\label{Theorem ID of alpha Cauchy}
The $\alpha$-Cauchy variable $\mathcal{C}_\alpha$ is infinitely divisible if  $1 < \alpha \leq 6/5$.

\begin{theorem}\label{Kristiansen's Theorem}{\cite{Kri94}}
    The independent product $\G_2 \times \W $ is infinitely divisible for any non-negative random variable $\W$. See the beginning of section \ref{section Gamma type} for the definition of $\G_2 $.
\end{theorem}

\begin{equation}\label{Introduction X}
        \mathbb{E}(X^s) =  \frac{\sin(\pi/\a)}{\pi}\frac{ \Gamma\left(\frac{1}{\a} +\frac{ \varepsilon p s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{ \varepsilon p s}{\a}\right)}{ \Gamma(2 + s) },
    \end{equation}

\begin{theorem}\label{Theorem ID of alpha Cauchy}
The $\alpha$-Cauchy variable $\mathcal{C}_\a$ is infinitely divisible if  $1 < \a \leq 6/5$.
\end{theorem}

\begin{theorem}\label{thm Gamma type}
    For positive $a,b,c,d$, we consider the random variable $\X_{a,b,c,d}$ satisfying
       \begin{equation}\label{def Xabcd}
        \mathbb{E}(\X_{a,b,c,d}^s) = \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b.
    \end{equation}
\begin{itemize}
    \item[(I)] $\X_{a,b,c,d}$ does not exist if one of the following conditions holds: \vspace{1mm}
    \begin{itemize}
        \item[(1)]  $d > 2$; \vspace{1mm}
        \item[(2)]   $c < ad$; \vspace{1mm}
        \item[(3)]  $3a + b > c, \; d=2$;  \vspace{1mm}
        \item[(4)] $1 < d \leq 2,\; a +b \geq 1, \;c < ad -d + f(d)$.
    \end{itemize}
    \vspace{1mm}
    \item[(II)] $\X_{a,b,c,d}$ exists if one of the following conditions holds: \vspace{1mm}
    \begin{itemize}
        \item[(1)]  $d \leq 1, c \geq ad$;\vspace{1mm}
        \item[(2)]  $1 < d \leq 2,\; a +b \leq 1, \;c \geq ad -d + f(d)$;\vspace{1mm}
        \item[(3)] $1 < d \leq 2,\; \frac{2c}{d} \geq 3a+b, \;2(\frac{c}{d}-a)(\frac{c}{d}+\frac{1}{2}-a)\geq a+b$.\vspace{1mm}
    \end{itemize}
\end{itemize}
 where $f$ is the function defined in Theorem \ref{Theorem admissible parameter domain}.
\end{theorem}

For $\alpha > 1$, the $\alpha$-Cauchy random variable, denoted by $\mathcal{C}_\alpha$, was introduced by Yano, Yano and Yor \cite{YYY09} in 2009. The density of $\mathcal{C}_\alpha$ is
\begin{equation}\label{eq density alpha Cauchy}
 f_{\mathcal{C}_\alpha}(t) = \frac{\sin(\pi/\alpha)}{2\pi/\alpha}\frac{1}{1+|t|^\alpha}, \quad \alpha > 1, \,\, t \in \mathbb{R}.
\end{equation}
In particular, when $\alpha = 2$, it is
the standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
It is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\mathcal{C}_\alpha$.

A random variable is infinitely divisible (ID) if for any positive integer $n$, it is the sum of $n$ independent and identically distributed random variables. Typical examples of infinitely divisible distributions are normal, Cauchy, half Cauchy, Stable, Gamma, Poisson, and Student-t.
The theory of infinitely divisible distribution is a classic topic in probability theory. In 1929, de Finetti first introduced the concept of infinite divisibility, and then Kolmogorov, Lévy, and Khintchine further developed the theory. Infinitely divisible distributions are closely related to limit theorems and the theory of Lévy processes and have important applications in finance, insurance, biology, physics, and signal processing. Proving or disproving infinite divisibility of a certain distribution can sometimes be quite sophisticated. In some cases, it is more convenient to consider subclasses of infinitely divisible distributions, like self-decomposable (SD) distributions, generalized gamma convolutions (GGC) and hyperbolically completely monotone (HCM) distributions. We refer to Steutel and van Harn \cite{SH04}, Sato \cite{Sat99} and Bondesson \cite{Bon92} for abundant properties of these subclasses.

In 1987,
 Bondesson \cite{Bon87} proved that the half Cauchy variable $ |\mathcal{C}_2|$ is ID. In 1998, Di\'edhiou \cite{Die98} proved that $ |\mathcal{C}_2|$ is SD.
Yano, Yano, and Yor proved $ |\mathcal{C}_\alpha|$ is ID for any $1 < \alpha \leq 2$ by Bondesson's argument, see, Theorem 2.7 in \cite{YYY09}. Furthermore, they
proposed three open questions (\cite[Remark 2.9]{YYY09}):
\begin{enumerate}
 \item Is $\mathcal{C}_\alpha$ SD (or ID at least)?
 \item Is $|\mathcal{C}_\alpha|$ SD (or ID at least)?
 \item Is $|\mathcal{C}_\alpha|^{-p}$ SD (or ID at least) for $p > 0$?
\end{enumerate}

There are very few results on the above questions.
In the present paper, we study the infinite divisibility of
$ \mathcal{C}_\alpha$ and $ |\mathcal{C}_\alpha|^{p}, \, p \in \mathbb{R}$, and partially give positive answers to the questions of Yano, Yano, and Yor.

\subsection{Infinite divisibility of \texorpdfstring{$\alpha$}{}-Cauchy variables}
It is well-known that the Cauchy variable $\mathcal{C}_2$ is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\alpha$. The following theorem gives a positive answer to this question when $1 < \alpha \leq 6/5$.

\begin{abstract}
 We study the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$, $\alpha > 1$. The distribution of $\mathcal{C}_2$ is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\alpha$.
 In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if $1 < \alpha \leq 6/5$, and we give some sufficient conditions for $|\mathcal{C}_\alpha|^p, \, p\in \mathbb{R},$ to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs,
 a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence.
\end{abstract}

\section{Introduction}
For $\alpha > 1$, the $\alpha$-Cauchy random variable, denoted by $\mathcal{C}_\a$, was introduced by Yano, Yano and Yor \cite{YYY09} in 2009. The density of $\mathcal{C}_\a$ is
\begin{equation}\label{eq density alpha Cauchy}
 f_{\mathcal{C}_\a}(t) = \frac{\sin(\pi/\a)}{2\pi/\a}\frac{1}{1+|t|^\a}, \quad \alpha > 1, \,\, t \in \mathbb{R}.
\end{equation}
In particular, when $\alpha = 2$, it is
the standard Cauchy distribution, also called the Lorentzian distribution or Lorentz distribution. It is the ratio of two independent normal random variables. It is a continuous distribution describing resonance behavior. It also describes the distribution of horizontal distances at which a line segment tilted at a random angle cuts the x-axis.
It is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. In this paper, we study the infinite divisibility of $\mathcal{C}_\a$.

