On the growth and zeros of polynomials attached to arithmetic functions

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions $g$ and $h$, where $g$ is normalized, of moderate growth, and $0<h(n) \leq h(n+1)$. We put $P_0^{g,h}(x)=1$ and \begin{equation*} P_n^{g,h}(x) := \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{equation*} As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $\eta$-function. Here, $g$ is the sum of divisors and $h$ the identity function. Kostant's result on the representation of simple complex Lie algebras and Han's results on the Nekrasov--Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein's $j$-invariant, and Chebyshev polynomials of the second kind.

Here, q := e 2πiτ , Im (τ ) > 0 and r ∈ Z. The coefficients are special values of the D'Arcais polynomials P n (x) [DA13,Ne55,Co74,We06]. It has been recently noticed that the growth and vanishing properties of these polynomials have much in common with properties of other interesting polynomials [HLN19,HN20B]. These include special orthogonal polynomials as associated Laguerre polynomials and Chebyshev polynomials of the second kind. Also included are polynomials attached to reciprocals of the Klein's j-invariant and Eisenstein series [HN20A,HN20C].
In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h inspired by Rota [KRY09].
This definition includes all mentioned examples. Before providing examples and explicit formulas for these polynomials, we give one application for the coefficients of the Dedekind η-function. Let g(n) = σ(n) := d|n d, h(n) = id(n) = n and a n (r) be defined by (1), the nth coefficient of the rth power of the Dedekind ηfunction. Han [Ha10] observed that the Nekrasov-Okounkov hook length formula [NO06,We06] implies that a n (r) = 0 if r > n 2 − 1. This improves previous results by Kostant [Ko04]. In [HN20B] we proved that (3) a n (r) = 0 holds for r > κ · (n − 1) where κ = 15.
Numerical investigations show that κ has to be larger than 9.55 (see Table 5). In this paper we prove that (3) is already true for κ = 10.82. Since the definition of P g,h n (x) is quite abstract, we provide two examples of families of polynomials, to familiarize the reader with the types of polynomials we are studying. At first, they appear to have nothing in common.
Let us start with the Nekrasov-Okounkov hook length formula [NO06]. Let η(τ ) be the Dedekind η-function. Let λ be a partition of n and let |λ| = n. By H(λ) we denote the multiset of hook lengths associated with λ and by P, the set of all partitions. The Nekrasov-Okounkov hook length formula ([Ha10], Theorem 1.2) states that The identity (4) is valid for all z ∈ C. Note that the P σ n (x) are integer-valued polynomials of degree n. From the formula it follows that (−1) n P σ n (x) > 0 for all real x < −(n 2 + 1).
The second example is of a more artificial nature, discovered recently [HN20A], when studying the q-expansion of the reciprocals of Klein's j-invariant and reciprocals of Eisenstein series [BB05,BK17,HN20C]. Let denote Klein's j-invariant. Asai, Kaneko, and Ninomiya [AKN97] proved that the coefficients of the q-expansion of 1/j(τ ) are non-vanishing and have strictly alternating signs. This follows from their result on the zero distribution of the nth Faber polynomials ϕ n (x) and the denominator formula for the monster Lie algebra. The zeros of the Faber polynomials are simple and lie in the interval (0, 1728). They obtained the remarkable identity: Let c * (n) := c(n)/744. Define the polynomials Q j,n (x) by We have proved in [HN20A] that Q j,n (x) = Q γ 2 ,n (x) + 2xQ ′ γ 2 ,n (x) + x 2 2 Q ′′ γ 2 ,n (x), where Q γ 2 ,n (x) are polynomials attached to Weber's cubic root function γ 2 of j in a similar way. We have also proved that Q γ 2 ,n (z) = 0 for all |z| > 82.5. Hence, the identity restates and extends the result of [AKN97]. Now, let g(n) be a normalized arithmetic function with moderate growth, such that ∞ n=1 |g(n)| T n is analytic at T = 0. Then the illustrated examples are special cases of polynomials P g n (x) and Q g n (x) defined by Note that P id n (x) = x L (1) n−1 (−x) are associated Laguerre polynomials (see [HLN19]). Letting g(n) = σ(n), then we recover the polynomials provided by the Nekrasov-Okounkov hook length formula. The polynomials Q id n (x) are related to the Chebyshev polynomials of the second kind [HNT20].
It is easy to see that P g n (z) and Q g n (z) are special cases of polynomials P g,h n (x) defined by the recursion formula (2). Here, P g n (x) = P g,id n (x) and Q g n (x) = P g,1 n (x). In the next section, we state the main results of this paper.

