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Some Properties Involving 2-Variable Modified Partially Degenerate Hermite Polynomials Derived from Differential Equations and Distribution of Their Zeros
... The differential equations derived from the generating functions of special numbers and polynomials have been studied by many mathematicians; see [11][12][13][14][15][16][17][18][19][20][21]. ...
... By using these differential equations, we can obtain the explicit identities for these polynomials. Many authors studied differential equations derived in the generating functions of special polynomials in order to derive explicit identities for special polynomials, see [11][12][13][14][15][16][17][18][19][20]. ...
Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers.
On the basis of convolution neural network, deep learning algorithm can make the convolution layer convolute the input image to complete the hierarchical expression of feature information, which makes pattern recognition more simple and accurate. Now, in the theory of multimodal discourse analysis, the nonverbal features in communication are studied as a symbol system similar to language. In this paper, the author analyzes the deep learning complexity and multimodal target recognition application in English education system. Multimodal teaching gradually has its practical significance in the process of rich teaching resources. The large-scale application of multimedia technology in college English classroom is conducive to the construction of a real language environment. The simulation results show that the multi-layer and one-dimensional convolution structure of the product neural network can effectively complete many natural language problems, including the tagging of lexical and semantic roles, and thus effectively improve the accuracy of natural language processing. Multimodal teaching mode helps to memorize vocabulary images more deeply. 84% of students think that multi-modal teaching mode is closer to life. Meanwhile, multimedia teaching display is more acceptable. College English teachers should renew their teaching concepts and adapt themselves to the new teaching mode.
In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.
In this paper, we study differential equations arising from the generating function of the ( r , β ) -Bell polynomials. We give explicit identities for the ( r , β ) -Bell polynomials. Finally, we find the zeros of the ( r , β ) -Bell equations with numerical experiments.
In this paper, we study differential equations arising from the generating functions of Hermit Kamp e ´ de F e ´ riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp e ´ de F e ´ riet polynomials. Finally, use the computer to view the location of the zeros of Hermite Kamp e ´ de F e ´ riet polynomials.
In this paper, we study differential equations arising from the generating functions of the generalized Bell polynomials.We give explicit identities for the generalized Bell polynomials. Finally, we investigate the zeros of the generalized Bell polynomials by using numerical simulations.
In this paper, we study differential equations arising from the generating functions of the Bell-Carlitz polynomials. We give explicit identities for the Bell-Carlitz polynomials. Finally, we investigate the zeros of the Bell-Carlitz polynomials by using computer.
In this paper, we study differential equations arising from the generating functions of tangent numbers. We give explicit identities for the tangent numbers.
In this paper, we consider the degenerate tangent numbers and polynomials. We also obtain some explicit formulas for degenerate tangent numbers and polynomials.
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We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.