\subsection{Infinite divisibility of \texorpdfstring{$\a$}{}-Cauchy variables}
It is well-known that the Cauchy variable $\mathcal{C}_2$ is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\a$. The following theorem gives a positive answer to this question when $1 < \a \leq 6/5$.

Yano, Yano and Yor introduced the $\alpha$-Cauchy variable to study the first hitting times of point $T_{\{a\}}(X_\alpha)$ for one-dimensional \textbf{symmetric} stable L\'evy process of index $\alpha$ ($1< \alpha \leq 2$) starting from zero, denoted by $X_\alpha = (X_\alpha(t): t\geq 0)$.
In 2019, Letemplier and Simon \cite{LS19} studied the first hitting time of zero $\tau(\alpha, \rho)$ for a real strictly $\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\tau(\alpha, \frac{1}{2})$ coincides with $T_{\{-1\}}(X_\alpha)$ mentioned above. They conjectured that $\tau(\alpha, \rho)$ is infinitely divisible for $\alpha \in (1,2]$.

We note that both $\mathcal{C}_\a$ and $|\mathcal{C}_\a|$ are not infinitely divisible when $\a$ is large enough. Indeed,
the fractional moments of the half $\alpha$-Cauchy variable $ |\mathcal{C}_\a|$ are of the form
\begin{equation*}
 \mathbb{E}\left[|\mathcal{C}_\a|^s\right] = \frac{\sin(\pi/\a)}{\pi} \Gamma\left(\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right), \quad -1 < s < \a-1.
\end{equation*}
The right hand side tends to $1/(1+s)$ as $\a$ tends to infinity, thus
\begin{equation}
 |\mathcal{C}_\a| \claw \mathcal{U}, \qquad \text{as} \, \,\,\a \rightarrow \infty,
\end{equation}
where $\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \mathcal{U} $ is bounded, $\mathcal{U}$ is not ID; see Sato \cite[Corollary 24.4]{Sat99}. Similarly, as $\alpha$ tends to infinity, $\mathcal{C}_\a$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID.
It is probable that $\mathcal{C}_\a$ is ID if and only if $1 < \a \leq 2$.

\begin{theorem}\label{Theorem ID of half alpha Cauchy}
Let $p \in (0, \infty)$.
The random variable $|\mathcal{C}_\a |^{ \varepsilon p}, \varepsilon = \pm 1,$ is infinitely divisible if
\begin{equation}\label{eq range p}
p \geq
 \begin{cases}
 (\a + 1)/3 ,& \quad \varepsilon = 1, \, 1 < \a \leq 2. \\
 \a/2 ,& \quad \varepsilon = 1, \, \a > 2. \\
 \a/2 ,& \quad \varepsilon = - 1, \, 1 < \a \leq 2. \\
 (2\a - 1)/3 ,& \quad \varepsilon = -1, \, \a > 2. \\
 \end{cases}
\end{equation}
\end{theorem}

\section{Proofs of Theorems \ref{Theorem ID of alpha Cauchy} and \ref{Theorem ID of half alpha Cauchy}}
\label{section proofs}
\subsection{Proof of Theorem \ref{Theorem ID of alpha Cauchy}}
We apply Proposition \ref{Proposition Steutel coro VI.4.8} to prove Theorem \ref{Theorem ID of alpha Cauchy}. Let $Y$ be a positive random variable with density $f$ given by
$$ f(x) = \frac{1}{2}(1+x)e^{-x}, \qquad x >0. $$
The fractional moments of $Y$ are
\begin{equation*}
 \mathbb{E}[Y^s] = \frac{1}{2}(s+2)\Gamma(s+1), \quad s > -1.
\end{equation*}
 Let $1 < \alpha \leq 2$.
Recall that
\begin{equation}\label{eq Mellin alpha Cauchy}
 \mathbb{E}\left[|\mathcal{C}_\a|^s\right] = \frac{\sin(\pi/\a)}{\pi} \Gamma\left(\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right), \quad -1 < s < \a-1.
\end{equation}
Therefore, by Proposition \ref{Proposition Steutel coro VI.4.8}, if there exists a symmetric distribution
$V_\alpha$ having fractional moments,
\begin{equation}
 \mathbb{E} [|V_\alpha|^s] = \mathbb{E}\left[|\mathcal{C}_\a|^s\right] / \mathbb{E}[Y^s] = \frac{2\alpha \sin(\pi/\a)}{\pi} \frac{\Gamma\left(1+\frac{1}{\a} +\frac{s}{\a}\right)\Gamma\left(1 - \frac{1}{\a} - \frac{s}{\a}\right)}{\Gamma(3+s)},
\end{equation}
then $\mathcal{C}_\alpha$ is infinitely divisible. Compared with \eqref{def Xabcd}, we see that
\begin{equation}
 |V_\alpha| \elaw \X_{1+1/\alpha, 1-1/\alpha, 3, \alpha}^{1/\alpha}.
\end{equation}
By \eqref{fact equiv}, the existence of $V_\alpha$ is equivalent to the non-negativity of $E_{\alpha, 2+\alpha}^{2}(-t)$ for any $t>0$.
By formula (5.1.31) in \cite{GKMR20}, we have
\begin{equation}
\label{LT of 3ML}
 \int_0^{\infty} e^{-st} e^{-t} t^{\beta-1} E^\gamma_{\alpha, \beta}(-t^\alpha)dt = \frac{(s+1)^{-\beta}}{(1+(s+1)^{-\alpha})^\gamma}
\end{equation}
where $\alpha > 0, \beta > 0, \gamma > 0, s > 0$.
Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero, see, e.g. Proposition 1.2 in \cite{SSV12}. Moreover, a function is completely monotone if and only if it is the Laplace transform of positive measure on $[0, \infty)$ (see Theorem 1.4 in \cite{SSV12}), and we have the following equivalence
$$ \text{ $E_{\alpha, 2+\alpha}^{2}(-t) \geq 0$ for any $t>0$ $\Longleftrightarrow$
$ \frac{(s+1)^{-2-\alpha}}{(1+(s+1)^{-\alpha})^2} $ is completely monotone. } $$
The multiplication of completely monotone functions is still completely monotone, see \cite[Corollary 1.6]{SSV12}. Therefore, the
complete monotonicity of $\frac{(s+1)^{-1 - \alpha/2}}{1+(s+1)^{-\alpha} }$ implies that of $\frac{(s+1)^{-2-\alpha}}{(1+(s+1)^{-\alpha})^2}$.
 Using again \eqref{LT of 3ML}, we obtain that
$$ \text{$E_{\alpha, 1+\alpha/2}(-t) \geq 0$ for any $t>0$ } \text{$\Longleftrightarrow$
$\frac{(s+1)^{-1 - \alpha/2}}{1+(s+1)^{-\alpha} }$ is completely monotone.} $$
If $1< \alpha \leq 6/5$, then $1+\alpha/2 \geq 4\alpha/3$, by Theorem \ref{Theorem admissible parameter domain}, we have
$E_{\alpha, 1+\alpha/2}(-t) \geq 0$ for any $t>0$.
This completes the proof.

