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

TL;DR

This paper derives continued fraction expansions for generating functions of Bernoulli, Cauchy, Euler, and harmonic numbers, providing explicit convergents and exploring transformations, thus advancing understanding of their mathematical structure.

Contribution

It introduces new continued fraction expansions for various special number sequences and discusses their transformations, enriching the analytical tools for these functions.

Findings

Explicit continued fraction forms for Bernoulli, Cauchy, Euler, and harmonic numbers.

Derivation of convergents for these continued fractions.

Analysis of linear fractional transformations of the continued fractions.

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 expansion for different numbers and types, in particular, on Cauchy numbers.

Equations

$$f(x)=1-\frac{h_{1}x}{g_1+h_{1}x-\frac{g_1{h}_{2}x}{g_2+h_{2}x-\frac{g_2{h}_{3}x}{g_3+h_{3}x-\frac{g_3{h}_{4}x}{g_4+h_{4}x-\genfrac{}{}{0ex}{}{}{\ddots }}}}}.$$

$$P_n(x)$$

$$Q_n(x)$$

$$P_n=g_1\cdots g_{n}and{Q}_n=g_1\cdots g_n\sum _{j=0}^n\frac{h_1\cdots h_j}{g_1\cdots g_j}{x}^j,$$

$$Q_{n}f(x)\sim P_n(x)(n\to \infty )$$

$$\frac{P_n(x)}{Q_n(x)}\equiv f(x)(\mathrm{mod}{x}^{n+1})$$

$$f(x)=\underset{n\to \infty }{\lim }\frac{P_n}{Q_n}={\left(\sum _{j=0}^{\infty }\frac{h_1\cdots h_j}{g_1\cdots g_j}{x}^j\right)}^{-1}.$$

$$f(x):=\sum _{n=i}^{\infty }{f}_i{x}^i={\left(\sum _{j=0}^{\infty }\frac{h_1\cdots h_j}{g_1\cdots g_j}{x}^j\right)}^{-1}$$

$$\frac{z}{\arctan z}={\left(\sum _{i=0}^{\infty }\frac{(-1{)}^i{z}^{2i}}{2i+1}\right)}^{-1},$$

$$P_m(x)=(2m+1)!!and{Q}_m(x)=(2m+1)!!\sum _{i=0}^m\frac{(-1{)}^i{z}^{2i}}{2i+1}$$

$$\frac{z}{\arctan z}=1+\frac{z^2}{3-z^2+\frac{(3z{)}^2}{5-3z^2+\frac{(5z{)}^2}{7-5z^2+\frac{(7z{)}^2}{9-7z^2+\genfrac{}{}{0ex}{}{}{\ddots }}}}}.$$

$$\arctan z=\frac{z}{1+\frac{z^2}{3-z^2+\frac{(3z{)}^2}{5-3z^2+\frac{(5z{)}^2}{7-5z^2+\frac{(7z{)}^2}{9-7z^2+\genfrac{}{}{0ex}{}{}{\ddots }}}}}}.$$

$$\arctan z=\frac{z}{1+\frac{z^2}{3+\frac{(2z{)}^2}{5+\frac{(3z{)}^2}{7+\frac{(4z{)}^2}{9+\genfrac{}{}{0ex}{}{}{\ddots }}}}}}$$

$$\sum _{n=1}^{\infty }{B}_n\frac{x^n}{n!}.$$

$${\left({}_2{F}_1(1,N-1/\lambda ;N+1;-\lambda x)\right)}^{-1}=\sum _{n=0}^{\infty }{\beta }_{N,n}(\lambda )\frac{x^n}{n!}$$

$${}_2{F}_1(a,b;c;z)=\sum _{n=0}^{\infty }\frac{(a{)}^{(n)}{b}^{(n)}}{(c{)}^{(n)}}\frac{z^n}{n!}$$

$${}_2{F}_1(1,N-1/\lambda ;N+1;-\lambda x)=\sum _{n=0}^{\infty }\frac{(1-N\lambda |\lambda {)}_n}{(N+n{)}_n}{x}^n,$$

$$P_m(x{)}^{\ast }=1and{Q}_m(x{)}^{\ast }=\sum _{n=0}^m\frac{(1-N\lambda |\lambda {)}_n}{(N+n{)}_n}{x}^n.$$

$$P_m(x)=(N+m{)}_{m}and{Q}_m(x)=(N+m{)}_m\sum _{n=0}^m\frac{(1-N\lambda |\lambda {)}_n}{(N+n{)}_n}{x}^n.$$

