Zeros transfer for recursively defined polynomials

Abstract

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.

1 Introduction

Powers of the Dedekind $\eta$-function $\eta \left( \omega \right) ^{r}$ and properties of its Fourier coefficients, especially their non-vanishing, play an important role in the theory of numbers [2, 27, 30], combinatorics [1, 15], and physics [7, 10, 25]. Let $q:=\text {e}^{2 \pi \text {i} \omega }$, where $\omega$ is in the complex upper half-plane. Let $r \in {\mathbb {N}}$ be a positive integer, and define

$$\begin{aligned} \sum _{n=0}^{\infty } a_n(r) \, q^n:=q^{-\frac{r}{24}}\,\, \eta \left( \omega \right) ^{r}= \prod _{n=1}^{\infty } \left( 1 - q^n\right) ^{r}. \end{aligned}$$

Let r be even. Then Serre [30] proved that $\{a_n(r)\}_n$ is lacunary ([22], (1.5)) if and only if

$$\begin{aligned} r \in S:= \{2,4,6,8,10,14,26\}. \end{aligned}$$

It is conjectured by Lehmer [2, 24, 28] that $a_n(24)$ never vanishes. Numerical experiments [22] suggest that $a_n(r) \ne 0$ for all $n,r \in {\mathbb {N}}$ with r even and $r \not \in S$, generalizing the Lehmer conjecture.

It is known that the coefficients $a_n(r)$ are special values of certain polynomials $P_n(z)$ at $z=-r$ of degree n called D’Arcais polynomials [6, 8, 17, 26], also known as Nekrasov–Okounkov polynomials [15, 33] in combinatorics. We have

$$\begin{aligned} \sum _{n=0}^{\infty } P_n(z) \, q^n = \prod _{n=1}^{\infty } \left( 1 - q^n\right) ^{-z} = \exp \left( z\sum _{n=1}^{\infty } \sigma _1(n) \,\frac{q^n}{n}\right) , \quad z \in {\mathbb {C}}, \end{aligned}$$

where $\sigma _d(n):= \sum _{\ell \mid n} \ell ^d$. Note that $P_n(z)$ is integer-valued and $n! \, P_n(z)$ is monic of degree n with non-negative integer coefficients and therefore, zeros are algebraic integers. The polynomials can also be defined recursively [21, 26], which enables us to study them using methods from difference equations:

$$\begin{aligned} P_n(z)= \frac{z}{n} \sum _{k=1}^n \sigma _1(k) \,P_{n-k}(z), \quad n \ge 1, \end{aligned}$$

(1.1)

with initial value $P_0(z)=1$. As a special case we obtain the well known recurrence relation of Euler for the partition numbers p(n):

$$\begin{aligned} n \, p(n) = \sum _{k=1}^n \sigma _1(k) \, p(n-k). \end{aligned}$$

Serre’s results [30] imply that for each $z \in S$, there are infinitely many n, such that $P_n(-z)=0$. It would be interesting to devise a combinatorial proof utilizing (1.1). Consider z as a parameter and (1.1) as a difference equation. We refer to Elaydi’s excellent introduction to difference equations [12]. The Eq. (1.1) has non-constant coefficients and is of hereditary type.

Poincaré and Perron ([12], section 8.2) offered a method to solve difference equations of fixed order with non-constant coefficients, to obtain the asymptotic behavior of the solutions. On the other hand, for some difference equations of hereditary type with constant coefficients, called Volterra difference equations of convolution type, one can utilize the discrete Laplace transform, also called Z-transform ([12], section 6.3).

In this paper, we develop a new method to study the solutions, and especially the zero distribution of (1.1). We associate $\{P_n(z)\}_n$ with another family of polynomials $\{Q_n(z)\}_n$. These polynomials satisfy a Volterra difference equation. Then we transfer properties of $Q_n(z)$ to the D’Arcais polynomials $P_n(z)$ (see also [34], introduction). In this paper, we provide evidence that $Q_n(z)$ is easier to study, rather than $P_n(z)$.

Definition 1.1

Let g be a normalized arithmetic function with non-negative values and let h be a normalized arithmetic function with positive values. Define

$$\begin{aligned} P_n^{g,h}(z):= \frac{z}{h(n)} \, \sum _{k=1}^n g(k) \, P_{n-k}^{g,h}(z), \qquad n \ge 1, \end{aligned}$$

(1.2)

with initial value $P_0^{g,h}(z)=1$. Let $h_s(n)=n^s$ for $s \in [0,1]$. The cases $s=0$ and $s=1$ are special. Therefore, we put $P_n^g(z):= P_n^{g,h_1}(z)$ and $Q_{n}^{g}(z):= P_n^{g,h_0}(z)$.