Let $\T_{\nu}$ be the Student-t random variable with density
$$ \frac{\Gamma(\frac{\nu +1}{2})}{\sqrt{\pi \nu} \Gamma(\frac{\nu }{2})} \left( 1 + \frac{t^2}{\nu}\right)^{-\frac{\nu +1}{2}}, \quad \nu > 0, \; t \in \mathbb{R}.$$
Note that $|\T_{1}| \elaw |\mathcal{C}_2|$.
\begin{theorem}\label{thm half student}
 For $0 < \nu \leq 1$, the half Student-t random variable $|\T_\nu|$ is infinitely divisible.
\end{theorem}

Yano, Yano and Yor introduced the $\alpha$-Cauchy variable to study the first hitting times of point $T_{\{a\}}(X_\alpha)$ for one-dimensional \textbf{symmetric} stable L\'evy process of index $\alpha$ ($1< \alpha \leq 2$) starting from zero, denoted by $X_\alpha = (X_\alpha(t): t\geq 0)$.
In 2019, Letemplier and Simon \cite{LS19} studied the first hitting time of zero $\tau(\alpha, \rho)$ for a real strictly $\alpha$-stable process (can be asymmetric) starting from one. The symmetric case $\tau(\alpha, \frac{1}{2})$ coincides with $T_{\{-1\}}(X_\alpha)$ mentioned above. They conjectured that $\tau(\alpha, \rho)$ is infinitely divisible for $\alpha \in (1,2]$.

More precisely, by formula (5.12) in \cite{YYY09}, the relation between the first hitting time of point for stable processes and the alpha Cauchy distributions is as follows
$$\hat{X}_\alpha ( T_{\{a\}}(X_\alpha)) \stackrel{(d)}{=} |a| \mathcal{C}_\alpha, $$
where $\hat{X}_\alpha$ is an independent copy of $X_\alpha$. Using Bochner’s subordination -see e.g. Theorem 30.1 in \cite{Sat99}- the infinite divisibility of $T_{\{a\}}(X_\alpha)$ would entail the infinite divisibility of $ \mathcal{C}_\alpha$. Vise versa, the infinite divisibility of $\mathcal{C}_\alpha$ would support the conjecture on the infinite divisibility of $T_{\{a\}}(X_\alpha)$.

We note that both $\mathcal{C}_\alpha$ and $|\mathcal{C}_\alpha|$ are not infinitely divisible when $\alpha$ is large enough. Indeed,
the fractional moments of the half $\alpha$-Cauchy variable $ |\mathcal{C}_\alpha|$ are of the form
\begin{equation*}
    \mathbb{E}\left[|\mathcal{C}_\alpha|^s\right] =  \frac{\sin(\pi/\alpha)}{\pi} \Gamma\left(\frac{1}{\alpha} +\frac{s}{\alpha}\right)\Gamma\left(1 - \frac{1}{\alpha} - \frac{s}{\alpha}\right), \quad -1 < s < \alpha-1.
\end{equation*}
The right hand side tends to $1/(1+s)$ as $\alpha$ tends to infinity, thus
\begin{equation}
    |\mathcal{C}_\alpha| \stackrel{(d)}{\longrightarrow} \mathcal{U}, \qquad \text{as} \, \,\,\alpha \rightarrow \infty,
\end{equation}
where $\mathcal{U}$ is a uniform random variable on $(0,1)$. Since the support of $ \mathcal{U} $ is bounded, $\mathcal{U}$ is not ID; see Sato \cite[Corollary 24.4]{Sat99}. Similarly, as $\alpha$ tends to infinity, $\mathcal{C}_\alpha$ tends to a uniform random variable in $(-1,1)$ in distribution, which is not ID.
It is probable that $\mathcal{C}_\alpha$ is ID if and only if  $1 < \alpha \leq 2$.

\subsection{Infinite divisibility of powers of half \texorpdfstring{$\alpha$}{}-Cauchy variable}
We know from Bondesson \cite{Bon87} that $ |\mathcal{C}_\alpha|$ is ID for $1 < \alpha \leq 2$. We have seen that $|\mathcal{C}_\alpha|$ is not infinitely divisible when $\alpha$ is large enough. It is also probable that $|\mathcal{C}_\alpha|$ is ID if and only if  $1 < \alpha \leq 2$.

The following theorem describes the infinite divisibility of powers of half $\alpha$-Cauchy variable.

\begin{theorem}\label{Theorem ID of half alpha Cauchy}
Let $p \in (0, \infty)$.
The random variable $|\mathcal{C}_\alpha |^{ \varepsilon p}, \varepsilon = \pm 1,$ is infinitely divisible if
\begin{equation}\label{eq range p}
p \geq
    \begin{cases}
   (\alpha + 1)/3 ,& \quad \varepsilon = 1, \, 1 < \alpha \leq 2. \\
     \alpha/2 ,& \quad \varepsilon = 1, \, \alpha > 2. \\
     \alpha/2 ,& \quad \varepsilon = - 1, \, 1 < \alpha \leq 2. \\
     (2\alpha - 1)/3 ,& \quad \varepsilon = -1, \, \alpha > 2. \\
    \end{cases}
\end{equation}
\end{theorem}

In particular, this theorem recovers and extends the result of Bondesson mentioned above.

\subsection{Sketch of the proofs} We first sketch the proof of Theorem \ref{Theorem ID of alpha Cauchy}.
We will prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition \ref{Proposition Steutel coro VI.4.8}.
This motivates us to study the existence condition of a class of positive random variables having moments of Gamma type satisfying
       \begin{equation}\label{Introduction Xabcd}
        \mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
    \end{equation}
for positive $a,b,c,d$. We finish the proof by non-negativity conditions of the Mittag-Leffler function.