$$\frac{P_0(x)}{Q_0(x)}$$

$$\frac{P_2(x)}{Q_2(x)}$$

$$\displaystyle=1-\cfrac{(1-N\lambda)x}{(N+1)+(1-N\lambda)x-\cfrac{(N+1)\big{(}1-(N+1)\lambda\bigr{)}x}{(N+2)+\bigl{(}1-(N+1)\lambda\bigr{)}x}}$$

$$\displaystyle P_{n}(x)=\left(N+n+\bigl{(}1-(N+n-1)\lambda\bigr{)}x\right)P_{n-1}(x)$$

$$\displaystyle\qquad\qquad\qquad\qquad\qquad-(N+n-1)\bigl{(}1-(N+n-1)\lambda\bigr{)}xP_{n-2}(x)\,,$$

$$\displaystyle Q_{n}(x)=\left(N+n+\bigl{(}1-(N+n-1)\lambda\bigr{)}x\right)Q_{n-1}(x)$$

$$\displaystyle\qquad\qquad\qquad\qquad\qquad-(N+n-1)\bigl{(}1-(N+n-1)\lambda\bigr{)}xQ_{n-2}(x)\,.$$

$$\sum _{n=0}^{\infty }{\beta }_{N,n}(\lambda )\frac{x^n}{n!}$$

$$\displaystyle=1-\cfrac{(1-N\lambda)x}{N+1+(1-N\lambda)x-\cfrac{(N+1)\bigl{(}1-(N+1)\lambda\bigr{)}x}{N+2+\bigl{(}1-(N+1)\lambda\bigr{)}x-\cfrac{(N+2)\bigl{(}1-(N+2)\lambda\bigr{)}x}{N+3+\bigl{(}1-(N+2)\lambda\bigr{)}x-{\atop\ddots}}}}\,.$$

$$\sum _{n=0}^{\infty }{B}_{N,n}\frac{x^n}{n!}=1-\frac{x}{N+1+x-\frac{(N+1)x}{N+2+x-\frac{(N+2)x}{N+3+x-\genfrac{}{}{0ex}{}{}{\ddots }}}}.$$

$$\sum _{n=0}^{\infty }{B}_n\frac{x^n}{n!}=1-\frac{x}{2+x-\frac{2x}{3+x-\frac{3x}{4+x-\frac{4x}{5+x-\genfrac{}{}{0ex}{}{}{\ddots }}}}}.$$

$$\frac{P_4(x)}{Q_4(x)}$$

$$\frac{P_5(x)}{Q_5(x)}$$

$$\frac{P_6(x)}{Q_6(x)}$$

$$\sum _{n=0}^{\infty }{B}_n\frac{x^n}{n!}=1-\frac{x}{2}+\frac{x^2}{12}-\frac{x^4}{720}+\frac{x^6}{30240}-\frac{x^8}{1209600}+\cdots$$

$$e^{-x}=1-\frac{x}{1+x-\frac{x}{2+x-\frac{2x}{3+x-\frac{3x}{4+x-\genfrac{}{}{0ex}{}{}{\ddots }}}}}.$$

Full text

Keywords: continued fractions, convergents, Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, hypergeometric function

1 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 (see, e.g., [8, 16, 19]). Lando [14] discuss other similar continued fractions from the aspect of Combinatorics and generating functions. Loya [15, Chapter 8] exhibits intriguing ideas with many useful examples of continued fraction expansions of analytic functions and famous numbers. However, it does not seem that explicit forms of the convergents of these generalized continued fraction expansions have been studied in detail.

Assume that $f(x)$ is expanded by the Taylor expansion as $f(x)=\sum_{i=0}^{\infty}f_{i}x^{i}$ for complex numbers $f_{i}$ ($i=0,1,2,\dots$). In this paper, we focus on the following continued fraction expansion, which is related to $T$-fractions.

