Abstract
In the paper, by virtue of (1) the Stirling inversion theorem and the binomial inversion theorem, (2) the Faa di Bruno formula and two identities for the Bell polynomials of the second kind, (3) a formula of higher order derivative for the ratio of two differentiable functions, the authors (1) present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, (2) derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, (3) recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.
Highlights
In [5], it was pointed out that the unnamed polynomials hn(x) and the Bell polynomials Bn(x) are connected by the identity n n k
It was pointed out in [4, pp. 257–258] that the expression (1.2) had been applied in 1937 to the theory of hyperbolic differential equations. It was pointed out in [10] that there have been some studies on interesting applications of Bell polynomials Bk(x) in soliton theory, including links with bilinear and trilinear forms of nonlinear differential equations which possess soliton solutions
The identity (1.9) is obviously simpler than (1.5) in their forms
Summary
K=0 and the Stirling numbers of the second kind S(n, k) can be generated by (ex − 1)k ∞. 257–258] that the expression (1.2) had been applied in 1937 to the theory of hyperbolic differential equations It was pointed out in [10] that there have been some studies on interesting applications of Bell polynomials Bk(x) in soliton theory, including links with bilinear and trilinear forms of nonlinear differential equations which possess soliton solutions. Bell polynomials of the second kind B(n, k), and utilizing a higher order derivative formula for the ratio of two differentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for hn(x), derive two identities connecting hn(x) with Bn(x), and recover the identity (1.3).
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