Examples of arithmetic functions are provided by $g\left( n\right) =\sigma _{d}\left( n\right)$, $\psi _d(n)=n^d$, and $\textrm{id}(n)=n$. Note, that $P_n(z)= P_n^{\sigma _1, h_{1}}\left( z\right)$.

The main results are provided in Sect. 1.3. We invest in the impact of the smallest and largest real zeros of $\{Q_n^g(z)\}_n$ on the location of the real zeros of $\{P_n^{g,h}(z)\}_n$, Theorems 1.4 and 1.5, based on a Fundamental Transfer Lemma 1.6.

We begin in Sect. 1.1 to illustrate our results with an explicit example related to orthogonal polynomials [5, 11, 31] related to $g(n)= \textrm{id}(n)$. In Sect. 1.2 we focus on the D’Arcais polynomials. In Sect. 2 we give proofs and in Sect. 3 we provide some final remarks.

1.1 Zeros transfer from Chebyshev to Laguerre polynomials

We describe how properties of Chebyshev polynomials, given by $h(n)=h_0(n)=1$, give obstructions for the zero distribution of associated Laguerre polynomials $L_m^{(1)}(z)$. Orthogonal polynomials have real zeros, which are interlacing and well-studied. To quote Rahmann–Schmeisser ([29], introduction, p. 24): “the Chebyshev polynomials are the only classical orthogonal polynomials whose zeros can be determined in explicit form”. This makes our task even more appealing, since we claim that these explicit values can be utilized to study the zeros of Laguerre polynomials. We also refer to [14] for interesting results. Let $L_m^{(\alpha )}(z)$ denote the mth $\alpha$-associated Laguerre polynomial and $U_m(z)$ represent the mth Chebyshev polynomial of second kind. Then we have ([16], Lemma 3.3 and [21], Remark 2.8):

$$\begin{aligned} P_n^{\textrm{id}}(z) = \frac{z}{n} \, L_{n-1}^{(1)} (-z), \text { and } Q_n^{\textrm{id}}(z) = z \, U_{n-1}\left( \frac{z}{2}+1\right) . \end{aligned}$$

Theorem 1.2

Let $n \ge 2$. Let $\alpha _n$, $\beta _n$ be the smallest and largest real zeros of $Q_n^{\textrm{id}}(z)/ \, z= U_{n-1}\left( z/2+1\right)$, where

$$\begin{aligned} \alpha _n = 2 \, \cos \left( \frac{n-1}{n} \, \pi \right) -2, \qquad \beta _n= 2 \, \cos \left( \frac{1}{n} \, \pi \right) -2. \end{aligned}$$

Then the zeros of $n \, P_n^{\textrm{id}}(z)/\, z= L_{n-1}^{(1)}(-z)$ are contained in the interval

$$\begin{aligned} \left[ \alpha _n \, (n-1)\,, \, \beta _n \right] . \end{aligned}$$

Corollary 1.3

Let $m \ge 2$. Then the zeros of $L_m^{(1)}(z)$ are contained in the interval

$$\begin{aligned} \left[ 2-2 \, \cos \left( \frac{\pi }{m+1} \right) \,, \, \left( 2-2 \, \cos \left( \frac{m \, \pi }{m+1} \right) \right) \, m\right] . \end{aligned}$$

Let $\alpha _n$, $\beta _n$ be the smallest and largest real zeros of $Q_n^g(x)$. Similarly, let $P_n^g(x)$ be given, then $\tilde{\alpha }_{n}$, ${\tilde{\beta }}_{n}$ denote its smallest and largest real zeros.

Table 1 Approximative values for $\alpha _n$, $\beta _n$ the smallest and largest real zeros of $Q_n^{\textrm{id}}(z)/z$, compared to ${\tilde{\alpha }}_n$, ${\tilde{\beta }}_n$ the smallest and largest real zeros of $P_n^{ \textrm{id}}(z)/z$

Theorem 1.4 implies that $\frac{{\tilde{\alpha }}_{n}}{(n-1) \, \alpha _{n}} \le 1$ for $g(n)=n$ as illustrated by Table 1. The data for the first data column has for example been generated with PARI/GP using the following code:

It would be interesting to find out if

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{{\tilde{\alpha }}_{n}}{(n-1) \, \alpha _{n}} =1 \end{aligned}$$

holds true?

Szegő ([31], (6.32.6)) gives for $- \tilde{\alpha }_{n+1}$ the upper bound:

$$\begin{aligned} \gamma _n = \left( \sqrt{4n+4}-6^{-1/3 }\left( 4n+4\right) ^{-1/6}i_{1}\right) ^{2}, \end{aligned}$$

where $i_1$ denotes the smallest positive zero of Airy’s function $A_1(x)$.

