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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.
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Context 1
... example, if n = 20, zeros of the (r, β)-Bell equations G n (x, r, β) = 0 is red. Our numerical results for approximate solutions of real zeros of the (r, β)-Bell equations G n (x, r, β) = 0 are displayed (Tables 1 and 2). Plot of real zeros of G n (x, r, β) = 0 for 1 ≤ n ≤ 20 structure are presented (Figure 4). ...Context 2
... us use the following notations. R G n (x,r,β) denotes the number of real zeros of G n (x, r, β) = 0 lying on the real plane Im(x) = 0 and C G n (x,r,β) denotes the number of complex zeros of G n (x, r, β) = 0. Since n is the degree of the polynomial G n (x, r, β), we have R G n (x,r,β) = n − C G n (x,r,β) (see Table 1). ...Context 3
... can see a good regular pattern of the complex roots of the (r, β)-Bell equations G n (x, r, β) = 0 for r > 0 and β > 0. Therefore, the following conjecture is possible. As a result of investigating more r > 0 and β > 0 variables, it is still unknown whether the conjecture 1 is true or false for all variables r > 0 and β > 0 (see Figure 1 and Table 1). ...Context 4
... how many zeros do G n (x, r, β) = 0 have? We are not able to decide if G n (x, r, β) = 0 has n distinct solutions (see Tables 1 and 2). We would like to know the number of complex zeros C G n (x,r,β) of G n (x, r, β) = 0, Im(x) = 0. Conjecture 3. Prove or disprove that G n (x, r, β) = 0 has n distinct solutions. ...Context 5
... a result of investigating more n variables, it is still unknown whether the conjecture is true or false for all variables n (see Tables 1 and 2). We expect that research in these directions will make a new approach using the numerical method related to the research of the (r, β)-Bell numbers and polynomials which appear in mathematics, applied mathematics, statistics, and mathematical physics. ...Similar publications
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Citations
... Motivated by their importance and potential for applications in certain problems in probability, combinatorics, number theory, differential equations, numerical analysis and other areas of mathematics and physics, several kinds of some special numbers and polynomials were recently studied by many authors (see [1,2,3,4,5,6,7]). Many mathematicians have studied in the area of the degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Genocchi polynomials, and degenerate tangent polynomials (see [ 6,7,8,9,10]). ...
... Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [9][10][11][12][13][14][15]). Based on the results so far, in the present work, a new class of 2-variable modified partially degenerate Hermite polynomials are constructed. ...
... We remember that the classical Stirling numbers of the first kind S 1 (n, k) and the second kind S 2 (n, k) are defined by the relations (see [6][7][8][9][10][11][12][13]) ...
... 1asλ approaches to 0, it is apparent that (10) descends to (7). Mathematicians have studied the differential equations arising from the generating functions of special numbers and polynomials (see [10][11][12][13][14]). Now, a new class of 2-variable modified degenerate Hermite polynomials are constructed based on the results so far. ...
... Since 1 þ μ ðÞ xt μ ! e xt as μ ! 0, it is clear that (11) reduces to (6). Observe that degenerate Hermite polynomials H n x, y, μ ðÞ and 2-variable modified degenerate Hermite polynomials H n x, yjμ ðÞ are totally different. ...
... We remember that the classical Stirling numbers of the first kind S 1 n, k ðÞ and the second kind S 2 n, k ðÞ are defined by the relations (see [6][7][8][9][10][11][12][13]) ...
In this chapter, we introduce the 2-variable modified degenerate Hermite polynomials and obtain some new symmetric identities for 2-variable modified degenerate Hermite polynomials. In order to give explicit identities for 2-variable modified degenerate Hermite polynomials, differential equations arising from the generating functions of 2-variable modified degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the 2-variable modified degenerate Hermite equations.
... Since log(1 + λ) λ → 1 as λ approaches 0, it is apparent that Equation (4) descends to Equation (2). The differential equations induced from the generating functions of special numbers and polynomials have been studied (see [10][11][12][13][14][15][16]). Now, a new class of two-variable partially degenerate Hermite polynomials is constructed based on the results so far. ...
... We remember that S 1 (n, k) and S 2 (n, k) have these relations(see [6][7][8][9][10][11][12]) ...
In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations.
Starting with a (didactic) discussion on the saddle surface of the job flexicurity (viewed as an indifference surface between job security and job flexibility), this chapter is focused on identifying and examining the set of (macro)indicators aimed at measuring the mark of job flexicurity, within a logical framework called quadrilateral rigidity–security–flexibility–plasticity. The basic theoretical and methodological innovation proposed here is the concept of positional job flexicurity, as opposed to the nominal one, more exactly, a vectorial-type of job flexicurity, where the components of the vector are normalized within the real range [0, 1]. Based on this proposal, four criteria are identified (origin, aim, trend and connectivity) based on which a set of 12 (macro)indicators of job flexicurity are build up. Next, logical and qualitative analyses are developed regarding the indicators of job flexicurity.
