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 ) and one other is Bernoulli polynomials (see , ). 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.  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.  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.  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.  considered poly-Bernoulli numbers and polynomials with a q parameter and developed some aritmetical and number theoretical properties. Dattoli et al.  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.  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.  studied on the Hermite-Kampé de Fériet based second kind Genocchi polynomials and presented some relatieonships of them. Ozarslan  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  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  . The prob- lem of determination of the basic generating function for simple Laguerre 2D and Hermite 2D polynomials was solved in - -. 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 , 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 . 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., , 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 (, Table 1), we can obtain the polynomials and the numbers given in -.
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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. With the help of this method McBride, Chongdar, Ghosh, Das and Chatterjea, Sultan, Majumder, Viswanathan 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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. 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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. 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.