Note that $A_1(x)$ is a solution of the differential equation $y'' + \frac{1}{3} x \, y =0$ (Szegő [31], Eq. (1.81.2)). An approximation of the large zeros is provided by Tricomi [32] (see also Gatteschi [13]). We obtain $i _{1}\approx 3.37213$ for the 1st zero of the Airy function $A_ {1}\left( x\right)$. Then Tricomi’s theorem ([32], Eq. (5)) offers for the largest zero $\lambda _n$ of $L_n^{(1)}(x)$ the approximation (as $n \rightarrow \infty$)

$$\begin{aligned} \lambda _n = 4n+4 - \root 3 \of {\frac{4}{3}} \, i _{1} \left( 4n+4\right) ^{\frac{1}{3}} + \frac{1}{5} \left( \root 3 \of {\frac{4}{3}} i_{1}\right) ^{2} \,\left( 4n+4\right) ^{- \frac{1}{3}} + O(n^{-1}). \end{aligned}$$

Compare also ([13], Theorem 5) but note that there Airy’s function is supposed to satisfy the differential equation $y^{\prime \prime }-xy=0$ resulting in a different zero $a_{1}=-3^{-1/3}i_{1}$. We omit the O-term (see also [13]) and obtain $\lambda _n'$. We refer to Table 2.

Table 2 Approximative values for the smallest zeros ${\tilde{\alpha }}_{n}$ of $P_{n}^{\textrm{id}}\left( z\right) /z$, compared with ([31], Eq. (6.36.6) and [13], Theorem 5)

Further, Theorem 1.5 implies that $\frac{\beta _n}{{\tilde{\beta }}_{n}}<1$. Nevertheless, Table 1 indicates that $\frac{{\tilde{\beta }}_n}{(n-1) \, \beta _n}$ converges to $\approx 0.37\ldots$. If this is the case, it would be interesting to identify this constant in the context of orthogonal polynomials. Thus, we raise the question, if the following limit exists:

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{{\tilde{\beta }}_n}{(n-1) \, \beta _n} = \beta . \end{aligned}$$

1.2 D’Arcais polynomials

Note that the non-trivial real zeros of the polynomials $\{ P_n^{g,h}(z)\}_n$ are negative, since the coefficients are non-negative real numbers. The real zeros dictate the sign changes of $\{P_n^{g,h}(z_0)\}_n$ for fixed $z_0 \in {\mathbb {R}}_{<0}$. We refer to [18], where the sign changes of the Ramanujan $\tau$-function are analyzed.

Kostant [23], building on representation theory of complex Lie algebras and Han [15] on the Nekrasov–Okounkov hook length formula, proved that $(-1)^n \, P_n^{\sigma _{1}}(z)> 0$ for $z \le -n^2 +1$ for $n \ge 4$. Utilizing the recursion formula (1.2) of $P_n^{g,h}(z)$ and assuming that $\sum _{n=1}^{\infty } g(n) \, q^n$ is regular at $q=0$ and h monotonically increasing, there exists a $\kappa >0$, only dependent on g, such that $P_n^{g,h}(z) \ne 0$ for all complex z with $\vert z \vert > \kappa \, h(n-1)$ (we refer to [19], Corollary 1). For example, we have shown [19] that the D’Arcais polynomials $P_n(z)$ are non-vanishing for all complex z with $\vert z \vert > 10.\overline{81} \, (n-1)$.

In [21], we obtained the following numerical result. Let $\alpha _n$ be the smallest real zero of $Q_n^{\sigma _{1}}(z)$. Then the smallest real zero ${\tilde{\alpha }}_n$ of $P_n^{\sigma _{1}}(z)$ satisfies $\alpha _n < \frac{{\tilde{\alpha }}_n}{n-1}$ for $n \le 1400$ (we refer to Fig. 1, [21]). For these n we also have $\alpha _n > \alpha _{n+1}$ and ${\tilde{\alpha }}_n > {\tilde{\alpha }}_{n+1}$. This implies the results displayed in Fig. 1, [21].

Let $\kappa _n$ be the maximum of $\{ \vert \alpha _1\vert , \ldots , \vert \alpha _n \vert \}$ and ${\tilde{\kappa }}_n$ be the maximum of $\{ \vert {\tilde{\alpha }}_1\vert , \ldots , \vert {\tilde{\alpha }}_n \vert \}$, where ${\tilde{\alpha }}_k$ is the smallest zero of $P_k^{\sigma _{1}}(x)$. Then we prove in this paper applying Theorem 1.4 that

$$\begin{aligned} \frac{{\tilde{\kappa }}_{n}}{\kappa _n \,(n-1)} < 1. \end{aligned}$$

Fig. 1

Minimal real zeros of $Q^{\sigma _{1}}_{n}\left( x\right)$ and $P_n^{\sigma _{1}} \left( (n-1)x \right)$

