Log-Concavity of Infinite Product and Infinite Sum Generating Functions

TL;DR

This paper investigates the log-concavity of coefficients derived from infinite product and sum generating functions related to partitions, plane partitions, and Fibonacci numbers, revealing patterns depending on residue classes modulo 3.

Contribution

It introduces new generating functions linked to classical combinatorial sequences and analyzes their log-concavity properties across different parameters and residue classes.

Findings

Coefficients are log-concave at n ≡ 0 mod 3 for almost all d.

Coefficients are not log-concave at n ≡ 2 mod 3 for almost all d.

Log-concavity flips for n ≡ 1 mod 3 depending on d.

Abstract

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $\sigma_d(n)= \sum_{\ell \mid n} \ell^d$ or $\psi_d(n)= n^d$. We associate double sequences $\left\{ p^{g_{d} }\left( n\right) \right\}$ and $\left\{ q^{g_{d} }\left( n\right) \right\} $, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} \mu(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\right) \, t^{n} & := & \frac{1}{1 - \sum_{n=1}^{\infty} g_d(n) \, t^{n} }. \end{eqnarray*} These coefficients are related to the number of partitions $\mathrm{p}\left( n\right) = p^{\sigma _{1 }}\left ( n\right) $, plane partitions $pp\left( n\right) = p^{\sigma _{2 }}\left( n\right) $ of $n$, and Fibonacci numbers $F_{2n} = q^{\psi _{1 }}\left( n\right) $. Let $n \geq 3$ and let $n \equiv 0 \pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential and geometric cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n \equiv 2 \pmod{3}$. Let $n\equiv 1 \pmod{3}$. Then the log-concave property flips for almost all $d$.

Tables

Table 1. Table 1. Exceptions for p σ d ​ ( n ) superscript 𝑝 subscript 𝜎 𝑑 𝑛 p^{\sigma_{d}}\left(n\right) for 0 ≤ d ≤ 17 0 𝑑 17 0\leq d\leq 17 and 1 ≤ n ≤ 14 1 𝑛 14 1\leq n\leq 14 .

Table 2. Table 2. Exceptions for p ψ d ​ ( n ) superscript 𝑝 subscript 𝜓 𝑑 𝑛 p^{\psi_{d}}\left(n\right) for 0 ≤ d ≤ 17 0 𝑑 17 0\leq d\leq 17 and 1 ≤ n ≤ 14 1 𝑛 14 1\leq n\leq 14 .

Table 3. Table 3. Exceptions for q σ d ​ ( n ) superscript 𝑞 subscript 𝜎 𝑑 𝑛 q^{\sigma_{d}}\left(n\right) for 0 ≤ d ≤ 17 0 𝑑 17 0\leq d\leq 17 and 1 ≤ n ≤ 14 1 𝑛 14 1\leq n\leq 14 .

Table 4. Table 4. Exceptions for q ψ d ​ ( n ) superscript 𝑞 subscript 𝜓 𝑑 𝑛 q^{\psi_{d}}\left(n\right) for 0 ≤ d ≤ 17 0 𝑑 17 0\leq d\leq 17 and 1 ≤ n ≤ 14 1 𝑛 14 1\leq n\leq 14 .

Equations

$$\sum _{n=0}^{\infty }{p}^{g_d}\left(n\right)t^n$$

$$\sum _{n=0}^{\infty }{q}^{g_d}\left(n\right)t^n$$

$$a_{n_0}^2-a_{n_0-1}{a}_{n_0+1}\ge 0.$$

$$\sum _{n=0}^{\infty }{p}^{g_d}\left(n\right)t^n$$

$$\sum _{n=0}^{\infty }{q}^{g_d}\left(n\right)t^n$$

$$a_n^2-a_{n-1}{a}_{n+1}<0.$$

$$E^{p^{{\sigma }_1}}=\{2k+1|0\le k\le 12\}.$$

$$E^{p^{{\sigma }_2}}=\{2k+1|0\le k\le 5\}.$$

$$E^{p^{{\sigma }_0}}=\left\{2k+1|0\le k\le 371\right\}\setminus \left\{717,723,729,735,741\right\}$$

