Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations

Abstract

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.

Some Identities Involving 2-Variable Modified Degenerate Hermite Polynomials Arising from Differential Equations and Distribution of Their Zeros

“…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])…”

“…The following basic properties of the 2-variable degenerate Hermite polynomials H n x, yjμ ðÞ are induced form (11). Therefore, it is enough to delete involved detail explanation.…”

“…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.…”

Some Properties Involving 2-Variable Modified Partially Degenerate Hermite Polynomials Derived from Differential Equations and Distribution of Their Zeros

“…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.…”