Tur\'an Inequalities for Infinite Product Generating Functions

In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function \func{pp}(n) for $n >11$ was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences $\{g_d(n)\}_{d,n}$ with $g_d(1)=1$ and $$0 \leq g_{d}\left( n\right) - n^{d}\leq g_{1}\left( n\right) \left( n-1\right) ^{d-1}$$ polynomials $\{P_n^{g_d}(x)\}_{d,n}$ 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 $p(n)= P_n^{\sigma_1}(1)$ and \func{pp}\left( n\right) = P_n^{\sigma_2}(1), where $\sigma_d (n):= \sum_{\ell \mid n} \ell^d$ and $f_d(n)= n^{d-1}$. Let $n \geq 6$. Then the sequence $\{P_n^{\sigma_d}(1)\}_d$ is log-concave for almost all $d$ if and only if $n$ is divisible by $3$. Let \func{id}(n)=n. Then P_n^{\func{id}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x), where $L_{n}^{\left( \alpha \right) }\left( x\right)$ denotes the $\alpha$-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 $n \geq 6$ and $0 \leq x < 2 - \frac{12}{n+4}$. Then $n$ is divisible by $3$ if and only if $\Delta_{n}^{g_d}(x) \geq 0$ for almost all $d$. Let $n \geq 6$ and $n \not\equiv 2 \pmod{3}$. Then the condition on $x$ can be reduced to $x \geq 0$. 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 $x \geq 0$ and all $n$

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 $A$ of positive integers, we associate the $A$-partition function $p_A(n)$ (that is the number of partitions of $n$ whose parts belong to $A$). We also consider the analogue of the $k$-colored partition function, namely, $p_{A,-k}(n)$. Further, we define a family of polynomials $f_{A,n}(x)$ which satisfy the equality $f_{A,n}(k)=p_{A,-k}(n)$ for all $n\in\mathbb{Z}_{\geq0}$ and $k\in\mathbb{N}$. This paper concerns the polynomization of the Bessenrodt--Ono type inequality for $f_{A,n}(x)$: \begin{align*} f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{align*} where $a$ and $b$ are arbitrary positive integers; and delivers some efficient criteria for its solutions. Moreover, we also investigate a few basic properties related to both functions $f_{A,n}(x)$ and $f_{A,n}'(x)$.Comment: 18 pages, 2 figure

Effect of corticosteroids during ongoing drug exposure in pantoprazole-induced interstitial nephritis

Acute interstitial nephritis (AIN) represents a significant cause of acute renal failure in hospital practice. An increasing number of drugs are known to cause AIN. Due to the lack of prospective, randomized clinical trials, the most effective management is still uncertain, especially the role of steroids in the resolution of interstitial nephritis remains to be further defined. We report on a case with pantoprazole-induced interstitial nephritis and on the effect of steroids during ongoing drug exposure. In spite of ongoing drug exposure, steroids led to almost complete resolution of the inflammatory infiltrates. Early diagnosis of interstitial nephritis by renal biopsy and identification of the causative drug and its withdrawal remains the mainstay of treatment. However, the additional use of steroids has the potential to eradicate inflammatory infiltrates more rapidly and completely and may thus be important to minimize subsequent chronic damag