Continued fraction expansions of the generating functions of Bernoulli and related numbers

doi:10.1016/j.indag.2020.06.006

Indagationes Mathematicae

Volume 31, Issue 4, July 2020, Pages 695-713

https://doi.org/10.1016/j.indag.2020.06.006 Get rights and content

Abstract

We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansions for different numbers and types, in particular, on Cauchy numbers.

Section snippets

Continued fractions

Given an analytic function $f (x)$ , several types of general continued fractions have been known and studied. J.H. Lambert expanded $log (1 - x)$ , $arctan x$ and $tan x$ in continued fractions in 1768. Some special types are known as $J$ -fractions, $C$ -fractions, $T$ -fractions, $M$ -fractions and Hankel continued fractions (see, e.g., [7], [9], [16], [19]). Lando [14] discusses other similar continued fractions from the aspect of Combinatorics and generating functions. Loya [15, Chapter 8] exhibits intriguing ideas

Bernoulli numbers

Bernoulli numbers $B_{n}$ are defined by $\sum_{n = 1}^{\infty} B_{n} \frac{x^{n}}{n!} .$ Many kinds of continued fraction expansions of the generating functions of Bernoulli numbers have been known and studied (see, e.g., [1, Appendix],[6]). However, those of generalized Bernoulli numbers seem to be few, though there exist several generalizations of the original Bernoulli numbers. In particular, direct generalizations from the continued fraction expansions seem to be hard.

Hypergeometric degenerate Bernoulli numbers $β_{N, n} (λ)$ are defined

Cauchy numbers

Cauchy numbers are defined by $\frac{x}{log (1 + x)} = \sum_{n = 0}^{\infty} c_{n} \frac{x^{n}}{n!} .$ Note that these numbers are also called Gregory coefficients, reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind. In [4], new identities are found by using the continued fraction

(see, e.g., [19, (90.1)]). In this section, we find a different type of continued fraction expansion related to Cauchy numbers.

Define the hypergeometric degenerate Cauchy numbers $γ_{N, n} (λ)$ by the generating

Euler numbers

Hypergeometric Euler numbers $E_{N, n}$ [13] are defined by $\frac{1}{_{1} F_{2} (1; N + 1, (2 N + 1) ∕ 2; x^{2} ∕ 4)} = \sum_{n = 0}^{\infty} E_{N, n} \frac{x^{n}}{n!},$ where $_{1} F_{2} (a; b, c; z)$ is the hypergeometric function defined by $_{1} F_{2} (a; b, c; z) = \sum_{n = 0}^{\infty} \frac{{(a)}^{(n)}}{{(b)}^{(n)} {(c)}^{(n)}} \frac{z^{n}}{n!} .$ When $N = 0$ , then $E_{n} = E_{0, n}$ are the classical Euler numbers, defined by $\frac{1}{cosh x} = \sum_{n = 0}^{\infty} E_{n} \frac{x^{n}}{n!} .$ Since by (9) $_{1} F_{2} (1; N + 1, (2 N + 1) ∕ 2; x^{2} ∕ 4) = \sum_{n = 0}^{\infty} \frac{(2 N)! x^{2 n}}{(2 N + 2 n)!},$ consider the convergents $P_{m} (x) ∕ Q_{m} (x)$ with $P_{m} (x) = \frac{(2 N + 2 m)!}{(2 N)!} and Q_{m} (x) = (2 N + 2 m)! \sum_{n = 0}^{m} \frac{x^{2 n}}{(2 N + 2 n)!} .$ Now, $\frac{P_{0} (x)}{Q_{0} (x)} = \frac{1}{1} = 1,$ $\frac{P_{1} (x)}{Q_{1} (x)} = \frac{(2 N + 1) (2 N + 2)}{(2 N + 1) (2 N + 2) + x^{2}} = 1 - x$

Harmonic numbers

There are several generalizations of harmonic numbers $H_{n}$ , defined by $H_{n} = \sum_{k = 1}^{n} \frac{1}{k} (n \geq 1) with H_{0} = 0 .$ In [20], the generalized harmonic numbers of order $m$ ( $m \geq 1$ ) are defined by $h_{n}^{(m)} (a, b) = \sum_{k = 1}^{n} \frac{1}{{((k - 1) a + b)}^{m}} (n \geq 1) with h_{0}^{(m)} (a, b) = 0,$ where $a$ and $b$ are positive real numbers. When $a = b = 1$ , $H_{n}^{(m)} = h_{n}^{(m)} (1, 1)$ are the $m$ -order harmonic numbers. When $m = a = b = 1$ , $H_{n} = h_{n}^{(1)} (1, 1)$ are the original harmonic numbers.