Statement of main results
Let g, h be arithmetic functions. Assume that g be normalized and 0 < h(n) ≤ h(n + 1). It is convenient to extend h by h(0) := 0.
2.1. Improvement A. The following result reproduces our previous result (9), if we choose ε = 1 2 . Theorem 1. Let 0 < ε < 1. Let R > 0 be the radius of convergence of Tε . Then This result can be reformulated in the following way, which is more suitable for applications to growth and non-vanishing properties.
We note that the smallest possible κ is independent of the function h(n). It is also possible to provide a lower bound for the best possible κ.
Proposition 1. The constant κ ε obtained in Theorem 1 has the following lower bound: As a lower bound independent of ε we have 4 |g (2)|.
Proof. If we consider only the first order term of the power series (1−ε)ε in the proposition depending on ε. The minimal value of this lower bound is at ε = 1 2 because of the inequality of arithmetic and geometric means ( Theorem 3. Let 0 < ε < 1. Let R > 0 be the radius of convergence of Theorem 4. Let 0 < ε < 1. Let R > 0 be the radius of convergence of Let 0 < T ε < R be such that G 2 (T ε ) ≤ ε and if |x| > κ h(n − 1) for all n ≥ 1.
Corollary 2. Let κ be chosen as in Theorem 3 or as in Theorem 4. Then . Proposition 2. The constant κ ε obtained in Theorem 3 has the following lower bound: As a lower bound independent of ε we have 3 Proof. If we consider only the second order term of the power series Applying the last inequality now to It is clear that To estimate κ ε independent of ε we consider the right hand side of the last inequality as a function in ε. Thus, we are interested in the minimal value of this function for 0 < ε < 1. The inequality of arithmetic and geometric means yields We obtain 3 2 3 (g (2)) 2 − g (3) + |g (2)|.
2.3. Comparing Improvement A and Improvement B. Let 0 < ε 1 < 1 and T ε 1 as in Theorem 1. For all T ≥ 0 we have that Let ε 2 be such that This shows that we can choose T ε 2 = T ε 1 . Let κ 1,ε and κ 2,ε be the respective constants from Theorems 1 and 3. Then This shows that the minimal value of the κ 2,ε is never larger than the minimal value of the κ 1,ε .
In the previous proof we showed that G 2 (T ε ) < 1 982 < 1 250 for T ε = 87 20000 and κ 2 < 240. This leads to the following  The lower bound is quite close to the optimal value e π √ 3 = 230.764588 . . ..

Associated Laguerre polynomials and Chebyshev polynomials of the second kind.
We briefly recall the definition of associated Laguerre polynomials L (α) n (x) and Chebyshev polynomials U n (x) of the second kind [RS02,Do16]. Both are orthogonal polynomials. We have The Chebyshev polynomials are uniquely characterized by (25) U n (cos(t)) = sin((n + 1)t) sin(t) (0 < t < π).
The Chebyshev polynomials are of special interest in the context of applications, since they are the only classical orthogonal polynomials whose zeros can be determined in explicit form (see Rahman and Schmeisser [RS02], Introduction). Let g(n) = id(n) = n. Then The generating series of the Chebyshev polynomial of the second kind is given by |x|, |q| < 1.
With this we can prove equation (27). We have From this we obtain the following values:  Table 3. Case g(n) = n If we consider the special case ε 1 = 1/2 in Improvement A, we can chose T ε 1 = 2/11 and finally get κ 1 = 11. This leads to several applications. For example, let |x| > (20/3) n then L (1) n (x) = 0 and the estimates hold 3.4. Powers of the Dedekind η-function. Let us recall the well-known identity: The q-expansion of the −zth power of the Euler product defines the D'Arcais polynomials where P σ 0 (x) = 1 and P σ n (x) = x n n k=1 σ(k)P σ n−k (x), as polynomials. Note that these polynomials evaluated at −24 are directly related to the Ramanujan τfunction: τ (n) = P σ n−1 (−24), which gives also a link to the Lehmer conjecture [Le47].
Note only minor further improvements can be achieved. c) Corollary 3 improves our previous result [HN20B], where κ = 15.

Proof of Theorem 1 and Theorem 2
Proof of Theorem 1. The proof will be by induction on n. The case n = 1 is obvious: P g,h 1 (x) − x h(1) P g,h 0 (x) = 0 < ε |x| h(1) P g,h 0 (x) for |x| > κ h(0). Let now n ≥ 2. Then The basic idea for the induction step is to use the inequality We estimate the sum by the following property for 1 ≤ j ≤ n − 1: for |x| > κ h(n − 1). Thus, Further, we have for |x| > κ h(n − 1) ≥ κ h(n − k) for all 2 ≤ k ≤ n by assumption. Using this, we can now estimate the sum by Estimating the sum using the assumption from the theorem we obtain Tε which is equivalent to (1−ε)|x| h(n−1) > 1 Tε and G 1 increases on [0, R) as |g (k + 1)| ≥ 0 for all k ∈ N.
Proof of Theorem 2. Consider the following upper and lower bounds: Applying (10) leads to the desired result.
Proof of Theorem 4. This basically follows from Theorem 3 (see also the proof of Theorem 2).