$$0\le g_d\left(n\right)-n^d\le \left(g_1\left(n\right)-n\right){\left(n-1\right)}^{d-1}.$$

$$0\le g_d(n)-n^d\le \left(g_0\left(n\right)-1\right)(n-1{)}^d.$$

$$D^g\left(n\right)$$

$${\log }_{9/8}\left(2\right)-\left(n+1\right){\log }_{9/8}\left(r\right).$$

$$\frac{{\left(q^{g_d}\left(n\right)\right)}^2}{q^{g_d}\left(n-1\right)q^{g_d}\left(n+1\right)}<1\text{, if and only if }n\equiv 1(\mathrm{mod}3).$$

$$D^{\psi}\left(n\right):=\left\{\begin{array}[]{ll}2n\log_{9/8}\left(2\right),&n\equiv 0\pmod{3},\\ 2n\log_{9/8}\left(2\right)+\log_{9/8}\left(3\right)-\log_{9/8}\left(n+2\right),&n\equiv 1\pmod{3},\\ \left(n+2\right)\log_{9/8}\left(2\right),&n\equiv 2\pmod{3},n\neq 5,\\ \log_{9/8}\left(512\right),&n=5.\end{array}\right.$$

$$\frac{{\left(q^{{\psi }_d}\left(n\right)\right)}^2}{q^{{\psi }_d}\left(n-1\right)q^{{\psi }_d}\left(n+1\right)}<1\text{, if and only if }n\equiv 1(\mathrm{mod}3).$$

$$\left\{\begin{array}[]{ll}2\log_{9/8}\left(3\right)n,&n\equiv 0\pmod{3},\\ \left(2n+1\right)\log_{9/8}\left(3\right)-\log_{9/8}\left(n+2\right),&n\equiv 1\pmod{3},\\ \log_{9/8}\left(2\right)+\left(n+1\right)\log_{9/8}\left(3\right),&n\equiv 2\pmod{3}.\end{array}\right.$$

$$\frac{{\left(q^{{\sigma }_d}\left(n\right)\right)}^2}{q^{{\sigma }_d}\left(n-1\right)q^{{\sigma }_d}\left(n+1\right)}<1\text{, if and only if }n\equiv 1(\mathrm{mod}3).$$

$$\frac{1}{1-{\sum }_{n=1}^{\infty }{g}_d\left(n\right)t^n}=1+\sum _{n=1}^{\infty }\sum _{k\le n}\sum _{\begin{array}{c}{m}_1,\dots ,m_k\ge 1\\ m_1+\dots +m_k=n\end{array}}{g}_d\left(m_1\right)\cdots g_d\left(m_k\right)t^n=\sum _{n=0}^{\infty }{q}_d\left(n\right)t^n.$$

$$q_d\left(n\right)=\sum _{k\le n}\sum _{\begin{array}{c}{m}_1,\dots ,m_k\ge 1\\ m_1+\dots +m_k=n\end{array}}{g}_d\left(m_1\right)\cdots g_d\left(m_k\right)$$

$$\max_{\begin{subarray}{c}m_{1},\ldots,m_{k}\geq 1\\ m_{1}+\ldots+m_{k}=n\end{subarray}}m_{1}\cdots m_{k}=\left\{\begin{array}[]{ll}3^{n/3},&n\equiv 0\pmod{3},\\ 4\cdot 3^{\left(n-4\right)/3},&n\equiv 1\pmod{3},\\ 2\cdot 3^{\left(n-2\right)/3},&n\equiv 2\pmod{3}.\end{array}\right.$$

$$8\cdot 3^{\left(n-6\right)/3},n$$

$$10\cdot 3^{\left(n-7\right)/3},n$$

$$3^{dn/3}$$

$${\left(4\cdot 3^{\left(n-4\right)/3}\right)}^d\frac{\left(n-1\right)\left(n+8\right)}{18}$$