Let ${(x | r)}^{(n)} = x (x + r) (x + 2 r) \dots (x + (n - 1) r)$ ( $n \geq 1$ ) be the generalized rising factorial with ${(x | r)}^{(0)} = 1$ . When $r = 1$ , ${(x)}^{(n)} = (x$

Functions associated with the Riemann zeta function

Let $μ (n)$ be Möbius function. From the property $\sum_{d | n} μ (d) = \{\begin{matrix} 1 & if n = 1, \\ 0 & if n \geq 2 \end{matrix},$ the Dirichlet series that generates the Möbius function is the multiplicative inverse of the Riemann zeta function: $\sum_{n = 1}^{\infty} \frac{μ (n)}{n^{s}} = \frac{1}{ζ (s)},$ where $s$ is a complex number with real part larger than $1$ . Then we consider the convergents $P_{m} (x) ∕ Q_{m} (x)$ as $P_{m} (x) = {(m!)}^{s}, Q_{m} (x) = {(m!)}^{s} \sum_{k = 1}^{m} \frac{x^{k}}{k^{s}} (m \geq 1) .$ Now,

and

P_{n} (x)

and

Q_{n} (x)

(

n \geq 3

) satisfy the recurrence relations

P_{n} (x) = (n^{s} + {(n - 1)}^{s} x) P_{n - 1} (x) - {(n - 1)}^{2 s} x P_{n - 2} (x),

Q_{n} (x) = (n^{s} + {(n - 1)}^{s} x) Q_{n - 1} (x) - {(n - 1)}^{2 s} x Q_{n - 2} (x) .

Transforms of continued fractions

Raney [18] established a method to yield the simple continued fraction expansions of $β (x) = \frac{a x + b}{c x + d}$ with $a d - b c \neq 0$ from the simple continued fraction expansion

where

a_{0}

is an integer and

a_{1}, a_{2}, \dots

are positive integers. In this section, we shall consider the linear fractional transformation

β (x)

for the continued fraction expansion in (1). Since the convergents

P_{n} (x) ∕ Q_{n} (x)

(

n \geq 0

) are given in (2) in our case, we have

\frac{{\tilde{P}}_{n} (x)}{{\tilde{Q}}_{n} (x)} ≔ β (\frac{P_{n} (x)}{Q_{n} (x)}) = \frac{g_{1} \dots g_{n} ((a + b) + b \sum_{j = 1}^{n} \frac{h_{1} \dots h_{j}}{g_{1} \dots g_{j}} x^{j})}{g_{1} \dots g_{n} ((c + d) + d \sum_{j = 1}^{n} \frac{h_{1} \dots h_{j}}{g_{1} \dots g_{j}} x)}

Continued fractions of the ordinary generating functions

A continued fraction expansion of the ordinary generating function of Bernoulli numbers is given by

([1, A.5]). However, any beautiful continued fraction expansion for Cauchy numbers has not been known yet. Nevertheless, in order to satisfy the approximation property (3), we have the following expansion.

Theorem 8

Concluding remarks

In this paper, we deal with continued fraction in the aspects of convergents. Such techniques and ideas can be applied to more different types. For example, we can have a more complicated continued fraction expansion than that in Corollary 4.

Theorem 9

For $N \geq 1$ ,

When $N = 1$ , this is a direct generalization of the continued fraction expansion

(see, e.g., [19, (90.1)]).

Similarly, the $n$ th convergent and the generating function of $c_{n}$ coincide up to the $n$ th term in their Taylor expansions. However, the structure

Acknowledgment

The author thanks the anonymous referee for his/her detailed comments and suggestions, which improved the quality of the paper.

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Continued fraction expansions of the generating functions of Bernoulli and related numbers

Abstract

Section snippets

Continued fractions

Bernoulli numbers

Cauchy numbers

Euler numbers

Harmonic numbers

Functions associated with the Riemann zeta function

Transforms of continued fractions

Continued fractions of the ordinary generating functions

Concluding remarks

Acknowledgment

Discrete Math.

Adv. Math.

Results Math.

Continued fractions and transformations of integer sequences

J. Integer Seq.

Some identities of Cauchy numbers associated with continued fractions

Results Math.

Classroom notes: Continued fractions and matrices

Amer. Math. Monthly

The Hankel power sum matrix inverse and the Bernoulli continued fraction

Math. Comp.

Lectures on Number Theory