417 research outputs found
Records on the vanishing of Fourier coefficients of Powers Of the Dedekind Eta Function
In this paper we significantly extend Serre's table on the vanishing
properties of Fourier coefficients of odd powers of the Dedekind eta function.
We address several conjectures of Cohen and Str\"omberg and give a partial
answer to a question of Ono. In the even-power case, we extend Lehmer's
conjecture on the coefficients of the discriminant function to all
non-CM-forms. All our results are supported with numerical data. For example
all Fourier coefficients of the -th power of the Dedekind eta
function are non-vanishing for . We also relate the
non-vanishing of the Fourier coefficients of to Maeda's conjecture.Comment: 13 page
Tur\'an Inequalities for Infinite Product Generating Functions
In the s, Nicolas proved that the partition function is
log-concave for . In \cite{HNT21}, a precise conjecture on the
log-concavity for the plane partition function \func{pp}(n) for was
stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we
provide a general picture. We associate to double sequences
with and polynomials given
by \begin{equation*} \sum_{n=0}^{\infty} P_n^{g_d}(x) \, q^n :=
\func{exp}\left( x \sum_{n=1}^{\infty} g_d(n) \frac{q^n}{n} \right)
=\prod_{n=1}^{\infty} \left( 1 - q^n \right)^{-x f_d(n)}. \end{equation*} We
recover and \func{pp}\left( n\right) =
P_n^{\sigma_2}(1), where and
. Let . Then the sequence
is log-concave for almost all if and only if is divisible by . Let
\func{id}(n)=n. Then P_n^{\func{id}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x),
where denotes the
-associated Laguerre polynomial. In this paper, we invest in Tur\'an
inequalities \begin{equation*} \Delta_{n}^{g_d}(x) := \left( P_n^{g_d}(x)
\right)^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \geq 0. \end{equation*} Let and . Then is divisible by if
and only if for almost all . Let and
. Then the condition on can be reduced to . We determine explicit bounds. As an analogue to Nicolas' result, we have
for g_1= \func{id} that \Delta_{n}^{\func{id}}(x) \geq 0 for all and all
Polynomization of the Chern--Fu--Tang conjecture
Bessenrodt and Ono's work on additive and multiplicative properties of the
partition function and DeSalvo and Pak's paper on the log-concavity of the
partition function have generated many beautiful theorems and conjectures. In
January 2020, the first author gave a lecture at the MPIM in Bonn on a
conjecture of Chern--Fu--Tang, and presented an extension (joint work with
Neuhauser) involving polynomials. Partial results have been announced.
Bringmann, Kane, Rolen and Tripp provided complete proof of the Chern--Fu--Tang
conjecture, following advice from Ono to utilize a recently provided exact
formula for the fractional partition functions. They also proved a large
proportion of Heim--Neuhauser's conjecture, which is the polynomization of
Chern--Fu--Tang's conjecture. We prove several cases, not covered by Bringmann
et.\ al. Finally, we lay out a general approach for proving the conjecture
Polynomization of the Bessenrodt-Ono type inequalities for A-partition functions
For an arbitrary set or multiset of positive integers, we associate the
-partition function (that is the number of partitions of whose
parts belong to ). We also consider the analogue of the -colored
partition function, namely, . Further, we define a family of
polynomials which satisfy the equality
for all and . This paper concerns the
polynomization of the Bessenrodt--Ono type inequality for :
\begin{align*}
f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{align*} where and are
arbitrary positive integers; and delivers some efficient criteria for its
solutions. Moreover, we also investigate a few basic properties related to both
functions and .Comment: 18 pages, 2 figure
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