$${\left(2\cdot 3^{\left(n-2\right)/3}\right)}^d\frac{n+1}{3}$$

$$q_d\left(n\right)$$

$$\sum _{k\le n}\sum _{\begin{array}{c}{m}_1,\dots ,m_k\ge 1\\ m_1+\dots +m_k=n\end{array}}{\left(m_1\cdots m_k\right)}^d\ge S\left(n\right)\underset{k\le n}{\max }\underset{\begin{array}{c}{m}_1,\dots ,m_k\ge 1\\ m_1+\dots +m_k=n\end{array}}{\max }{\left(m_1\cdots m_k\right)}^d$$

$$S\left(n\right)=\left\{\begin{array}[]{ll}1,&n\equiv 0\pmod{3},\\ \frac{n-1}{3}+\binom{\left(n+2\right)/3}{2}=\frac{\left(n-1\right)\left(n+8\right)}{18},&n\equiv 1\pmod{3},\\ \frac{n+1}{3},&n\equiv 2\pmod{3}.\end{array}\right.$$

$$g_d\left(m_1\right)\cdots g_d\left(m_k\right)$$

$$q_d\left(n\right)\le {\left(\underset{k\le n}{\max }\underset{\begin{array}{c}{m}_1,\dots ,m_k\ge 1\\ m_1+\dots +m_k=n\end{array}}{\max }{m}_1\cdots m_k\right)}^{d}S\left(n\right)+{\left(Z\left(n\right)\right)}^d{q}_0\left(n\right)$$

$$\frac{{\left(q_d\left(n\right)\right)}^2}{q_d\left(n-1\right)q_d\left(n+1\right)}$$

Full text

[TABLE]

These coefficients are related to the number of partitions $\mathrm{p}\left(n\right)=p^{\sigma_{1}}\left(n\right)$, plane partitions $\mathop{\rm pp}\left(n\right)=p^{\sigma_{2}}\left(n\right)$ of $n$, and Fibonacci numbers $F_{2n}=q^{\psi_{1}}\left(n\right)$. Let $n\geq 3$ and let $n\equiv 0\pmod{3}$. Then the coefficients are log-concave at $n$ for almost all $d$ in the exponential (0.1) and geometric (0.2) cases. The coefficients are not log-concave for almost all $d$ in both cases, if $n\equiv 2\pmod{3}$. Let $n\equiv 1\pmod{3}$. Then the log-concave property flips for almost all $d$.

1. Introduction

In this paper, we study log-concave properties of families of sequences related to infinite product and infinite sum generating functions [Br89, St89, HN22a].

Log-concavity is an important property. For polynomials with positive coefficients, real-rootedness entails log-concavity of all internal coefficients, which implies unimodality. Recent breakthrough works by Huh and his collaboraters, using methods in algebraic geometry, have proven the Mason and Heron–Rota–Welsh conjecture on the log-concavity of the chromatic polynomials of graphs, and finally the characteristic polynomials of matroids [AHK18, BHMPW22, Hu12]. We refer to the survey by Kalai [Ka22] on the work by Huh. Note that Zhang [Zh22] proved that the coefficients of the Nekrasov–Okounkov polynomials are almost all unimodal, building on the work by Odlyzko and Richmond [OR85] and Hong and Zhang [HZ21].

We offer an approach for sequences associated with generating functions, where in general, not all coefficients are log-concave. For example, it is well-known that the partition numbers $\mathrm{p}\left(n\right)$ are log-concave for $n\geq 26$. We encounter $\mathrm{p}\left(n\right)$ and the number of plane partition numbers $\mathop{\rm pp}\left(n\right)$ of $n$ [An98, Kr16] and Fibonacci numbers $F_{n}$. A sequence $\{a_{n}\}_{n}$ is called log-concave at $n_{0}$, if

[TABLE]

Let $\{g_{d}(n)\}_{d\geq 0,n\geq 1}$ be a double sequence of positive integers. We examine the coefficients of the associated generating functions of exponential (1.1) and geometric type (1.2):

[TABLE]

Here $\alpha_{d}(n)=\sum_{\ell\mid n}\mu(\ell)\,g_{d}(n\,/\,\ell)$, where $\mu$ is the Möbius function.

