Are $\cosh(1), \sinh(1), $ and $e^{1/r}-1 $ the only known constants with regular Engel expansions?

Question

Background

If we define the Engel expansion of a number $x$ as the unique increasing sequence $x_{E}:= \{a_{1}, a_{2}, a_{3}, \dots \} $ such that $$x = \frac{1}{a_{1}} + \frac{1}{a_{1}a_{2}} + \frac{1}{a_{1} a_{2} a_{3}} + \dots .$$ We have $$\sinh(1)_{E} = \lim_{n \to \infty} \{ 1, 6, 20, 42, 72, \dots , 2(n-1) (2n-1) \} ,$$ and $$\cosh(1)_{E} = \lim_{n \to \infty} \{1, 2, 12, 30, 56 ,\dots, 2(n-1)(2n-3) \} ,$$ and finally $$(e^{1/r}-1)_{E} := \lim_{n \to \infty} \{1r, 2r, 3r, 4r, 5r, \dots, nr \} .$$ The Engel expansions of these constants display regularity in the sense that they have a nice pattern that can easily be described with a formula. I've found these on MathWorld, the wiki article, and on OEIS A118239. Besides these constants, I have not encountered any well-known constants with regular Engel expansions.

Questions

1. Are these the only well-known constants that have regular Engel expansions?
2. If not: which others are there, and what are their expansions? Pointers to relevant articles are appreciated.

Answer 1

One other case is Liouville's constant $L = 0.110001000000 000000 000001\dots$ whose decimal expansion has a $1$ in the $k!^{\text{th}}$ digit for every $k\ge 1$, and $0$'s everywhere else. This expression $$ L = \frac1{10^{1!}} + \frac1{10^{2!}} + \frac1{10^{3!}} + \frac1{10^{4!}} + \dots $$ gives us an Engel expansion of $$L_E = \lim_{n \to \infty} \{10^{1!}, 10^{2!-1!}, 10^{3!-2!}, 10^{4!-3!}, \dots, 10^{n!-(n-1)!}\}.$$

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The modified Bessel function of the first kind $I_\alpha(x)$ is given by the sum $$I_\alpha(x) = \sum_{m=0}^\infty \frac1{m! \Gamma(m+\alpha+1)} \left(\frac x2\right)^{2m+\alpha}$$ which lends itself to a nice Engel expansion when $\alpha$ is an integer and $x=2$. For example, $$ I_1(2) = \sum_{m=0}^\infty \frac1{m!(m+1)!} $$ which gives us an Engel expansion of $$ (I_1(2))_E = \lim_{n \to \infty} \{1, 2, 6, 12, \dots, n(n+1)\}. $$ As with the Engel expansions of $e^{1/r}$ or $\sinh(\frac1r)$ or $\cosh(\frac1r)$, we can also generalize this to other values of $x$: $$ (I_1(\tfrac2r))_E = \lim_{n \to \infty} \{r, 2r^2, 6r^2, 12r^2, \dots, n(n+1)r^2\}. $$ We get similar results for other values of $\alpha$; $I_0(2)$ and $I_0(\frac2r)$ are particularly straightforward.

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The sequence A001566 given by $a_1 = 3$ and $a_{n+1} = a_n^2 - 2$ for $n \ge 1$ is the Engel expansion of $\frac{3-\sqrt5}{2}$, the reciprocal of the square of the golden ratio.

Some other initial conditions give other constants with closed forms. Starting with $a_1 = 4$, for example, gives A003010, which is the Engel expansion of $2 - \sqrt3$. In general, starting with $a_1 = p$ gives us the Engel expansion of $\frac{p - \sqrt{p^2-4}}{2}$, the number with continued fraction expansion $[0;p-1,1,p-2,1,p-2,1,p-2,\dots]$. The sequence starting with $p$ has closed form $a_n = 2 \cos (2^{n-1} \arccos \frac p2)$, so we may write $$ \left(\frac{p - \sqrt{p^2-4}}{2}\right)_E = \lim_{n \to \infty} \left\{p, p^2-2, \dots, 2 \cos \left(2^{n-1} \arccos \frac p2\right)\right\}. $$

For more details, see Liardet and Stambul, Séries de Engel et fractions continuées (in French).

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Finally, it almost feels like a joke to say that many special cases of hypergeometric functions have Engel expansions, because that notation is so much more general. Following the notation in Graham et al.'s Concrete Mathematics, we have $$ F\left({1 \atop b_1,\dots,b_n}\,\middle|\,z\right) = \sum_{k \ge 0} \frac{1}{b_1^{\overline k} \cdots b_n^{\overline k}} z^k $$ where $b^{\overline k}$ is the rising power $b(b+1)\cdots (b+k-1)$. This is already an Engel expansion if $\frac1z = r$ is a positive integer, in which case $F\left({1 \atop b_1,\dots,b_n}\,\middle|\,\frac1r\right)_E$ is given by $$ \lim_{k \to \infty} \{1, b_1 \cdots b_n r, (b_1+1)\cdots(b_m+1)r, \dots, (b_1+k)\cdots (b_n+k)r\}. $$ In particular, special cases of this hypergeometric function that give us examples we've already seen are:

• $F\left({1 \atop 1,1}\middle|\frac1r\right) = e^{1/r}$,
• $F\left({1 \atop 1, \frac32}\middle|\frac1r\right) =\frac{\sqrt r}{2} \sinh \frac2{\sqrt r}$,
• $F\left({1 \atop 1,\frac12}\middle|\frac1r\right) = \cosh \frac2{\sqrt r}$, and
• $F\left({1 \atop 1,1+\alpha}\middle|\frac1r\right) = \alpha! r^{\alpha/2} I_\alpha(\frac2{\sqrt r})$.

(Not all hypergeometric functions have the form above; in general, $F\left({a_1, \dots, a_m \atop b_1, \dots, b_n} \middle| z\right)$ has rising powers in its numerator as well as its denominator, and doesn't have an integer Engel expansion.)

Answer 2

From the Oeis select other examples are:

• A059193, $\frac{1}{e}$
• A068379, $\sinh\left(\frac{1}{2}\right)$
• A068380, $\sinh\left(\frac{1}{3}\right)$
• A028310, $e$
• A008486, $\sqrt[3]{e}$
• A086570, $1 + \frac{\sqrt{2 \pi} \, e^{1/8}}{5} \, \text{erf}\left(\frac{1}{2 \, \sqrt{2}}\right)$
• A000330, $\frac{1}{1^2} + \frac{1}{1^2} \cdot \frac{1}{1^2 + 2^2} + \frac{1}{1^2} \cdot \frac{1}{1^2 + 2^2} \cdot \frac{1}{1^2 + 2^2 + 3^3} + \cdots$
• A253909, $\text{I}_{0}(2)$.

Other examples exist of this type. Then there are classes of values that have recurrence definitions like A220338 or A137507. The Oeis has a good selection of Engel and Pierce expansions but could always use more.