Abstract

In the paper, the author notes several families of ordinary differential equations related to the generating function for the Bernoulli numbers of the second kind.

It is well known ^{1, 2, 3, 4} that the Bernoulli numbers of the second kind for can be generated by

This means that the reciprocal of the logarithmic function is related to the Bernoulli numbers of the second kind for

In [ ⁵, Theorem 2.1], it was obtained inductively and recursively that the nonlinear differential equations

(1)

have a solution where and

(2)

for

In [ ⁶, Lemma 2.1], it was obtained also inductively and recursively that the family of differential equations

(3)

for has a solution where denotes the Stirling numbers of the second kind. In [ ⁶, Corollary 2.2], it was claimed that

(4)

In [ ⁷, Theorem 1], it was established still inductively and recursively that the family of differential equations

for has a solution

where and

(5)

for and

is called ⁸ the rising factorial.

The proofs of [ ⁷, Theorem 1], [ ⁵, Theorem 2.1], and [ ⁶, Lemma 2.1] are almost the same and have the same shortcoming: they are rather long and tedious. More importantly, the expressions (2) and (5) are difficult to be understood, to be remembered, and to be computed.

By virtue of the Faà di Bruno formula

for where the Bell polynomials of the second kind for are defined [ ⁹, p. 134, Theorem A] and [ ⁹, p. 139, Theorem C] by

in view of two identities

and

in [ ⁹, p. 135], where a and b are any complex numbers and stands for the Stirling numbers of the first kind, and in light of an inversion theorem [ ¹⁰, p. 171, Theorem 12.1] which reads that

(6)

two families of differential equations

(7)

and

(8)

for and were standardly established in [ ¹¹, Theorem 1], where the falling factorial of for is defined ⁸ by

Since taking in (8) derives

(9)

Comparing (9) with (1) results in the equality (4). Replacing t by in (9) yields the differential equation (3).

Letting r = 1 in (7) arrives at

(10)

for which is an inversion formula of (1). This can also be deduced from applying (6) to (1).

Replacing t by in (10) gives

for which is an inversion formula of (3). This can also be acquired by applying (6) to (3).

Remark. The motivations in the papers ^{4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} are same as the one in this paper. This paper is a slightly modified version of the preprint ³⁷.

Acknowledgements

The author thanks Professor D. V. Dolgy at Far Eastern Federal University and Kwangwoon University for his e-mailing the paper ⁴ to the author on 7 December 2017.

References