of two Laguerre 2D **polynomials**. The corresponding **generating** func- tions with general 2D matrices U as parameters in these **polynomials** are fairly complicated [12]. In present paper, we derive by an operational approach the **generating** **functions** for products of two special Laguerre 2D **polynomials**, for products of two **Hermite** 2D **polynomials** and for the mixed case of a product of a Laguerre 2D with a **Hermite** 2D polynomial (also called bilinear **generating** **functions**). This corresponds to the formula of Mehler (e.g., [1], 10.13 (22) and below in Section 3) which is the bilinear **generating** function for the product of two usual **Hermite** **polynomials**. We begin in next Section with a short representation of the analogical 1D case of **Hermite** **polynomials** and discuss in Section 3 their bilinear **generating** function and continue in Sections 4-7 with the corresponding derivations for the Laguerre 2D and **Hermite** 2D cases. In Section 8 we derive a summa- tion formula over Laguerre 2D **polynomials** which can be considered as intermediate step to the mentioned ge- nerating **functions** but possesses also its own importance in applications. Sections 9 and 10 are concerned with the further illumination of two factorizations of two different bilinear **generating** **functions**.

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Various properties and possible generalizations of the above **polynomials** will be discussed in the following sections, where we will also consider the possibility of developing an approach to the theory of a new form of Bessel-like **functions**, which can be developed from nonexponential **generating** **functions**.

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Such transformations may easily be written down in similar way for all other explicitly given **polynomials**. As a rule, such truncated series are not introduced as Special **functions** with a special function symbol and, clearly, problems of their convergence also do not exist. In contrast, the case of infinite series, in par- ticular, the Sumudu transformation of **generating** **functions** can make problems. We now consider the Sumudu transformation of the basic **Generating** func- tion for **Hermite** **polynomials**

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In this paper we introduce Humbert matrix **polynomials** of two variables. Some hypergeometric matrix representations of the Humbert matrix **polynomials** of two variables, the double **generating** matrix **functions** and expansions of the Humbert matrix **polynomials** of two variables in series of **Hermite** **polynomials** are given. Results of Gegenbauer matrix **polynomials** of two variables follow as particular cases of Humbert matrix **polynomials** of two variables.

respectively. It is therefore clear that by making use of relation (4.3) in some other **generating** **functions** ob- tained in Section 2, we may get a number of interesting results for the 2VHMaP of the second form H n ( x, y; A ) . In this article, **generating** relations involving the **Hermite** matrix **polynomials** are introduced by making use of operational identities for decoupling of exponential operators. The approach presented here can be explored further to derive the results for some other suitable families of special matrix **functions**.

The manuscript of this paper as follows: In section 2, we consider generat- ing **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**. 2. A new class of **Hermite**-Fubini numbers and **polynomials**

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Abstract. 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**.

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The manuscript of this paper as follows: In section 2, we consider **generating** **functions** for Laguerre-based **Hermite**-Fubini numbers and **polynomials** and give some properties of these numbers and **polynomials**. In section 3, we derive summation for- mulas of Laguerre-based **Hermite**-Fubini numbers and **polynomials**. In Section 4, we construct a symmetric identities of Laguerre-based **Hermite**-Fubini numbers and poly- nomials by using **generating** **functions**.

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Abstract. Here, in this work we present a generalization of the Weierstrass Approxima- tion Theorem for a general class of **polynomials**. Then we generalize it for two variable continuous function F (x, t) and prove that on a rectangle [a, b] × (−1, 1), a ≤ x ≤ b, |t|<1, a, b, t ∈ R , it uniformly converges into a **generating** function.As a result,we are able to apply our theorems to derive a number of **generating** **functions**.

M − k a b c α = M − k a b c α (2.2) The **generating** function in (2.1) gives many types of **polynomials** as special cases, for example, see Table 1. Remark 2.2. From NO. 13 in Table 1 and ([9], Table 1), we can obtain the **polynomials** and the numbers given in [12]-[16].

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Abstract. A uniﬁed presentation of a class of Humbert’s **polynomials** in two variables which generalizes the well known class of Gegenbauer, Humbert, Legendre, Cheby- cheﬀ, Pincherle, Horadam, Kinnsy, Horadam-Pethe, Djordjevi´ c , Gould, Milovanovi´ c and Djordjevi´ c, Pathan and Khan **polynomials** and many not so called ’named’ **polynomials** has inspired the present paper and the authors deﬁne here general- ized Humbert-**Hermite** **polynomials** of two variables. Several expansions of Humbert- **Hermite** **polynomials**, **Hermite**-Gegenbaurer (or ultraspherical) **polynomials** and **Hermite**- Chebyshev **polynomials** are proved.

Although, in the literature, one can ﬁnd extensive investigations related to the gener- ating **functions** for the Bernoulli, Euler and Genocchi numbers and **polynomials** and also their generalizations, the λ-Stirling numbers of the second kind, the array **polynomials** and the Eulerian **polynomials**, related to nonnegative real parameters, have not been studied yet. Therefore, this paper deal with new classes of **generating** function for generalized λ-Stirling type numbers of the second kind, generalized array type **polynomials** and gen- eralized Eulerian **polynomials**, respectively. By using these **generating** **functions**, we derive many functional equations and diﬀerential equations. By using these equations, we inves- tigate and introduce fundamental properties and many new identities for the generalized λ-Stirling type numbers of the second kind, the generalized array type **polynomials** and the generalized Eulerian type **polynomials** and numbers. We also derive multiplication formulas and recurrence relations for these numbers and **polynomials**. We derive many new identities related to these numbers and **polynomials**.

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dx + n(2λ + 3n)]u = 0. (1.2) Several **generating** **functions** for Gegenbauer **polynomials** have been derived by different method namely classical, theory of Lie groups etc. Here we are mainly interested in group theoretic method as intro- duced by L. Weisner[5]. With the help of this method McBride[4], Chongdar[2], Ghosh[3], Das and Chatterjea[7], Sultan[8], Majumder[9], Viswanathan[1] and others have derived a large number of gen- erating **functions** for Gegenbauer **polynomials**.

In conclusion, it is obvious that not only the theorems stated in the paper 1,2 but also their extensions can be easily derived by using the operator in the paper 4 , which is in the paper 1 . Furthermore, by using the theorems (1-3), we canimmediately generalize any known result of the form (2.1) or (2.5) from therelations (2.2) or (2.6). Thus a large number of **generating** relations can beeasily obtained by attributing different suitable values to in (2.1) or (2.5).

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is given. It is worth recalling that this method has produced for the Gegenbaur **polynomials**, six **generating** **functions** in general, one at a time by various other methods. These **generating** **functions**, in turn yield, the Legendre **polynomials** as special case for α = 1/2 . Many results obtained for Gegenbaur **polynomials** are known but some of them are believed to be new.

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[5]. P. Reddaiah, Deriving shape **functions** for 8-noded rectangular serendipity element in horizontal channel geometry and verified, International Journal of Mathematics Trends and Technology (IJMTT), Volume 50, Number 2 , October 2017.

In this paper, we investigated some properties and identities for degenerate Euler poly- nomials in connection with degenerate Bernstein **polynomials** and operators which were recently introduced as degenerate versions of the classical Bernstein **polynomials** and op- erators. This has been done by means of fermionic p-adic integrals on Z p and **generating**

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[7]. P. Reddaiah, Deriving shape **functions** for Hexahedral element by natural coordinate system and Verified, International Journal of Mathematics Trends and Technology (IJMTT), Volume 51, Number 6, November 2017.