We then sketch the proof of Theorem \ref{Theorem ID of half alpha Cauchy}.
By Kristiansen's Theorem (see Theorem \ref{Kristiansen's Theorem}), the existence of a random variable $X$ satisfying
         \begin{equation}\label{Introduction X}
        \mathbb{E}(X^s) =  \frac{\sin(\pi/\alpha)}{\pi}\frac{ \Gamma\left(\frac{1}{\alpha} +\frac{ \varepsilon p s}{\alpha}\right)\Gamma\left(1 - \frac{1}{\alpha} - \frac{ \varepsilon p s}{\alpha}\right)}{ \Gamma(2 + s) },
    \end{equation}
for $s$ in some interval, will imply the infinite divisibility of $|\mathcal{C}_\alpha|^{\varepsilon p}, \varepsilon = \pm 1$. Therefore, we thoroughly investigate the conditions for the existence of random variables of the form \eqref{Introduction Xabcd}, and formulate our results as Theorem \ref{thm Gamma type}. Applying this theorem to \eqref{Introduction X}, we can prove Theorem \ref{Theorem ID of half alpha Cauchy}.

The rest of this paper is organized as follows.
 In Section \ref{section Preliminaries}, we introduce the three-parametric Mittag-Leffler function. We then state some
useful known propositions and theorems that will be used for the proofs of our theorems.
In Section \ref{section Gamma type},
we explicitly state and prove some necessary conditions and some sufficient conditions for the existence of some random variables having moments of Gamma type satisfying \eqref{Introduction Xabcd}, which serve as the foundation for proving the above two theorems. In Section \ref{section proofs}, we prove Theorems \ref{Theorem ID of alpha Cauchy} and \ref{Theorem ID of half alpha Cauchy}. In section \ref{section apllication of new Gamma type}, we give three other applications of Theorem \ref{thm Gamma type}.

To prove Theorem~\ref{Theorem ID of alpha Cauchy}, they "prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture; see Proposition~\ref{Proposition Steutel coro VI.4.8}." This leads them to "study the existence condition of a class of positive random variables having moments of Gamma type" with
\[
\mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
\]
for positive $a,b,c,d$. They "finish the proof by non-negativity conditions of the Mittag-Leffler function."

To prove the main theorem, they "prove the infinite divisibility of $\alpha$-Cauchy ($1<\alpha \leq 6/5$) by proving that it is a sym-gamma(2) mixture". We use the following proposition.

\begin{proposition}
    \label{Proposition Steutel coro VI.4.8}{\cite[Corollary VI.4.8]{SH04}}
    A random variable $X$ with $X \elaw Z \times Y$ is infinitely divisible
if $Z$ and $Y$ are independent, $Z$ has a symmetric distribution and $Y$ has the
density $f$ given by
$$ f(x) = \frac{1}{2}(1+x)e^{-x}, \qquad x >0.  $$
\end{proposition}

This leads them to "study the existence condition of a class of positive random variables having moments of Gamma type" with
\[
\mathbb{E}(\X_{a,b,c,d}^s) =  \frac{ \Gamma(c) }{ \Gamma(a)\Gamma(b)} \frac{ \Gamma(a + s)\Gamma(b-s)}{ \Gamma(c+ d s) }, \quad -a < s < b,
\]
for positive $a,b,c,d$. They "finish the proof by non-negativity conditions of the Mittag-Leffler function."

Question:
Let \(\alpha\) satisfy \(1<\alpha\le 6/5\), and let \(\mathcal C_\alpha\) denote the \(\alpha\)-Cauchy random variable with density
\[
f_{\mathcal C_\alpha}(t)=\frac{\sin(\pi/\alpha)}{2\pi/\alpha}\frac{1}{1+|t|^\alpha},\qquad t\in\mathbb R.
\]
Which of the following conclusions about \(\mathcal C_\alpha\) is valid under these hypotheses?

The random variable \(\mathcal C_\alpha\) is infinitely divisible; that is, for every positive integer \(n\), its distribution can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.

Choices:
B. The random variable \(\mathcal C_\alpha\) is infinitely divisible for every \(\alpha>1\); that is, for every positive integer \(n\), its distribution can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.
C. There exists at least one positive integer \(n\ge 2\) such that the distribution of \(\mathcal C_\alpha\) can be written as the distribution of a sum of \(n\) independent and identically distributed random variables.
D. The random variable \(\mathcal C_\alpha\) is self-decomposable; equivalently, for every \(c\in(0,1)\), there exists an independent random variable \(Y_c\) such that \(\mathcal C_\alpha\stackrel{d}=c\mathcal C_\alpha+Y_c\).
E. For each \(1<\alpha\le 6/5\), there exists a decomposition \(\mathcal C_\alpha\stackrel{d}=Z\times Y\) with \(Z\) and \(Y\) independent, \(Y\) having density \(\tfrac12(1+x)e^{-x}\) on \((0,\infty)\), and \(Z\) symmetric and itself infinitely divisible.

for-every-n quantifier in infinite divisibility weakened to existence of one n

ID conclusion replaced by stronger self-decomposability

Steutel product criterion only requires symmetric Z, not Z itself infinitely divisible

The stem states only the definition and parameter range; it does not explicitly reveal infinite divisibility or otherwise give away choice A. The threshold 6/5 may hint at a known theorem, but there is no direct answer leakage.

Choice A is essentially a direct restatement of the underlying theorem for the given hypothesis range, so the item mainly tests recognition/recall rather than selecting among genuinely competing derived conclusions.

There is some reasoning pressure because one must distinguish the exact theorem statement from a stronger overgeneralization (B), a weaker true statement (C), and stronger/adjacent properties (D, E). However, the main task is still theorem identification rather than substantial generation of a conclusion.

The distractors are mathematically meaningful and target common errors: extending the parameter range too far, settling for a weaker consequence, confusing infinite divisibility with self-decomposability, or accepting an overly strong technical decomposition statement.

A mathematically polished item with strong distractors and little answer leakage, but it is largely tautological because the correct option restates the theorem almost verbatim.