Table 3 illustrates that $\left| \alpha _{n}\left( n-1\right) \right|$ is an upper bound for $\left| {\tilde{\alpha }}_{n}\right|$. Let $\beta _n$ be the largest non-trivial real zero of $Q_n^{\sigma _{1}}(x)$ and ${\tilde{\beta }}_{n}$ be the largest non-trivial zero of $P_{n}^{\sigma _{1}}\left( x\right)$, the D’Arcais polynomial of degree n. Let n be a prime, then we observed for $n \le 257$ that

$$\begin{aligned} \frac{\tilde{\beta _n}}{(n-1) \beta _n} <1. \end{aligned}$$

For general n, this is not always the case. For example, for $n=18$ we obtain $\frac{{\tilde{\beta }}_{n}}{\left( n-1\right) \beta _{n}}\approx 1.878282$. Moreover, it seems that the quotient for a prime number n converges to a positive real number, if n goes to infinity. This indicates that the D’Arcais polynomials with prime number degree have some special properties among all D’Arcais polynomials.

Table 3 Approximative values for the smallest $\alpha _{n}$ and the largest real zeros $\beta _{n}$ of $Q_{n}^{\sigma _{1}}\left( z\right) /z$, compared to the smallest ${\tilde{\alpha }}_{n}$ and the largest real zeros $\tilde{\beta }_{n}$ of $P_{n}^{\sigma _{1}}\left( z\right) /z$

Figure 2 compares the locations of the largest real parts of the zeros of $Q_n^{\sigma _{1}}(x)$ and $P_n^{\sigma _{1}}(x)$. To illustrate a certain dominant role of the real zeros, we displayed for $Q^{\sigma _{1}}_{n}\left( z\right)$ both the real zeros and the real parts of the non-real zeros with largest values. Real zeros have a blue color and non-real zeros (their real part) are plotted in red.

Fig. 2

Non-trival zeros of $Q_{n}^{\sigma _{1}} (z)$ with largest real part (left plot) for real and non-real zeros, and $P_n^{\sigma _{1}} (x)$ with largest real part (right plot), where the dashed line denotes the convex hull of the zeros with largest real part of $Q_n^{\sigma _{1}}(z)$ for $1\le n\le 100$

1.3 Main results

We describe the impact of the smallest and largest real zeros of $\{Q_n^g(z)\}_n$ on the location of the real zeros of $\{P_n^{g,h}(z)\}_n$. Let $H(n):= \max \{h(k)\,: \, 1 \le k \le n\}$ and $H(0):=0$.

Theorem 1.4

Let g and h be normalized arithmetic functions with $g,h: {\mathbb {N}} \longrightarrow {\mathbb {R}}_{\ge 0}$ and h positive. Let us fix $n \ge 1$ and suppose there exists $\kappa _n > 0$, such that $(-1)^m \, Q_m^g(x) >0$ for all real $x < - \kappa _n$ and $1 \le m \le n$. If $x < - \kappa _n$ and

$$\begin{aligned} y \le x \, H(n-1), \end{aligned}$$

(1.3)

then we have for $1 \le m \le n$ the inequalities

$$\begin{aligned} (-1)^m P_m^{g,h}(y) \ge (-1)^m \frac{y}{x \, h\left( m\right) } \, Q_m^{g}(x) >0. \end{aligned}$$

(1.4)

If we replace (1.3) by $y < \ - \kappa _n \, H(n-1)$, then we have for $1 \le m \le n$:

$$\begin{aligned} (-1)^m P_m^{g,h}(y) >0. \end{aligned}$$

(1.5)

Further, we obtain a result on the influence of the largest non-trivial real zeros of $Q_m^g(z)$ for $1 \le m \le n$ on the zeros close to 0 of $P_n^{g,h}(z)$.

Theorem 1.5

Let g and h be normalized positive valued arithmetic functions, $h(n) \ge 1$ for all $n \in {\mathbb {N}}$. Let $n \ge 1$ be fixed. Suppose there exists a $\mu _n <0$, such that for all $\mu _n< x <0$: $Q_m^g(x) < 0$ for $1 \le m \le n$. Then $P_m^{g,h}(x) < 0$ for all $1 \le m \le n$.

These results follow from an identity, which we consider as fundamental in studying properties of $P_n^{g,h}\left( z\right)$.

