An Asymptotic Form of the Generating Function $\prod_{k=1}^\infty (1+x^k/k)$

TL;DR

This paper investigates the asymptotic behavior of the generating function product involving harmonic terms, showing the convergence of its coefficients to a positive limit expressed through the Riemann zeta function.

Contribution

It provides an explicit expression for the limit of the sequence generated by an infinite product involving harmonic terms, linking it to the Riemann zeta function.

Findings

The sequence of coefficients converges to a positive constant C.

The constant C is expressed as an exponential involving the Riemann zeta function.

The generating function's asymptotic form is characterized explicitly.

Abstract

It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function $\prod_{k=1}^\infty (1+x^k/k)$ converges to a limit $C > 0$. $C$ can be expressed as $C = \exp\Bigl(-\sum_{k = 2}^\infty \frac{(-1)^k}{k}\ \zeta(k) \Bigr)$ where $\zeta()$ denotes the Riemann zeta function.

Tables

Table 1. ζ ​ ( k ) → k → ∞ 1 → 𝑘 → 𝜁 𝑘 1 \zeta(k)\xrightarrow{k\to\infty}1

$k$	$\zeta \left(k\right)-1$
2	$\frac{{\pi }^2}{6}-1$	0.644934
3	-	0.202057
4	$\frac{{\pi }^4}{90}-1$	0.082323
5	-	0.036928
6	$\frac{{\pi }^6}{945}-1$	0.017343
7	-	0.008349
8	$\frac{{\pi }^8}{9450}-1$	0.004077
9	-	0.002008
10	$\frac{{\pi }^{10}}{93555}-1$	0.000995
11	-	0.000494

Table 2. Table 1: Approximation of C 𝐶 C

$m$	${\Delta }_m$	$\frac{2}{\exp \left(1+{\Delta }_m\right)}$
1	0.0	0.7357589
2	0.3224670	0.5329542
3	0.2551147	0.5700863
4	0.2756955	0.5584734
5	0.2683100	0.5626133
6	0.2712005	0.5609894
7	0.2700078	0.5616589
8	0.2705174	0.5613727
9	0.2702943	0.5614980
10	0.2703937	0.5614421
11	0.2703488	0.5614674
12	0.2703693	0.5614559
13	0.2703599	0.5614612

Table 3. Table 2: Upper left section of the q n ​ ( k ) subscript 𝑞 𝑛 𝑘 q_{n}(k) field [ 0 ≤ n ≤ 5 , 0 ≤ k ≤ 16 ] delimited-[] formulae-sequence 0 𝑛 5 0 𝑘 16 [\ 0\leq n\leq 5\ ,\ 0\leq k\leq 16\ ]

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	1	1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0
3	1	1	1	2	1	1	1	0	0	0	0	0	0	0	0	0	0
4	1	1	1	2	2	2	2	2	1	1	1	0	0	0	0	0	0
5	1	1	1	2	2	3	3	3	3	3	3	2	2	1	1	1	0

Table 4. Table 3: Upper left section of the r n ​ ( k ) subscript 𝑟 𝑛 𝑘 r_{n}(k) field [ 0 ≤ n ≤ 5 , 0 ≤ k ≤ 16 ] delimited-[] formulae-sequence 0 𝑛 5 0 𝑘 16 [\ 0\leq n\leq 5\ ,\ 0\leq k\leq 16\ ]

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	1	1	$\frac{1}{2}$	$\frac{1}{2}$	0	0	0	0	0	0	0	0	0	0	0	0	0
3	1	1	$\frac{1}{2}$	$\frac{5}{6}$	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{6}$	0	0	0	0	0	0	0	0	0	0
4	1	1	$\frac{1}{2}$	$\frac{5}{6}$	$\frac{7}{12}$	$\frac{5}{12}$	$\frac{7}{24}$	$\frac{5}{24}$	$\frac{1}{12}$	$\frac{1}{24}$	$\frac{1}{24}$	0	0	0	0	0	0
5	1	1	$\frac{1}{2}$	$\frac{5}{6}$	$\frac{7}{12}$	$\frac{37}{60}$	$\frac{59}{120}$	$\frac{37}{120}$	$\frac{1}{4}$	$\frac{19}{120}$	$\frac{1}{8}$	$\frac{7}{120}$	$\frac{1}{24}$	$\frac{1}{60}$	$\frac{1}{120}$	$\frac{1}{120}$	0

Equations

$$R(x)\ :=\ \prod\limits_{k=1}^{\infty}\Big{(}1+\frac{x^{k}}{k}\Big{)}\ =\ \sum\limits_{k=0}^{\infty}r(k)\ x^{k}$$

$$Q(x)\ :=\ \prod\limits_{k=1}^{\infty}\Big{(}1+x^{k}\Big{)}\ =\ \sum\limits_{k=0}^{\infty}q(k)\ x^{k}$$