[TABLE]

(Each concrete case needs some modifications.) Since $a_{n}(x)=g_{n}+h_{n}x$ and $b_{n}(x)=-g_{n-1}h_{n}x$ ($n\geq 1$) with $g_{0}=1$ and $a_{0}(x)=1$, the convergents $P_{n}(x)/Q_{n}(x)$ of the continued fraction expansion of (1) satisfy the recurrence relations

[TABLE]

with $P_{-1}(x)=1$, $Q_{-1}(x)=0$, $P_{0}(x)=Q_{0}(x)=1$. By using the induction on $n$, we can know the explicit forms of $P$’s and $Q$’s as

[TABLE]

respectively. For convenience, $g_{1}\cdots g_{j}=h_{1}\cdots h_{j}=1$ when $j=0$. By the approximation property,

[TABLE]

or as polynomials

[TABLE]

Namely,

[TABLE]

In other words, the analytic function

[TABLE]

can be expanded as a continued fraction of type (1). In addition, Taylor expansion of $P_{n}/Q_{n}$ matches the terms of $f(x)$, up to $x^{n}$.

Example. Since

[TABLE]

the convergents $P_{m}(x)/Q_{m}(x)$ ($m\geq 0$) with

[TABLE]

yield the continued fraction expansion

[TABLE]

In other words, we put $g_{n}=2n+1$ ($n\geq 0$) and $h_{n}=-g_{n-1}=-(2n-1)$ ($n\geq 1$) and $x=z^{2}$ in (1). Thus, we get a famous generalized continued fraction of $\arctan z$, which is believed to be due to Leonhard Euler:

[TABLE]

In fact, the continued fraction expansion

[TABLE]

is more well-known ([8, (6.1.14)],[19, (90.3)]), and is due to C. F. Gauss. However, the convergents of this continued fraction is more complicated. This is another reason why we do not consider different types of continued fraction expansions other than (1) in this paper.

2 Bernoulli numbers

Bernoulli numbers $B_{n}$ are defined by

[TABLE]

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 $\beta_{N,n}(\lambda)$ are defined by

[TABLE]

where

[TABLE]

is the Gauss hypergeometric function with the rising factorial $(x)^{(n)}=x(x+1)\cdots(x+n-1)$ ($n\geq 1$) and $(x)^{(0)}=1$. Denote the generalized falling factorial by (x|r)_{n}=x(x-r)\cdots\bigl{(}x-(n-1)r\bigr{)} ($n\geq 1$) with $(x|r)_{0}=1$. $(x)_{n}=(x|1)_{n}$ is the original falling factorial. Since

[TABLE]

we consider the convergents $P_{m}(x)^{\ast}/Q_{m}(x)^{\ast}$ with

[TABLE]

In order to treat with integer coefficients for $Q$’s, we can also consider the convergents $P_{m}(x)/Q_{m}(x)$, where

[TABLE]

Then

[TABLE]

and $P_{n}(x)$ and $Q_{n}(x)$ ($n\geq 2$) satisfy the recurrence relations

[TABLE]

By the recurrence relations, we know that a_{n}(x)=N+n+\bigl{(}1-(N+n-1)\lambda\bigr{)}x ($n\geq 1$) and b_{n}(x)=-(N+n-1)\bigl{(}1-(N+n-1)\lambda\bigr{)}x ($n\geq 2$) with $a_{0}(x)=1$ and $b_{1}(x)=-(1-N\lambda)x$. Therefore, we have the following continued fraction expansion of the generating function of the hypergeometric degenerate Bernoulli numbers.

Theorem 1**.**

[TABLE]

When $\lambda\to 0$ in Theorem 1, we get a continued fraction expansion of the generating function of hypergeometric Bernoulli numbers.

Corollary 1**.**

[TABLE]

When $\lambda\to 0$ and $N=1$ in Theorem 1, we get a continued fraction expansion of the generating function of the original Bernoulli numbers defined in (4).

Corollary 2**.**

[TABLE]

We shall see the approximation property concerning Corollary 2. For example, for $n=4,5,6$ the convergents have Taylor expansions as

[TABLE]

We see that

[TABLE]

Thus, as expected, the coefficients of the Taylor expansion of $P_{n}(x)/Q_{n}(x)$ match those of the generating function of Bernoulli numbers up to the term of $x^{n}$.

Finally, when $\lambda\to 0$ and $N=0$ in Theorem 1, we get a continued fraction expansion of $e^{-x}$.

Corollary 3**.**

[TABLE]

3 Cauchy numbers

Cauchy numbers are defined by

[TABLE]

In [4], new identities are found by using the continued fraction

[TABLE]

(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 $\gamma_{N,n}(\lambda)$ by the generating function

[TABLE]

When $\lambda\to 0$ in (7), $c_{N,n}=\gamma_{N,n}(0)$ are the hypergeometric Cauchy numbers ([11]). When $N=1$ in (7), $\gamma_{n}(\lambda)=\gamma_{1,n}(\lambda)$ are the degenerate Cauchy numbers (see, e.g., [3, (2.13)]). When $\lambda\to 0$ and $N=1$ in (7), $c_{n}=\gamma_{1,n}(0)$ are the classical Cauchy numbers in (6). $b_{n}=c_{n}/n!$ are often called Bernoulli numbers of the second kind.

