Expansions of the exponential and the logarithm of power series and applications

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.


Introduction
Throughout this paper, we understand an empty sum to be 0 and regard an empty product as 1.
Let us recall definitions for an asymptotic expansion.
Definition 1.1 [11, p. 31, Definition 2.1] Let F be a function of a real or complex variable z; let ∞ n=0 a n z n denote a convergent or divergent formal power series, of which the sum of the first n terms is denoted by S n (z); and let In other words, a k z k + R n (z) = S n (z) + R n (z), n ≥ 0, where we assume that when n = 0 we have F(z) = R 0 (z). Next, assume that for each n ≥ 0 the relation holds in some unbounded domain . Then, ∞ n=0 a n z n is called an asymptotic expansion of the function F(z) and we denote this by F(z) ∼ ∞ n=0 a n z n , z → ∞, z ∈ . We denote the fact that the series is the asymptotic expansion of f (z) by writing On the asymptotic expansion of the composite of a function and an asymptotic expansion, we recite the following four results.
and if |a 0 | < r, then the function also possesses an asymptotic representation, and this is again calculated exactly as if ∞ n=0 a n x n were convergent, where, since F(x) → a 0 as x → ∞, and since |a 0 | < r, the function (x) is obviously defined for every sufficiently large x.
In [6, p. 541], it was stated that, when taking g(z) = e z in Theorem 1.3, we obtain, without any restrictions, where α 0 = e a 0 and In [3], many useful conclusions on asymptotic expansions were obtained. We note that no accurate and explicit references were given in [9] to cite the formula (1.1) in Theorem 1.4. We observe that the constant term did not appear in (1.2) in Theorem 1.5 and that the formulas (1.3) and (1.4) are recurrences.
In Sect. 2 of this paper, we will add a constant term into (1.2) and acquire a modified version of Theorem 1.5. In Sect. 3, we will derive from the recurrences (1.3) and (1.4) explicit formulas for the above a n and b n . In Sect. 4, we will simply confirm explicit formulas for power series expansions of the exponential and the logarithm of a power series expansion. In Sect. 5, we will apply the explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion to find explicit formulas for the Bell numbers and logarithmic polynomials extensively studied in combinatorics and number theory.

A slightly modified version of Theorem 1.5
Now we are in a position to give a slightly modified version of Theorem 1.5. and First proof Differentiation yields which can be written as: Equating coefficients of 1 x k+1 results in (2.2). Taking the limit x → ∞ on both sides of E(x) = e D(x) , we arrive at (2.1). The proof of Theorem 2.1 is complete.
Then, by virtue of Theorem 1.5, we have where β 0 = 1 and This means that the Eq. (2.1) is valid and that e k = e d 0 β k for k ∈ N. Hence, the sequence e k for k ∈ N satisfies Consequently, the recurrence relation (2.2) follows. The proof of Theorem 2.1 is complete.

Explicit formulas for a n and e n
In this section, we derive explicit formulas for a n and e n from recurrence relations (1.4) and (2.2).
Proof From the recurrence relation (1.4) and by induction, it follows that and a n = c n c 0 The proof of Theorem 3.1 is complete.
Proof From the recurrence relation (2.2) and by induction, it follows that The proof of Theorem 3.2 is complete.

Expansions of the exponential and logarithm of power series
By similar arguments as in the proofs of Theorems 3.1 and 3.2, we can obtain the following power series expansions of the exponential and the logarithm of a power series expansion. For simplicity, we do not write down their proofs in details.  where the coefficients a n for n ∈ N satisfy (1.4) and (3.1).

Applications of Theorems 4.1 and 4.2
In this section, we will apply Theorems 4.1 and 4.2, respectively, to the Bell numbers and logarithmic polynomials which are extensively studied in combinatorics and number theory.

An application of Theorems 4.1 to the Bell numbers
In combinatorics, Bell numbers, usually denoted by B n for n ∈ {0} ∪ N, count the number of ways a set with n elements can be partitioned into disjoint and nonempty subsets. Every Bell number B n can be generated by

3)
and This is equivalent to for k ∈ N. From (2.1) and (3.2) in Theorem 4.1, it follows that B 0 = 0!e 0 = 0!e d 0 = 1 and This can be easily rewritten as (5.4). The required proof is complete.

An application of Theorem 4.2 to logarithmic polynomials
According to the monograph [2, pp. 140-141], the logarithmic polynomials L n can be defined by ln ∞ n=0 g n t n n! = ∞ n=1 L n t n n! , (5.5) where g 0 = G(a) = 1, g n = G (n) (a) for n ∈ N, and G(x) is an infinitely differentiable function at x = a, and they are expressions for the nth derivative of ln G(x) at the point x = a.

Theorem 5.2
For n ∈ N, the logarithmic polynomials L n can be computed by and Proof Applying Theorem 4.2 to the Eq. (5.5) gives c 0 = g 0 = 1, c n = g n n! , a n = L n n! , n ∈ N.
These two concrete results are the same as corresponding ones listed in (5.1).

Remark 6.4
Finding various expressions for a mathematical quantity is meaningful and significant in mathematics. For example, mathematicians find several infinite product expressions for π. From this point of view, the formulas (5.3) and (5.4) are all useful somewhat. Remark 6.5 In [2, pp. 140-141, Theorem A], it was given that the logarithmic polynomials L n equal L n = L n (g 1 , g 2 , . . . , g n ) = n k=1 (−1) k−1 (k − 1)!B n,k (g 1 , g 2 , . . . , g n−k+1 ), where B n,k denotes the Bell polynomials of the second kind, which are defined by for n ≥ k ≥ 0. This is different from the formulas (5.6) and (5.7). Remark 6.7 The Bell numbers B k are connected with the Lah numbers L(n, k) and the Stirling numbers of the second kind S(n, k). See, for example, the papers [5,7,8] and closely related references therein. Importantly, some inequalities for determinants and products of the Bell numbers B k were established in [8].
Remark 6.8 This paper is a revised version of the preprint [10].