$$\sum _{i\in S}i=k$$

$$\text{textsc}ip(S):=\frac{1}{\prod _{i\in S}i}$$

$$r(k)$$

$$\displaystyle\ln{R(x)}=\sum_{k\geq 1}\ln{\Big{(}1+\tfrac{x^{k}}{k}\Big{)}}$$

$$+\frac{x^2}{2}$$

$$+\frac{x^3}{3}$$

$$⋮$$

$$={\text{Li}}_1(x)$$

$$R(x)\stackrel{x\to 1-}{\to }\frac{C}{1-x}$$

$$C=\underset{x\to 1-}{\lim }(1-x)R(x)$$

$$\ln{C}\ =\ \lim_{x\to 1-}\big{(}\ln{(1-x)}+\ln{R(x)}\big{)}$$

$${\text{Li}}_1(x)=-\ln (1-x)$$

$$\displaystyle\ln{C}\$$

$$\displaystyle=\ -\ \tfrac{1}{2}\ \zeta(2)\$$

$$C=\exp{\Bigg{(}-\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}\ \zeta(k)\Bigg{)}}$$

$$\ln C$$

$$\displaystyle=\$$

$$\displaystyle-\ \tfrac{1}{2}\$$

$$\displaystyle=\$$

$$\Delta$$

$$C=\frac{2}{e^{1+\Delta }}$$

$${\Delta }_m$$

$$C_m=\frac{2}{\exp (1+{\Delta }_m)}:m\ge 1$$

$$\displaystyle R_{n}(x)\$$

$$\displaystyle Q_{n}(x)\$$

$$\displaystyle q_{n}(k)\$$

$$\displaystyle\sum_{k\geq 0}\ q_{n}(k)\$$

$$q_0(k)$$

$$q_n(k)$$

$$+q_1(k-2)+q_0(k-1):k\ge n>0$$

$$\displaystyle r_{n}(k)\$$

$$\displaystyle\sum_{k\geq 0}\ r_{n}(k)\$$

$$r_0(k)$$

Full text

The ordinary generating function (OGF) {ceqn}

[TABLE]

$Q(x)$ generates the sequence of counters for the number of integer partitions with distinct parts [Wil]. $q(k)$ is equal to the number of partitions of $k$ into distinct parts for each $k\geq 0$ [Int].

A partition with distinct parts of integer $k$ can be regarded as a finite set $S$ of (distinct) positive integers $i\geq 1$ whose sum equals $k$. Let $\mathcal{P}(k)$ denote the set of all such partitions of $k$ and let $S\in\mathcal{P}(k)$. We then have

[TABLE]

With each partition $S\in\mathcal{P}(k)$ we can associate the inverse of the product of its (distinct) elements

[TABLE]

With this in mind $r(k)$ can be written as

[TABLE]

In other words, $r(k)$ is equal to the sum over all partitions $S\in\mathcal{P}(k)$ of the reciprocal of the product of the elements of $S$ .

How does the sequence $r(k)$ given in (4a) and (4b) behave? Does it converge to some limit $C>0$? Taking the logarithm of (1a), applying the Mercator series expansion [Wol], and summing up columns first gives {fleqn}

[TABLE]

Here $\text{Li}_{s}(x)$ denotes the so-called polylogarithm [Wikb], a Dirichlet type series [Wik].

We are looking for an asymptotic relation of the form {ceqn}

[TABLE]

for some constant $C>0$. This is equivalent to the existence of the limit

[TABLE]

Taking the logarithm of (7) gives

[TABLE]

If we insert (5), observe the identity [Wikc]

[TABLE]

and finally set $x=1$, we arrive at the condition

[TABLE]

where $\zeta(s)$ denotes the Riemann Zeta function [Wikd]. We therefore have {ceqn}

[TABLE]

We observe that $\zeta(k)$ converges rapidly towards 1 [Wika]:

This motivates the decomposition of (10) {fleqn}

[TABLE]

where $\Delta$ is defined as

[TABLE]

We therefore have from (12) {ceqn}

[TABLE]

From (13) we derive the sequence of corrections $\Delta_{m}$ as follows {ceqn}

[TABLE]

This creates the sequence {ceqn}

[TABLE]

of approximations of $C$ whose first elements are listed in table 1.

Useful recurrence relations for computation

For $n>0$ we define the finite products {ceqn}

[TABLE]

The integer numbers $q_{n}(k)$ in (19b) count the number of partitions of $k$ with distinct parts where no part exceeds $n$. The coefficients $q_{n}(k)$ clearly have 3 basic properties:

[TABLE]

where (20c) follows from evaluation of $Q_{n}(1)$. The $q_{n}(k)$ obey the recurrence relations

[TABLE]

Initial values are prescribed in row $n=0$ (21c). The values in any subsequent row $n\geq 1$ are determined by values in previous rows $m<n$.

Analogous properties and relations hold for the rational numbers $r_{n}(k)$ in (19a):

[TABLE]

[TABLE]

Figure 1 assembles some instances of $r(k)$ which have been computed on the R platform for statistical computing [RPr] using recurrence relations (23c), (23d), and (23e). The plot shows that the $r(k)$ approach the asymptotic value {ceqn}

[TABLE]

from above as $k$ increases. The constant $C$ is determined by (11) and (14) and is marked by a dashed horizontal line.

Conclusion

It has been shown that the function {ceqn}

[TABLE]

is an asymptotic form of the generating function (1a) in the sense that the sequence of rational numbers $r(k)$ generated by (1a) converges towards $C>0$ which is determined by (11) and (14).