Lemma 1.6

(Fundamental Transfer Lemma) Let g and h be normalized positive valued arithmetic functions. Let $x,y\in {\mathbb {C}}\setminus \left\{ 0\right\}$. Then we have for $n \ge 1$:

$$\begin{aligned} \frac{h\left( n\right) }{y} \, P_{n}^{g,h}\left( y\right) -\frac{1}{x} \,Q_{n}^{g}\left( x\right) = \sum _{k=1}^{n-1}\left( \frac{1}{x}-\frac{h\left( k\right) }{y}\right) P_{k}^{g,h}\left( y\right) Q_{n-k}^{g}\left( x\right) . \end{aligned}$$

(1.6)

The expression on the left hand side of (1.6) is equal to

$$\begin{aligned} \sum _{k=1}^{n-1} g(k) \, \left( P_{n-k}^{g,h}(y) - Q_{n-k}^{g}\left( x\right) \right) . \end{aligned}$$

To prove Theorems 1.4 and 1.5, we examine the identity

$$\begin{aligned} \frac{ x h\left( n\right) }{y} \, P_{n}^{g,h}\left( y\right) -Q_{n}^{g}\left( x\right) = \sum _{k=1}^{n-1}\left( 1 -\frac{ x \, h\left( k\right) }{y}\right) P_{k}^{g,h}\left( y\right) Q_{n-k}^{g}\left( x\right) \end{aligned}$$

(1.7)

for $\frac{x \, h(k)}{y} > 1$ or $\frac{x \, h(k)}{y} <1$ for $1 \le k \le n-1$.

2 Proof of the Fundamental Transfer Lemma 1.6, Theorem 1.4, and Theorem 1.5

We verify Theorems 1.4 and 1.5 for $n=1$ and $n=2$. The Fundamental Transfer Lemma is trivial for $n=1$ and reduces to the term $y-x$ on the left and right hand side for $n=2$, since $h(2) P_2^{g,h}(x) = Q_2^{g,h}(x)= x ( x + g(2))$.

Let $n=1$. Let $\kappa _1, - \mu _1>0$ and $\mu _1 <0$ be any positive real numbers. Then $y< x < - \kappa _1$ implies (1.4). Let $y < 0$. Then $-y>0$ (1.5). The claim of Theorem 1.5 for $n=1$ is trivial.

Let $n=2$. Then any $\kappa _2 > g(2)$ works. Let $y<x < - \kappa _2$. Then we obtain (1.4) for $m=1$ and (1.4) for $m=2$ since $y \ge x$. Further, let $y < - \kappa _2$. Then (1.5) for $m=1$ is obvious and (1.5) follows for $m=2$, since $y< -\kappa _2 < -g(2)$. It is sufficient and necessary to choose $\mu _2$ as $-g(2)< \mu _2 <0$. Then Theorem 1.5 follows for $n=1,2$.

2.1 Proof of the Fundamental Transfer Lemma

Proof of Lemma 1.6

To make the proof transparent, we first assume that all involved sums are regular at $q=0$. We have

$$\begin{aligned} x\sum _{k =1}^{\infty }g\left( k\right) q^{ k} \sum _{m =0}^{\infty }P_{m }^{g,h}\left( y\right) q^{ m}= & {} \sum _{n=1}^{\infty }x\sum _{\begin{array}{c} k\ge 1,m\ge 0 \\ k+m=n \end{array}}g\left( k\right) P_{m}^{g,h}\left( y\right) q^{k+m}\\= & {} \sum _{n=1}^{\infty }\frac{xh\left( n\right) }{y}\frac{y}{h\left( n\right) }\sum _{k=1}^{n}g\left( k\right) P_{n-k}^{g,h}\left( y\right) q^{n}\\= & {} \sum _{n=1}^{\infty }\frac{xh\left( n\right) }{y}P_{n}^{g,h}\left( y\right) q^{n}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left( 1-x\sum _{n=1}^{\infty }g\left( n\right) q^{n}\right) \sum _{n=0}^{\infty }P_{n}^{g,h}\left( y\right) q^{n}=\sum _{n=0}^{\infty }\left( 1-\frac{xh\left( n\right) }{y}\right) P_{n}^{g,h}\left( y\right) q^{n}. \end{aligned}$$

Here, we extended the arithmetic function h by $h(0)=0$. Since

$$\begin{aligned} \left( 1-x\sum _{n=1}^{\infty }g\left( n\right) q^{n}\right) \sum _{n=0}^{\infty }Q_{n}^{g}\left( x\right) q^{n}=1, \end{aligned}$$

we obtain

$$\begin{aligned} \sum _{n=0}^{\infty }P_{n}^{g,h}\left( y\right) q^{n}= & {} \sum _{n=0}^{\infty }\left( 1-\frac{xh\left( n\right) }{y}\right) P_{n}^{g,h}\left( y\right) q^{n} \sum _{m=0}^{\infty }Q_{m}^{g}\left( x\right) q^{m}\\= & {} \sum _{n=0}^{\infty }\sum _{k=0}^{n}\left( 1-\frac{xh\left( k\right) }{y}\right) P_{k}^{g,h}\left( y\right) Q_{n-k}^{g}\left( x\right) q^{n}. \end{aligned}$$