Since

[TABLE]

we consider the convergents $P_{m}(x)/Q_{m}(x)$ as

[TABLE]

Then, we have

[TABLE]

and the convergents $P_{n}(x)/Q_{n}(x)$ ($n\geq 2$) satisfy the recurrence relations

[TABLE]

Since $a_{n}(x)=N+n+(\lambda-N-n+1)x$ ($n\geq 1$) and $b_{n}(x)=-(N+n-1)(\lambda-N-n+1)x$ ($n\geq 2$) with $a_{0}(x)=1$ and $b_{1}(x)=-(\lambda-N)x$, we have the following continued fraction expansion of the generating function of hypergeometric degenerate Cauchy numbers.

Theorem 2**.**

[TABLE]

When $\lambda\to 0$ in Theorem 2, we have a continued fraction expansion of hypergeometric Cauchy numbers.

Corollary 4**.**

[TABLE]

When $\lambda\to 0$ and $N=1$ in Theorem 2, we have a continued fraction expansion of the original Cauchy numbers.

[TABLE]

([15, Chapter 8]).

4 Euler numbers

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Hypergeometric Euler numbers $E_{N,n}$ ([13]) are defined by

[TABLE]

where ${}_{1}F_{2}(a;b,c;z)$ is the hypergeometric function defined by

[TABLE]

When $N=0$, then $E_{n}=E_{0,n}$ are classical Euler numbers, defined by

[TABLE]

Since by (9)

[TABLE]

consider the convergents $P_{m}(x)/Q_{m}(x)$ with

[TABLE]

Now,

[TABLE]

and $P_{n}(x)/Q_{n}(x)$ ($n\geq 2$) satisfy the recurrence relations

[TABLE]

Since $a_{n}(x)=(2N+2n-1)(2N+2n)+x^{2}$ ($n\geq 1$) and $b_{n}(x)=-(2N+2n-3)(2N+2n-2)x^{2}$ ($n\geq 2$) with $a_{0}(x)=1$ and $b_{1}(x)=-x^{2}$, we have the following continued fraction expansion of hypergeometric Euler numbers.

Theorem 3**.**

[TABLE]

When $N=0$ in Theorem 3, we get a continued fraction expansion of the classical Euler numbers.

Corollary 5**.**

[TABLE]

Hypergeometric Euler numbers of the second kind $\widehat{E}_{N,n}$ ([12, 13]) are defined by

[TABLE]

When $N=0$, $\widehat{E}_{n}=\widehat{E}_{0,n}$ are Euler numbers of the second kind or complementary Euler numbers, defined by

[TABLE]

([12]). Similarly, we have the following continued fraction expansions.

Theorem 4**.**

[TABLE]

Corollary 6**.**

[TABLE]

5 Harmonic numbers

There are several generalizations of harmonic numbers $H_{n}$, defined by

[TABLE]

In [20], the generalized harmonic numbers of order $m$ ($m\geq 1$) are defined by

[TABLE]

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+2r)\cdots\bigl{(}x+(n-1)r\bigr{)} ($n\geq 1$) be the generalized rising factorial with $(x|r)^{(0)}=1$. When $r=1$, $(x)^{(n)}=(x|1)^{(n)}$ is the original rising factorial. From the definition in (10), we have

[TABLE]

Thus,

[TABLE]

or

[TABLE]

yield that

[TABLE]

or

[TABLE]

Now,

[TABLE]

and $P_{n}(x)$ and $Q_{n}(x)$ ($n\geq 3$) satisfy the recurrence relations

[TABLE]

It is proved by induction. The right-hand side of $P$’s is equal to

[TABLE]

The relation for $Q$’s is trivially checked.

Since a_{n}(x)=\bigl{(}(n-1)a+b\bigr{)}^{m}+\bigl{(}(n-2)a+b\bigr{)}^{m}x ($n\geq 2$) and b_{n}(x)=-\bigl{(}(n-2)a+b\bigr{)}^{2m}x ($n\geq 3$) with $a_{0}(x)=0$, $a_{1}(x)=b^{m}(1-x)$, $b_{1}(x)=x$, $b_{2}(x)=-b^{2m}x(1-x)$, we have the following continued fraction expansion.