The approach offered in this paper, is incited by Andrews’ remark ([An98], chapter 6, page 99) in the context of Meinardus’ theorem: “Unfortunately not much is known about problems when a series rather than a product is involved”. We call $n$ an exception related to a sequence $\left\{a_{n}\right\}_{n}$, if

[TABLE]

The set of all exceptions is denoted by ${E}^{a}$.

To this point only the exponential cases have been studied in the literature. Let $g_{d}=\sigma_{d}$. For fixed $d=1$, we have the number of partitions $\mathrm{p}\left(n\right)=p^{\sigma_{1}}\left(n\right)$.

Nicolas [Ni78] proved in 1978, that the partition function $\mathrm{p}\left(n\right)$ is log-concave, if and only if $n$ is not an element of the finite set

[TABLE]

This was proved again by DeSalvo and Pak [DP15]. Both proofs utilize the Rademacher formula for $\mathrm{p}\left(n\right)$. In [HNT22], we have proven that the plane partition function $\mathop{\rm pp}\left(n\right)=p^{\sigma_{2}}\left(n\right)$ is log-concave for almost all $n$. Finally, based on numerical experiments, we conjectured that

[TABLE]

Recently, the conjecture was proven by Ono, Pujahari, and Rolen [OPR22].

In this paper, we study the similarities between log-concavity properties of the coefficients obtained by the generating function of exponential (1.1) and geometric type (1.2).

1.1. Landscape of Exceptions in the

Exponential Cases

We consider log-concavity for $\left\{p^{g_{d}}\left(n\right)\right\}$. We recall the results obtained in [HN22a] and [HN22b]. Note, the information on $d=0$ is new. Numerical investigations indicate that

[TABLE]

tested up to $n=2500$. Further, for $0\leq d\leq 5$ the cardinality of ${E}^{p^{\sigma_{d}}}$ seems to be decreasing: $367>13>6>4>2\geq 2$. But $\left|{E}^{p^{\sigma_{6}}}\right|\geq 3$. We refer to Table 1. The case $g_{d}=\psi_{d}$, if we see Table 2, reveals the similar pattern.

Now, fixing $n$ and studying log-concavity, reveals a new phenomenon. Let $n\geq 3$. Let $g_{d}(n)=\sigma_{d}(n)$ or $\psi_{d}(n)$. Then the set of all exceptions for all $d$ is finite, if and only if $n\equiv 0\pmod{3}$. More generally [HN22b], let $\{g_{d}(n)\}_{d\geq 1,n\geq 1}$ be positive real numbers satisfying $g_{d}(1)=1$ and

[TABLE]

Let $n\geq 3$. Then for almost all $d$, $p^{g_{d}}\left(n\right)$ is log-concave at $n$, if and only if $n$ is divisible by $3$. Moreover, explicit bounds are given. It would be interesting to examine the results of this paper in the context of generalized Laguerre-Pólya functions and Jensen polynomials [Wa22].

1.2. Landscape of Exceptions in the Geometric Cases

At first glance, the geometric case, see Table 3 and Table 4, seems not to reveal much structure. Nevertheless, we recall that $q^{\psi_{1}}(n)=F_{2n}$ can be identified with the sequence of the $2n$th Fibonacci numbers, which is log-concave for $n>1$. This follows from the fact that $q^{\psi_{1}}(n)=U_{n-1}(\frac{3}{2})$, where $U_{n}(x)$ is the $n$th Chebyshev polynomial of the second kind. Thus, we have some kind of analogue to Nicolas’ result. Thus far, for $d=2$ and $\psi_{2}(n)$, we expect infinitely many exceptions. Nevertheless, by fixing $n\geq 3$ we obtain the following new result.

We have the geometric cases for $g_{d}(n)=\sigma_{d}(n)$ in Table 3 and $g_{d}(n)=n^{d}$ in Table 4.