Comparing coefficients of the last power series yields

$$\begin{aligned} P_{n}^{g,h}\left( y\right) =\sum _{k=0}^{n}\left( 1-\frac{xh\left( k\right) }{y}\right) P_{k}^{g,h}\left( y\right) Q_{n-k}^{g}\left( x\right) . \end{aligned}$$

Subtracting the terms in the last sum for $k=0$ and $k=n$ yields (1.7)

$$\begin{aligned} \frac{xh\left( n\right) }{y}P_{n}^{g,h}\left( y\right) -Q_{n}^{g}\left( x\right) =\sum _{k=1}^{n-1}\left( 1-\frac{xh\left( k\right) }{y}\right) P_{k}^{g,h}\left( y\right) Q_{n-k}^{g}\left( x\right) . \end{aligned}$$

Let $n \ge 1$. We could truncate all involved sums to obtain (1.7). Therefore, the regularity of $\sum _{n=1}^{\infty } g(n) \, q^n$ at $q=0$ is not needed. $\square$

2.2 Proof of Theorem 1.4 and Theorem 1.5

Proof of Theorem 1.4

We prove the theorem by mathematical induction. The base case $n=1$ is already proven. Next, let $n>1$ and let the theorem hold true for all $1 \le n_0 <n$. Suppose $\kappa _{n_0} >0$ is given, such that for each $1 \le m_0 \le n_0$ and $x < -\kappa _{n_0}$: $(-1)^{m_0} Q_{m_0}^g(x) >0$. By induction hypothesis we have for $y < -\kappa _{n_0} \, H(n_0-1)$ and $1 \le m_0 \le n_0$:

$$\begin{aligned} (-1)^{m_0} \, P_{m_0}^{g,h}(y) >0. \end{aligned}$$

By (1.6) we obtain

$$\begin{aligned}{} & {} \left( -1\right) ^{n }\left( \frac{xh\left( n \right) }{y}P_{n }^{g,h}\left( y\right) -Q_{n }^{g}\left( x\right) \right) \\= & {} \sum _{n_0=1}^{n-1}\left( 1-\frac{x h\left( n_0\right) }{y}\right) \left( -1\right) ^{ n_0}P_{n_0 }^{g,h}\left( y\right) \left( -1\right) ^{ n-n_0 }Q_{ n-n_0 }^{g}\left( x\right) \ge 0 \end{aligned}$$

for $x < -\kappa _{n} \le 0$ and $y \le xH\left( n-1\right) \le xH \left( n_0 \right)$ for all $n_0\le n -1$ as

$$\begin{aligned} H\left( n_0 \right) =\max \left\{ h\left( n_0\right) ,H\left( n_0 -1\right) \right\} . \end{aligned}$$

This implies

$$\begin{aligned} (-1)^n \frac{x \, h(n)}{y} \, P_n^{g,h}(y) \ge (-1)^n \, Q_n^g(x) >0, \end{aligned}$$

and finally, the theorem is proven.

Proof of Theorem 1.5

Let $n \ge 2$ be given. Let $\frac{x \, h(k)}{y} >1$ for $1 \le k \le n-1$. Then for each $1 \le k \le n-1$, the term $1 - \frac{x \, h(k)}{y}$ is always negative. Since $h(n) \ge 1$, this is guaranteed since $\left| x \right| > \left| y \right|$. We prove the theorem by mathematical induction. Theorem 1.5 holds true for $n=1$ and $n=2$. Therefore, let $n \ge 3$ and $\mu _n<0$ be given, such that for $\mu _n< x <0$: $Q_k^g(x) <0$ for all $1 \le k \le n$. Then $P_k^{g,h}(x) < 0$ for $1\le k \le n-1$ by induction hypothesis. We examine (1.7). Let $\mu _n<y <0$. Let x be given, such that $\mu _n< x<y <0$. Then we obtain from (1.7) that

$$\begin{aligned} \frac{x \, h(n)}{y} P_n^{g,h}(y) - Q_n^g(x) \le 0. \end{aligned}$$

Since $Q_{n}^{g}\left( x\right) <0$, the theorem is proven.

3 Final remarks

We begin with an example, which illustrates the refinement of previously known results on the zero domain of $P_{n}^{g,h}\left( x\right)$. Then we state some identities deduced from the Fundamental Transfer Lemma. Finally, we give a reformulation of Lehmer’s conjecture and apply Theorem 1.4 to Hermite polynomials $H_n(x)$.