[$=$\,Theorem~\ref{thm:QK-Whitney_A}\,$+$\,Theorem~\ref{thm:QK-Whitney_B}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\underline{\BC}^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}

\begin{thm} \label{thm:Whitney_presentation}
There exists an $R(T)$-algebra isomorphism $R_{Q}^{\SW}/I_{Q}^{\SW} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{thm}

\begin{thm}[{\cite[Theorem~4.1]{GMSXZZ2}}] \label{thm:Nakayama}

Assume that we are given an isomorphism of $R(T)$-algebras
\begin{equation}
\Phi: R(T)[m_{1}, \ldots, m_{s}]/\langle P_{1}, \ldots, P_{r} \rangle \xrightarrow{\sim} K_{T}(G/P)
\end{equation}
where $m_{1}, \ldots, m_{s}$ are indeterminates, and the $P_{i}$ are polynomials in the variables $m_{j}$ with coefficients in $R(T)$.
Consider any power series
\begin{equation}
\ti{P}_{i} = \ti{P}_{i}(m_{1}, \ldots, m_{s}; Q_{1}, \ldots, Q_{k}) \in (R(T)[m_{1}, \ldots, m_{s}])\bra{Q_{1}, \ldots, Q_{k}}
\end{equation}
such that
\begin{itemize}
\item $\ti{P}_{i}(m_{1}, \ldots, m_{s}; 0, \ldots, 0) = P_{i}(m_{1}, \ldots, m_{s})$ for all $m_{1}, \ldots, m_{s}$ and all $i$, and
\item $\ti{P}_{i}(m_{1}, \ldots, m_{s}; Q_{1}, \ldots, Q_{k}) = 0$ in $QK_{T}(G/P)$, where the $m_{i}$ are regarded as elements in $QK_{T}(G/P)$ via the isomorphism $\Phi$.
\end{itemize}
Then $\Phi$ lifts to an isomorphism of $R(T)\bra{Q_{1}, \ldots, Q_{k}}$-algebras
\begin{equation}
\ti{\Phi}: (R(T)\bra{Q_{1}, \ldots, Q_{k}})[m_{1}, \ldots, m_{s}]/\langle \ti{P}_{1}, \ldots, \ti{P}_{r} \rangle \xrightarrow{\sim} QK_{T}(G/P).
\end{equation}
\end{thm}

\begin{ithm}[$=$\,Theorem~\ref{thm:Whitney_presentation}] \label{ithm:Whitney_presentation}
Define a ring $R_{Q}^{\SW}$ by
\begin{equation}
R_{Q}^{\SW} := (R(T)\bra{Q_{1}, \ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \mid 1 \le k \le n, \ 1 \le d \le k, \ 1 \le j \le n].
\end{equation}
Take an ideal $I_{Q}^{\SW}$ of $R_{Q}^{\SW}$ generated by
\begin{align}
& F_{d}^{k} - \left( F_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
& G_{d}^{k} - \left( G_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
\begin{split}
&\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \Biggl( e_{d}(T_{1}, T_{2}, \ldots, T_{n}, T_{n}^{-1}, \ldots, T_{2}^{-1}, T_{1}^{-1}) \\
& \quad - \sum_{p = 1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} \sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \Biggr) \quad \text{for $1 \le d \le n$},
\end{split} \\
& F_{1}^{1} - x_{1}, \\
& G_{1}^{1} - y_{1}, \\
& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \quad \text{for $1 \le j \le n$};
\end{align}
here, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \in R(T)$ for $1 \le k \le 2n$ are obtained from the decomposition of $\BC^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \ldots$, $T_{2n} = T_{1}^{-1}$.
In addition, we understand that
$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \le k \le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \le k \le n$.

Then there exists an $R(T)$-algebra isomorphism $R_{Q}^{\SW}/I_{Q}^{\SW} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{ithm}

\begin{ithm}[$=$\,Theorem~\ref{thm:QK-Whitney_A}\,$+$\,Theorem~\ref{thm:QK-Whitney_B}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}
\end{ithm}

Let $X$ be a partial flag manifold of arbitrary type.
Givental \cite{Givental} and Lee \cite{Lee} introduced the \emph{torus-equivariant quantum $K$-ring} $QK_{T}(X)$ of the flag manifold $X$,
which is a deformation of the ordinary torus-equivariant $K$-ring $K_{T}(X)$; here $T$ denotes the torus.
The product $\star$ of $QK_{T}(X)$, called the \emph{quantum product}, is defined by using \emph{$K$-theoretic Gromov--Witten invariants}.
To understand the quantum product, presentations of $QK_{T}(X)$ with generators and relations have been studied, including Borel-type, Coulomb branch, and Whitney presentations. In this paper, we focus on the Whitney presentation.

For a $T$-equivariant vector bundle $\mathcal{E}$ over $X$, we denote by $[\mathcal{E}]$ the class of $\mathcal{E}$ in $K_{T}(X)$.
The \emph{Hurzebruch $\lambda_{y}$-class} $\lambda_{y}(\mathcal{E}) \in K_{T}(X)[y]$ of a $T$-equivariant vector bundle $\mathcal{E}$, introduced in \cite{Hirzebruch}, is defined as
\begin{equation}
\lambda_{y}(\mathcal{E}) := 1 + y [\mathcal{E}] + y^{2} \left[ \bigwedge\nolimits^{2} \mathcal{E} \right] + \cdots + y^{\rk(\mathcal{E})} \left[ \bigwedge\nolimits^{\rk(\mathcal{E})} \mathcal{E} \right],
\end{equation}
where $\rk(\mathcal{E})$ denotes the rank of $\mathcal{E}$.
For a short exact sequence $0 \rightarrow \CE_{1} \rightarrow \CE_{2} \rightarrow \CE_{3} \rightarrow 0$ of torus-equivariant vector bundles over $X$, we have
\begin{equation} \label{eq:Whitney}
\lambda_{y}(\CE_{1}) \cdot \lambda_{y}(\CE_{3}) = \lambda_{y}(\CE_{2})
\end{equation}
in $K_{T}(X)[y]$, where $\cdot$ denotes the product of $K_{T}(X)$ (or $K_{T}(X)[y]$).
The identity \eqref{eq:Whitney} is called the \emph{$K$-theoretic Whitney relation}.