Theorem 5**.**

[TABLE]

When $a=b=1$ in Theorem 5, we have a continued fraction expansion concerning the $m$-order harmonic numbers.

Corollary 7**.**

[TABLE]

When $m=a=b=1$ in Theorem 5, we have a continued fraction expansion concerning the original harmonic numbers.

Corollary 8**.**

[TABLE]

6 Functions associated with the Riemann zeta function

Let $\mu(n)$ be Möbius function. From the property

[TABLE]

the Dirichlet series that generates the Möbius function is the multiplicative inverse of the Riemann zeta function:

[TABLE]

where $s$ is a complex number with real part larger than $1$. Then we consider the convergents $P_{m}(x)/Q_{m}(x)$ as

[TABLE]

Now,

[TABLE]

and $P_{n}(x)$ and $Q_{n}(x)$ ($n\geq 3$) satisfy the recurrence relations

[TABLE]

Since $a_{n}(x)=$ and $b_{n}(x)=$ with $a_{1}(x)=x$, $a_{2}(x)=2^{s}$, $b_{1}(x)=1$ and $b_{2}(x)=x^{2}$, we have the continued fraction expansion

[TABLE]

Notice that

[TABLE]

Putting $x=1$, we have a continued fraction expansion of the Dirichlet series of Möbius function.

Theorem 6**.**

[TABLE]

For example, for $s=7$

[TABLE]

but

[TABLE]

7 Transforms of continued fractions

Raney [18] established the method to yield the simple continued fraction expansions of $\displaystyle\beta(x)=\frac{ax+b}{cx+d}$ with $ad-bc\neq 0$ from the simple continued fraction expansions

[TABLE]

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 $\beta(x)$ 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

[TABLE]

Since

[TABLE]

and the convergents $\tilde{P}_{n}(x)/{\tilde{Q}}_{n}(x)$ ($n\geq 3$) satisfy the recurrence relations

[TABLE]

Since $a_{n}(x)=g_{n}+h_{n}x$ ($n\geq 2$) and $b_{n}(x)=-g_{n-1}h_{n}x$ ($n\geq 3$) with $a_{0}(x)=(a+b)/(c+d)$, $a_{1}(x)=(c+d)g_{1}+dh_{1}x$, $b_{1}(x)=-(ad-bc)/(c+d)h_{1}x$ and $b_{2}(x)=-(c+d)g_{1}h_{2}x$, we have the following continued fraction expansion.

Theorem 7**.**

If $f(x)$ has a continued fraction expansion (1), we have

[TABLE]

Examples. Applying Theorem 7 to the continued fraction expansions in Corollary 2, Corollary 3 and (8), we obtain the following expansions. When $a=-1$ and $b=c=d=1$, we get

[TABLE]

and

[TABLE]

When $b=-1$ and $a=c=d=1$, we get

[TABLE]

7.1 Linear fractional transformation

Theorem 7 can be proved by tools by Raney to manipulate expansions. This method is due to Hurwitz (see, e.g., [7]), Frame [5], and Kolden [10], independently, but popularized by van der Poorten (see, e.g., [17]). For simplicity, put $P_{n}=P_{n}(x)$, $Q_{n}=Q_{n}(x)$, $a_{n}=a_{n}(x)$ and $b_{n}=b_{n}(x)$. Since the recurrence relation can be explained by matrices as

[TABLE]

the linear fractional transformation $\beta(x)=(ax+b)/(cx+d)$ implies the multiplication of the matrix

[TABLE]

from the left. Namely, if

[TABLE]

then

[TABLE]

Now, we see that

[TABLE]

Therefore, we obtain the continued fraction expansion in Theorem 7.

There are different types of transformations. In [2], the $r$-th binomial transform $b_{n}=\sum_{k=0}^{n}r^{n-k}\binom{n}{k}a_{k}$ and some more variations are performed in the sense of Riordan arrays $(1/(1-rx),x/(1-rx))$ and so on.

8 Continued fractions of the ordinary generating functions

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

[TABLE]

([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**.**

[TABLE]

9 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 $N\geq 1$,

[TABLE]

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

[TABLE]

(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 of the continued fraction expansion in Theorem 9 is more complicated than that in Corollary 4. In addition, the corresponding Euler continued fractions in Theorem 3 and Theorem 4 have not been found yet. The details will be discussed in the following papers.