3.1 Example

Let $g(n)=n$ if n is odd and $g(n)=\frac{n}{2}$ if n is even. Let $h(n)=n$. We deduce from [19] that for $\kappa = 5.71$, we have

$$\begin{aligned} P_n^{g,h} (z) \ne 0 \text { for all } \vert z \vert > \kappa \, (n-1). \end{aligned}$$

Numerical experiments indicate that $\kappa =4$ seems to be possible (Fig. 3 and Table 4). We note that $Q_n^g(z)$ satisfies a 5-term recursion. From [21], we obtain

$$\begin{aligned} Q_n^g(z) = z \, Q_{n-1}^g(z) + (z+2) \, Q_{n-2}^g(z) + z \, Q_{n-3}^g(z) - Q_{n-4}^g(z), \qquad n \ge 5, \end{aligned}$$

(3.1)

with initial values $Q_1^g(z) = z$, $Q_2^g(z) = z^2 +z$, $Q_3^g(z)= z^3 + 2 z^2 + 3 z$, and $Q_{4}^{g}\left( z\right) =z^4 + 3z^3 + 7z^2 + 2z$. One of the reviewers kindly pointed out the relevance of [4] to attack such questions on the limit of zeros for recursively defined sequences of polynomials (see also [20]). Therefore, we calculate the characteristic polynomials $\chi _z(\lambda )$ of (3.1). We obtain:

$$\begin{aligned} \chi _z(\lambda )= \lambda ^4 - z \, \lambda ^3 - (z+2)\, \lambda ^2 - z \, \lambda +1. \end{aligned}$$

Let $\chi _z(\lambda _0)=0$. Then $1 / \lambda _0$ is also a zero. Moreover, the discriminant is given by:

$$\begin{aligned} -3 \, z^2 \, (z^2+4z + 16)^2 \end{aligned}$$

with the double zeros $0, -2 \pm \textrm{i} \sqrt{12}$. Therefore, to apply ([4], main result in Sect. 2) needs some additional investigation.

Fig. 3

Real parts of zeros of $Q_{n}^{g} (x)$ and $P_n^g (x)$ for $1\le n\le 10$,  here blue denotes that the zero is a real number and red denotes that it is not a real one

Table 4 Approximative largest absolute values of the zeros of $Q_{n}^{g}\left( x\right)$

In the following let $n \le 100$. Then $-1$ is the smallest real zero among all $Q_n^g(x)$. Then Theorem 1.4 implies that all real zeros of $P_n^{g}(z)$ are larger than or equal to $-(n-1)$. We also see that the largest real zeros of $\{Q_n^g(z)\}$ are monotonically increasing and tend to converge to 0. Therefore, Theorem 1.5 implies that for $\beta _n$ the largest non-trivial real zero of $\{Q_m^g(z)\}_{m \le n}$, the largest non-trivial real zero ${\tilde{\beta }}_n$ of $\{P_n^g(z)\}_{m \le n}$ satisfies

$$\begin{aligned} {\tilde{\beta }}_{n} \le \beta _n. \end{aligned}$$

Note, since $Q_n^g(z)/z$ and n odd has no real zeros one may deduce in general that this implies $P_n^g(z)/z$ not to have real zeros as well. But this is not the case in general. Since our main focus in this paper is on Theorems 1.4 and 1.5 and the underlying Fundamental Transfer Lemma, we leave further investigations to the reader.

Counterexample Let $g(n) = \left( {\begin{array}{c}\alpha \, n -1\\ n-1\end{array}}\right)$ for $\alpha \ge 1$. Then by a straightforward calculation we obtain

$$\begin{aligned} P_3^g(z) = \frac{z}{3!} \left( P_2^g(z) + g(2)\, z + g(3)\right) = \frac{z}{3!} \left( z + 3(\alpha -1) +1\right) \,\left( z + 3(\alpha -1) +2 \right) . \end{aligned}$$

Let $\alpha = \frac{3}{2}$. Then the zeros of $Q_3^g(z)/z$ are not real, but $P_3^g(z)/z$ has obviously two real zeros.

3.2 Identities

Let $F_n$ be the Fibonacci numbers, where $F_0=0$, $F_1=1$, $F_2=1$, $F_3=2$, $F_4 = 3$. It is known that $Q_n^{\textrm{id}}(1)= F_{2n}$ for $n \ge 1$ [3]. We obtain from the Fundamental Transfer Lemma and special properties of $P_n^{\textrm{id}}(z)$ and $Q_n^{\textrm{id}}(z)$:

$$\begin{aligned} L_{n-1}^{(1)}(y) - F_{2n} = - \sum _{k=1}^{n-1} \left( \frac{y}{k} +1 \right) L_{k-1}^{(1)}(y) \, F_{2(n-k)}. \end{aligned}$$

3.3 Lehmer’s conjecture

Let $\Delta \left( \omega \right) := q \prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{24}$. Then the Ramanujan $\tau$-function is provided by the Fourier coefficients of $\Delta (\omega )$:

$$\begin{aligned} \Delta (\omega ) = \sum _{n=1}^{\infty } \tau (n) \, q^n. \end{aligned}$$

Lehmer conjectured that $\tau (n)$ never vanishes. Suppose that n is the smallest natural number satisfying $\tau (n) =0$, then n is a prime [24]. Moreover, it is known that Lehmer’s conjecture is true for $n \le 10^{23}$ [9].