Our interest is to quantize the above $K$-theoretic Whitney relations.
More precisely, we study the quantum product $\lambda_{y}(\CE_{1}) \star \lambda_{y}(\CE_{3})$ in $QK_{T}(X)[y]$.
Let us consider the case $X = \Fl(r_{1}, \ldots, r_{k}; n)$ is a $k$-step flag manifold of type $A$ parametrizing all sequences $(0 \subset V_{1} \subset \cdots \subset V_{k} \subset \mathbb{C}^{n})$ of vector subspaces of $\mathbb{C}^{n}$ with $\dim V_{i} = r_{i}$ for $1 \le i \le k$.
Then we can take a tautological filtration $0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{k} \subset \underline{\BC}^{n}$ of the trivial bundle $\underline{\BC}^{n} = X \times \mathbb{C}^{n}$.
Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ}) conjectured the result of the quantum product of $\lambda_{y}$-classes associated to tautological vector bundles as follows:
\begin{equation} \label{eq:QK-Whitney_typeA}
\lambda_{y}(\CS_{j}) \star \lambda_{y}(\CS_{j+1}/\CS_{j}) = \lambda_{y}(\CS_{j+1}) - y^{r_{j+1}-r_{j}} \dfrac{Q_{j}}{1-Q_{j}} \det(\CS_{j+1}/\CS_{j}) \star (\lambda_{y}(\CS_{j}) - \lambda_{y}(\CS_{j-1}))
\end{equation}
for $1 \le j \le k$, here $Q_{1}, \ldots, Q_{k}$ are Novikov variables of $QK_{T}(X)$, and $\lambda_{y}(\CS_{n+1}) = \lambda_{y}(\underline{\BC}^{n}) = \prod_{1 \le j \le n} (1 + yT_{j})$, where $T_{j} \in R(T)$ for $1 \le j \le n$ are given by the decomposition of $\mathbb{C}^{n}$ into one-dimensional $T$-modules (see \cite[Conjecture~1.1]{GMSXZZ}).
This identity is called the \emph{quantum $K$-theoretic Whitney relation} (the \emph{$QK$-Whitney relation} for short).
This $QK$-Whitney relation was proved for Grassmannians $X = \mathrm{Gr}(k, n) := \Fl(k; n)$ by Gu, Mihalcea, Sharpe, and Zou (\cite{GMSZ2}), for $1 \le k \le n$, for the incidence variety $X = \Fl(1, n-1; n)$ by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ}), and for arbitrary partial flag manifolds in type $A$ by Huq-Kuruvilla (\cite{HK}).
In addition, Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou showed in \cite[Theorem~1.4]{GMSXZZ} that this conjecture holds under the quantum $K$-theoretic divisor axiom stated by Buch and Mihalcea, the identity for three-pointed $K$-theoretic Gromov--Witten invariants.
Recently, Lenart, Naito, Sagaki, and Xu proved the quantum $K$-theoretic divisor axiom in \cite{LNSX} (not only for $X = \Fl(r_{1}, \ldots, r_{k}; n)$ but also for arbitrary $X$).
As a benefit of describing $QK$-Whitney relations, we can obtain a presentation of $QK_{T}(\Fl(r_{1}, \ldots, r_{k}; n))$ as a quotient ring of a polynomial ring. In fact, $QK$-Whitney relations give a complete set of defining relations of $QK_{T}(\Fl(r_{1}, \ldots, r_{k}, n))$ (\cite[Corollary~5.4]{GMSXZZ2}).

Since the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$.
Let $X = \FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences
\begin{equation}
(0 \subset V_{1} \subset \cdots \subset V_{n} \subset \mathbb{C}^{2n})
\end{equation}
of isotropic subspaces $V_{1}, \ldots, V_{n}$ with respect to a fixed symplectic form with $\dim V_{k} = k$ for $1 \le k \le n$.
Then we can take a tautological filtration
\begin{equation}
0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{n} \subset \underline{\BC}^{2n}
\end{equation}
of subbundles of the trivial bundle $\underline{\BC}^{2n} = \FlC{n} \times \mathbb{C}^{2n}$ over $\FlC{n}$ such that $\rk(\CS_{k}) = k$ for $1 \le k \le n$.
For a $T$-equivariant vector bundle $\mathcal{E}$ over $\FlC{n}$, we denote by $\mathcal{E}^{\vee}$ its dual bundle.
We denote by $Q_{1}, \ldots, Q_{n}$ the Novikov variables of $QK_{T}(\FlC{n})$. We understand that $Q_{0} = 0$ and $\CS_{-1} = 0$.
Our main results are as follows.

Since the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$.
Let $X = \FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences
\begin{equation}
(0 \subset V_{1} \subset \cdots \subset V_{n} \subset \BC^{2n})
\end{equation}
of isotropic subspaces $V_{1}, \ldots, V_{n}$ with respect to a fixed symplectic form with $\dim V_{k} = k$ for $1 \le k \le n$.
Then we can take a tautological filtration
\begin{equation}
0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{n} \subset \udBC^{2n}
\end{equation}
of subbundles of the trivial bundle $\udBC^{2n} = \FlC{n} \times \BC^{2n}$ over $\FlC{n}$ such that $\rk(\CS_{k}) = k$ for $1 \le k \le n$.
For a $T$-equivariant vector bundle $\CE$ over $\FlC{n}$, we denote by $\CE^{\vee}$ its dual bundle.
We denote by $Q_{1}, \ldots, Q_{n}$ the Novikov variables of $QK_{T}(\FlC{n})$. We understand that $Q_{0} = 0$ and $\CS_{-1} = 0$.
Our main results are as follows.

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

Then we observe that relations \eqref{eq:QK-Whitney_1_intro}, \eqref{eq:QK-Whitney_2_intro}, and \eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\FlC{n})$, which we call the \emph{Whitney-type presentation}.

Note that the above identities are similar to those for the type $A$ flag manifold; see \eqref{eq:QK-Whitney_typeA}.
On the other hand, the following identity, analogous to \eqref{eq:Whitney_classical_3}, is specific to the type $C$ case.
\begin{thm} \label{thm:QK-Whitney_B}
In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}
\end{thm}
We call identities \eqref{eq:QK-Whitney_1}, \eqref{eq:QK-Whitney_2}, and \eqref{eq:QK-Whitney_3} \emph{quantum $K$-theoretic Whitney relations} (\emph{$QK$-Whitney relations} for short) for $\FlC{n}$.

\subsection{Proof of Theorem~\ref{thm:QK-Whitney_A}}
First, we give a proof \eqref{eq:QK-Whitney_1}.
For this purpose, we show the following identity.
\begin{lem}
In $QK_{T}(\FlC{n})$, for $1 \le k \le n$ and $1 \le d \le k$, we have
\begin{equation} \label{eq:QK-Whitney_lem_1}
\left[ \extprod^{d} \CS_{k} \right] = \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star \left( \left[ \extprod^{d-1} \CS_{k-1} \right] - Q_{k-1} \left[ \extprod^{d-1} \CS_{k-2} \right] \right).
\end{equation}
\end{lem}
\begin{proof}
We compute as follows:
\begin{align}
& \left[ \extprod^{d} \CS_{k} \right] \\
& = \sum_{\substack{J \subset [1, k] \\ |J| = d}} [\CO_{G/B}(-\ve_{J})] \\
& = \sum_{\substack{J \subset [1, k-1] \\ |J| = d}} [\CO_{G/B}(-\ve_{J})] + \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J \sqcup \{k\}})] \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] + \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{k})] \star [\CO_{G/B}(-\ve_{J})] \\
& \qquad + \frac{1}{1-Q_{k-1}} \sum_{\substack{J \subset [1, k-1] \\ k-1 \in J \\ |J| = d-1}} [\CO_{G/B}(-\ve_{k})] \star [\CO_{G/B}(-\ve_{J})]
\end{split} \qquad \text{by \eqref{eq:multiple_1}} \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] \\
& \qquad - \frac{Q_{k-1}}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})]
\end{split} \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] \\
& \qquad + \frac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \left( \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] - Q_{k-1} \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] \right)
\end{split} \\
&= \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star \left( \left[ \extprod^{d-1} \CS_{k-1} \right] - Q_{k-1} \left[ \extprod^{d-1} \CS_{k-2} \right] \right),
\end{align}
as desired.
\end{proof}