1.3. Main

Results

In this paper, we prove the following:

Theorem 1.1**.**

Let $\{g_{d}(n)\}_{d\geq 0,n\geq 1}$ be a double sequence of positive real numbers with $g_{d}(1)=1$ for all $d$ and

[TABLE]

Suppose there is an $0<r\leq 1$, such that $q^{g_{0}}\left(n\right)\leq r^{-n}$. Let $n\geq 3$ and $D^{g}\left(n\right)$ be defined by

[TABLE]

for $n\equiv 0\pmod{3}$ in (1.4) and $n\equiv 1\pmod{3}$ in (1.5). Further, let $n\equiv 2\pmod{3}$ and $n\neq 5$. Then $D^{g}\left(n\right)$ is defined as

[TABLE]

Moreover, $D^{g}\left(5\right):=\log_{9/8}\left(2q^{g_{0}}\left(4\right)q^{g_{0}}\left(6\right)\right)$.

Let $d>D^{g}\left(n\right)$. Then

[TABLE]

The double sequences $\{g_{d}(n)\}$ given by $\psi_{d}(n)=n^{d}$ and $\sigma_{d}(n)=\sum_{\ell\mid n}\ell^{d}$ satisfy (1.3). In the case $\psi_{0}\left(n\right)$, we have $\frac{t}{1-t}=\sum_{n=1}^{\infty}t^{n}$ and $\frac{1}{1-\frac{t}{1-t}}=1+\sum_{n=1}^{\infty}2^{n-1}t^{n}$. Therefore, $q^{\psi_{0}}\left(n\right)\leq 2^{n}$ for $n\geq 1$. We can apply Theorem 1.1 with $r=\frac{1}{2}$.

Let

[TABLE]

Corollary 1.2**.**

Let $n\geq 3$. Let $d>D^{\psi}(n)$. Then

[TABLE]

For $g_{0}=\sigma_{0}$, we obtain $\frac{1}{1-\sum_{n=1}^{\infty}\sigma_{0}\left(n\right)t^{n}}=1+t+3\*t^{2}+7\*t^{3}+18\*t^{4}+43\*t^{5}+108\*t^{6}+\ldots$. Obviously, $\sigma_{0}\left(n\right)\leq n$. Then $\sum_{n=1}^{\infty}nt^{n}=\frac{t}{\left(1-t\right)^{2}}$ and the radius of convergence of the series expansion of $\frac{1}{1-\frac{t}{\left(1-t\right)^{2}}}$ is $\left(3-\sqrt{5}\right)/2>1/3$. Analyzing the coefficients shows that we can choose any $0<r<\left(3-\sqrt{5}\right)/2$. For simplicity, we take $r=\frac{1}{3}$ and obtain $q^{\sigma_{0}}\left(n\right)\leq 3^{n}$. We define $D^{\sigma}(n)$ for $n\geq 3$ and $n\neq 5$ by

[TABLE]

Further, $D^{\sigma}\left(5\right):=\log_{9/8}\left(3888\right).$

Corollary 1.3**.**

Let $n\geq 3$. Let $d>D^{\sigma}(n)$. Then

[TABLE]

2. Proof of Theorem 1.1

Let $g_{d}$ be fixed satisfying (1.3). To simplify notation, we put $q_{d}\left(n\right)=q^{g_{d}}\left(n\right)$. We have

[TABLE]

Therefore,

[TABLE]

for $n\geq 1$.

2.1. Two Lemmata

It is known [HN22a] that:

Lemma 2.1**.**

Let $n\geq 2$. Then

[TABLE]

For $n\geq 6$, $n\mathrel{\hbox to0.0pt{/\hss}\kern-3.39996pt\equiv}2\pmod{3}$, the second largest products are

[TABLE]

and $3$ for $n=4$.

Further, we provide an extension of a result from [HN22a, HN22b].