There are several variations of Lehmer’s conjecture known, especially in the context of special values of certain polynomials [28]. In this spirit, we offer:

Lehmer’s Conjecture (Polynomial Version) For all $n \ge 1$ and for every $z \in {\mathbb {C}}{\setminus } \left\{ 0\right\}$:

$$\begin{aligned} \sum _{k=0}^{n-1} \left( \frac{1}{z} + \frac{k}{24}\right) \, \tau (k+1) \, Q_{n-k}^{\sigma _{1}}(z) \ne 0. \end{aligned}$$

It would be interesting to have Lehmer’s conjecture in terms of special values of $Q_n^{\sigma _{1}}(z)$, as $z=-24$ in the case of D’Arcais polynomials.

3.4 Bounds on the zeros of Hermite polynomials

We denote by $H_n(x)$ the nth Hermite polynomials defined by ([31], (5.5.4)):

$$\begin{aligned} H_n(x):=n! \, \sum _{k=0}^{\left\lfloor n/2 \right\rfloor } \frac{(-1)^k}{k!} \, \frac{(2x)^{n-2k}}{(n-2k)!}. \end{aligned}$$

We have $H_0(x)=1$, $H_1(x)=2x$, $H_2(x)=4x^2-2$, and $H_3(x)=8x^3-12x$. The Hermite polynomials are orthogonal. The zeros are real, simple, and interlacing. Theorem 1.4 leads to the following result. We also refer to ([31], Eq. (6.32.6))

Corollary 3.1

Let $n\ge 2$. Then any zero x of $H_n(x)$ satisfies

$$\begin{aligned} \left| x\right| \le \cos \left( \frac{\pi }{n+1}\right) \sqrt{ 2n-2 }. \end{aligned}$$

Proof

Let $g\left( 1\right) =g\left( 2\right) =1$ and $g\left( n\right) =0$ for $n\ge 3$. Then

$$\begin{aligned} \sum _{n=0}^{\infty }Q_{n}^{g}\left( -x^{2}\right) q^{n}=\left( 1 -2\left( x/2\right) \left( -xq\right) +\left( -xq\right) ^{2} \right) ^{-1}=\sum _{n=0}^{\infty }U_{n}\left( x/2\right) \left( -x\right) ^{n}q^{n}. \end{aligned}$$

Therefore, $Q_{n}^{g}\left( -x^{2}\right) =\left( -x\right) ^{n}U_{n}\left( x/2\right)$. Note, that $U_{m} \left( x/2\right) =0$ implies $\left| x\right| \le 2\cos \left( \frac{\pi }{n+1}\right)$ for all $m\le n$. This leads to $x ^{2}\le 4\left( \cos \left( \frac{\pi }{n+1}\right) \right) ^{2}$.

We obtain

$$\begin{aligned} \sum _{n=0}^{\infty }P_{n}^{g}\left( -2x^{2}\right) q^{n}=\exp \left( 2x\left( -x q\right) - \left( -xq \right) ^{2} \right) =\sum _{n=0}^{\infty } \frac{H _{n}\left( x\right) \left( -x \right) ^{n}}{ n!} q^{n}. \end{aligned}$$

Therefore, $P_{n}^{g}\left( -2x^{2}\right) = \left( -x\right) ^{n} H_{n}\left( x\right) /\left( n!\right)$. Theorem 1.4 now ensures that

$$\begin{aligned} 2x ^{2}\le 4\left( \cos \left( \frac{\pi }{n+1}\right) \right) ^{2}\left( n-1\right) . \end{aligned}$$

Finally, $\left| x\right| \le \cos \left( \frac{\pi }{n+1}\right) \sqrt{2n-2}$. $\square$

Remark

It follows from ([31], Eq. (6.32.6)) that the largest zero of $H_{n}\left( x\right)$ is bounded by $\sqrt{2n+1}-6^{-1/2} \left( 2n+1\right) ^{-1/6} i_{1}$ with $i_{1}$ the smallest positive zero of Airy’s function.

In our case, we have

$$\begin{aligned} \cos \left( \frac{\pi }{n+1}\right) \sqrt{2n-2}= \sqrt{2n}-\left( 2n\right) ^{-1/2}+O \left( n^{-3/2}\right) \end{aligned}$$

with Landau’s O notation. The estimate we obtain from the Transfer Lemma is slightly weaker, since the second order term has exponent $-1/2$ instead of $-1/6$.