\subsection{Proof of Theorem~\ref{thm:QK-Whitney_B}}
We prove \eqref{eq:QK-Whitney_3} by comparing coefficients of $y^{d}$ of both sides for each $1 \le d \le 2n$.
\begin{lem}
In $QK_{T}(\FlC{n})$, for $1 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Whitney_computation_lem1}
\begin{split}
& \sum_{\substack{k, l \ge 0 \\ k+l = d}} \left[ \extprod^{k} \CS_{n} \right] \star \left[ \extprod^{l} \CS_{n}^{\vee} \right] = \left[ \extprod^{d} \udBC^{2n} \right] \\
& \hspace{20mm} - \sum_{p = 1}^{n} \frac{Q_{p}Q_{p+1} \cdots Q_{n}}{1 - Q_{p}} \sum_{\substack{J, K \subset [1, n] \\ |J| + |K| = d \\ \max J = \max K = p}} [\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(\ve_{K})].
\end{split}
\end{equation}
\end{lem}
\begin{proof}
In what follows, we set $\overline{K} := \{\overline{j} \mid j \in K \}$ for $K \subset [1, n]$.
Let $J, K \subset [1, n]$.
By \eqref{eq:multiple_1}, we have
\begin{align}
[\CO_{G/B}(-\ve_{J})] &= \left( \prod_{\substack{2 \le j \le n \\ j-1 \in J, j \in J}} \frac{1}{1-Q_{j-1}} \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right) \\
&= \left( \prod_{1 \le j \le n} \varphi_{J}(j) \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right).
\end{align}
Also, by \eqref{eq:multiple_2}, we have
\begin{align}
[\CO_{G/B}(\ve_{K})] &= \left( \prod_{\substack{2 \le j \le n \\ j-1\in K, j \in K}} \frac{1}{1-Q_{j-1}} \right) \left( \qprod_{j \in K} [\CO_{G/B}(\ve_{j})] \right) \\
&= \left( \prod_{\overline{n} \le j \le \overline{1}} \varphi_{\overline{K}}(j) \right) \left( \qprod_{j \in \overline{K}} [\CO_{G/B}(-\ve_{j})] \right).
\end{align}
Hence we have
\begin{equation}
[\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(\ve_{K})] = \left( \prod_{1 \le j \le n} \varphi_{J}(j) \right) \left( \prod_{\overline{n} \le j \le \overline{1}} \varphi_{\overline{K}}(j) \right) \left( \ \qprod_{j \in J \sqcup \overline{K}} [\CO_{G/B}(-\ve_{j})] \right).
\end{equation}

\begin{prop}
In $QK_{T}(\FlC{n})$, for $1 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Whitney_lem_3}
\begin{split}
& \sum_{\substack{k, l \ge 0 \\ k+l = d}} \left[ \extprod^{k} \CS_{n} \right] \star \left[ \extprod^{l} \CS_{n}^{\vee} \right] = \left[ \extprod^{d} \udBC^{2n} \right] \\
& \qquad - \sum_{p = 1}^{n} \frac{Q_{p} Q_{p+1} \cdots Q_{n}}{1 - Q_{p-1}} \sum_{\substack{k, l \ge 0 \\ k+l = d-2}} \left( \left[ \extprod^{k} \CS_{p-1} \right] - Q_{p-1} \left[ \extprod^{k} \CS_{p-2} \right] \right) \\
& \hspace{80mm} \star \left( \left[ \extprod^{l} \CS_{p-1}^{\vee} \right] - Q_{p-1} \left[ \extprod^{l} \CS_{p-2}^{\vee} \right] \right).
\end{split}
\end{equation}
\end{prop}

While \eqref{eq:QK-Whitney_1_intro} and \eqref{eq:QK-Whitney_2_intro} are analogous to \eqref{eq:QK-Whitney_typeA}, \eqref{eq:QK-Whitney_3_intro} is specific to the type $C$ case.

Then we observe that relations \eqref{eq:QK-Whitney_1_intro}, \eqref{eq:QK-Whitney_2_intro}, and \eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\FlC{n})$, which we call the \emph{Whitney-type presentation}.

\begin{ithm}[$=$\,Theorem~\ref{thm:Whitney_presentation}] \label{ithm:Whitney_presentation}
Define a ring $R_{Q}^{\mathsf{W}}$ by
\begin{equation}
R_{Q}^{\mathsf{W}} := (R(T)\bra{Q_{1}, \ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \mid 1 \le k \le n, \ 1 \le d \le k, \ 1 \le j \le n].
\end{equation}
Take an ideal $I_{Q}^{\mathsf{W}}$ of $R_{Q}^{\mathsf{W}}$ generated by
\begin{align}
& F_{d}^{k} - \left( F_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
& G_{d}^{k} - \left( G_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
\begin{split}
&\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \Biggl( e_{d}(T_{1}, T_{2}, \ldots, T_{n}, T_{n}^{-1}, \ldots, T_{2}^{-1}, T_{1}^{-1}) \\
& \quad - \sum_{p = 1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} \sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \Biggr) \quad \text{for $1 \le d \le n$},
\end{split} \\
& F_{1}^{1} - x_{1}, \\
& G_{1}^{1} - y_{1}, \\
& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \quad \text{for $1 \le j \le n$};
\end{align}
here, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \in R(T)$ for $1 \le k \le 2n$ are obtained from the decomposition of $\mathbb{C}^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \ldots$, $T_{2n} = T_{1}^{-1}$.
In addition, we understand that
$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \le k \le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \le k \le n$.

Then there exists an $R(T)$-algebra isomorphism $R_{Q}^{\mathsf{W}}/I_{Q}^{\mathsf{W}} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \bigwedge\nolimits^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \bigwedge\nolimits^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{ithm}

To prove Theorem~\ref{ithm:QK-Whitney}, we use the theory of semi-infinite flag manifolds.
Kato (\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules.
Through this isomorphism, we can translate certain quantum products in $QK_{T}(\FlC{n})$ that appear in the computation of the left-hand side of $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold.
Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\FlC{n})$ (Corollary~\ref{cor:multiple_line_bundles}).
These identities enable us to compute $\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1})$ and $\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee})$.
To consider the product $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we use an additional method, the Borel-type presentation given in \cite{KN2}. This gives a relation in $QK_{T}(\FlC{n})$ (Proposition~\ref{prop:QK-Borel_rel}) which enables us to simplify such a product.