Lemma 2.2**.**

For $n\geq 2$

[TABLE]

Additionally, for $n\geq 6$, $n\mathrel{\hbox to0.0pt{/\hss}\kern-3.39996pt\equiv}2\pmod{3}$, we have

[TABLE]

and $q_{d}\left(4\right)\leq 2\cdot 4^{d}+3^{d}q_{0}\left(4\right)$.

Proof.

Since $m_{1}\cdots m_{k}\leq\max_{\begin{subarray}{c}m_{1},\ldots,m_{k}\geq 1\\ m_{1}+\ldots+m_{k}=n\end{subarray}}m_{1}\cdots m_{k}$, the upper bounds should be obvious as $g_{d}\left(n\right)\leq n^{d}+\left(g_{0}\left(n\right)-1\right)\left(n-1\right)^{d}\leq g_{0}\left(n\right)n^{d}$. For the lower bounds, we obtain

[TABLE]

where $S\left(n\right)$ is the number of $m_{1}+\ldots+m_{k}=n$, which yield the maximal product. Therefore,

[TABLE]

For the refined upper bounds, we consider $g_{d}\left(n\right)\leq n^{d}+\left(g_{0}\left(n\right)-1\right)\left(n-1\right)^{d}$. Then $g_{d}\left(m_{1}\right)\cdots g_{d}\left(m_{k}\right)\leq g_{0}\left(m_{1}\right)\cdots g_{0}\left(m_{k}\right)\left(m_{1}\cdots m_{k}\right)^{d}$ and for the maximal values

[TABLE]

Therefore,

[TABLE]

where $Z\left(n\right)$ is the second largest product $m_{1}\cdots m_{k}$ of all $m_{1}+\ldots+m_{k}=n$. ∎

2.2. Proof of Theorem 1.1

We consider the cases $n\equiv 0,1,2\pmod{3}$ and $n=5$ separately.

2.2.1. Let $n\equiv 0\pmod{3}$

Then

[TABLE]

for $d\geq-2\log_{9/8}\left(r\right)n$.

2.2.2. Let $n\equiv 1\pmod{3}$

Then

[TABLE]

for $d>\log_{9/8}\left(3\right)-2n\log_{9/8}\left(r\right)-\log_{9/8}\left(n+2\right)$.

2.2.3. Let $n\equiv 2\pmod{3}$ and $n\neq 5$

Then

[TABLE]

where

[TABLE]

Then as a lower bound for the expression on the right hand side of (2.1) we obtain the following:

[TABLE]

for

[TABLE]

for $0<r\leq 1$ as $0\leq-\log_{6/5}\left(r\right)\leq-\log_{9/8}\left(r\right)$ and $\log_{6/5}\left(9/4\right)<\log_{9/8}\left(2\right)$.

2.2.4. Let $n=5$

We have

[TABLE]

for $d\geq\log_{9/8}\left(2q_{0}\left(4\right)q_{0}\left(6\right)\right)$.

3. Final Remarks

Let us examine $\left\{q^{g_{d}}\left(n\right)\right\}$. There are no exceptions for $g_{d}=\sigma_{d}$ or $\psi_{d}$ for $n=2$, since

[TABLE]

Challenge 1*.*

We consider the exponential case for $d=0$ and $\sigma_{0}$. We expect $E^{p^{\sigma_{0}}}$ to be finite. Moreover, numerical experiments (tested up to $n=2500$) suggest that

[TABLE]

Challenge 2*.*

We consider the geometric case. We have $E^{q^{\psi_{0}}}=E^{q^{\psi_{1}}}=\left\{1\right\}$, since $q^{\psi_{0}}(n)=2^{n-1}$ and $q^{\psi_{1}}(n)=F_{2n}$. For $E^{q^{\psi_{2}}}$, we expect infinitely many exceptions and non-exceptions.

Challenge 3* (Geometric case).*

Let $\sigma_{d}$ for $0\leq d\leq 4$ be given. Then all the odd numbers $n$ up to $10^{4}$ are exceptions. Note that for $d=5$ some even numbers also appear as exceptions. For example,

[TABLE]

Nevertheless, it seems that the set of exceptions for each $\sigma_{d}$ is infinite.