To prove Theorem~\ref{ithm:Whitney_presentation}, we use Nakayama-type lemma established by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ2}). This lemma shows that the isomorphism for the classical $K$-ring $K_{T}(X)$ lifts to that for the quantum $K$-ring $QK_{T}(X)$ under conditions (Theorem~\ref{thm:Nakayama}). By applying this lemma, our Whitney-type presentation reduces to the well-known presentation of $K_{T}(\FlC{n})$, and hence we obtain the theorem.

This paper is organized as follows. In Section~\ref{sec:preliminaries}, we fix notation for root systems; in particular, for the type $C$ root system.
Also, we briefly review the flag manifold and its $K$-ring.
In Section~\ref{sec:QK-Whitney}, we give explicit descriptions of $QK$-Whitney relations in $QK_{T}(\FlC{n})$ (Theorem~\ref{ithm:QK-Whitney}) and the Whitney-type presentation of $QK_{T}(\FlC{n})$ (Theorem~\ref{ithm:Whitney_presentation}), which are the main results of this paper.
In Section~\ref{sec:method}, we review the semi-infinite flag manifold and its $K$-group. Then we compute some quantum products necessary to study $QK$-Whitney relations, where some proofs are deferred to Appendix~\ref{sec:appendix}.
In Section~\ref{sec:QK-Whitney_computation}, we give a proof of Theorem~\ref{ithm:QK-Whitney}. In the computation of $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we review Borel-type relations of $QK_{T}(\FlC{n})$.
In Section~\ref{sec:Nakayama}, we review the Nakayama-type lemma and give a proof of Theorem~\ref{ithm:Whitney_presentation}.

\subsection*{Acknowledgments}
The author is grateful to Takeshi Ikeda and Satoshi Naito for helpful discussions.
This work was supported by JSPS Grant-in-Aid for Research Activity Start-up 24K22842.

To prove the main theorem, we use the theory of semi-infinite flag manifolds. Kato (\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules. Through this isomorphism, we can translate certain quantum products in $QK_{T}(\FlC{n})$ that appear in the computation of the left-hand side of the $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold. Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\FlC{n})$. We first record the following corollary.

\begin{cor} \label{cor:multiple_line_bundles}
Assume that $G = \Sp_{2n}(\BC)$.
Let $2 \le k \le n$ and take $J \subset [1, k-1]$. Then we have
\begin{align}
[\CO_{G/B}(-\ve_{J \sqcup \{k\}})] &= \begin{cases} \dfrac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(-\ve_{k})] & \text{if $k-1 \in J$}, \\[5mm]
[\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(-\ve_{k})] & \text{if $k-1 \notin J$}, \end{cases} \label{eq:multiple_1} \\
[\CO_{G/B}(\ve_{J \sqcup \{k\}})] &= \begin{cases} \dfrac{1}{1-Q_{k-1}} [\CO_{G/B}(\ve_{J})] \star [\CO_{G/B}(\ve_{k})] & \text{if $k-1 \in J$}, \\[5mm]
[\CO_{G/B}(\ve_{J})] \star [\CO_{G/B}(\ve_{k})] & \text{if $k-1 \notin J$}. \end{cases} \label{eq:multiple_2}
\end{align}
\end{cor}

These identities enable us to compute $\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1})$ and $\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee})$. To consider the product $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we use an additional method, the Borel-type presentation. This gives a relation in $QK_{T}(\FlC{n}) enabling us to simplify such a product; more precisely, we use the following proposition.

\begin{prop}[{\cite[Corollary~6.4]{KN2}}] \label{prop:QK-Borel_rel}
In $QK_{T}(\FlC{n})$, for $0 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Borel_rel}
\sum_{\substack{J \subset [1, \overline{1}] \\ |J|=d}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}(j) \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right) = \left[ \extprod^{d} \udBC^{2n} \right].
\end{equation}
\end{prop}

[$=$\,\begin{thm} \label{thm:QK-Whitney_A}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}
\end{enu}
\end{thm}\,+$\,\begin{thm} \label{thm:QK-Whitney_B}
In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}
\end{thm}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\underline{\BC}^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}

Question: Let \(X=\mathrm{Fl}^C_n\) be the type \(C_n\) flag manifold parametrizing complete isotropic flags \(0\subset V_1\subset\cdots\subset V_n\subset \mathbb C^{2n}\), and let \(0=\mathcal S_0\subset \mathcal S_1\subset\cdots\subset \mathcal S_n\) be its tautological filtration. In the torus-equivariant quantum \(K\)-ring \(QK_T(X)\) with quantum product \(\star\) and Novikov variables \(Q_i\), define the Hirzebruch \(\lambda_y\)-class of a \(T\)-equivariant vector bundle \(\mathcal E\) by \(\lambda_y(\mathcal E)=\sum_{i=0}^{\operatorname{rk}(\mathcal E)} y^i[\wedge^i\mathcal E]\). Which explicit identities hold in \(QK_T(X)[y]\) for the tautological bundles, their quotients, and their duals?

The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_{p-1}\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]

Choices:
B.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_k}{1-Q_k}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_k}{1-Q_k}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_p}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_p\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_p\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]
C.
The following two identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr).
\]
D.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_{p-1}\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]
E.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr).
\]

shifted-index denominator and Novikov dependence

dropped the third identity for \(\lambda_y(\mathcal S_n)\star\lambda_y(\mathcal S_n^{\vee})\)

power of \(y\) in the global correction term

dual factor in the second parenthesis of the third identity

The stem does not explicitly reveal the correct option. It defines the setting and asks which identities hold, but does not contain the identities themselves or a direct hint toward choice A.

This is essentially a direct theorem-recall item: the correct choice reproduces the explicit identities verbatim, and the task is to recognize the exact statement rather than infer a conclusion from premises.

There is some reasoning/pattern-checking needed because the distractors alter indices, powers of y, omitted identities, or dual factors. However, the item mainly tests precise recall or formula matching, not genuine generation of a mathematical conclusion.

The distractors are mathematically plausible and distinct: shifted Novikov indices, a weaker-true partial statement, an incorrect power of y, and confusion between a bundle and its dual. These align with realistic error modes.

Technically well-constructed in terms of distractors and lack of answer leakage, but it is mostly a verbatim theorem-recognition question rather than a non-tautological or genuinely generative reasoning task.