A Class of
Multiparameter Signless Stirling Numbers of the First Kind and their \(q\)-Analogues
January 01, 1970
In this work, we provide a probabilistic derivation of a class of multiparameter signless Stirling numbers of the first kind and their \(q\)-analogues, and study the associated multivariate discrete distributions.
The present work is motivated by earlier contributions of Cacoullos and Papageorgiou [@CacoulPap] and Koutras [@Koutras1990], as well as more
recent developments by Charalambides [@Charal1; @Charal2; @Charal3; @Charal4], Kyriakoussis and Vamvakari [@kyrvam] and Vamvakari
[@malvina]. Inspired by these works, we introduce a class of multiparameter Stirling numbers and their \(q\)-analogues through probabilistic models.
In the first part, we extend to the multivariate setting the univariate Stirling distributions from which the corresponding signless Stirling numbers of the first kind arise, as presented in Charalambides [@Charal1], together with the associated class of multiparameter signless Stirling numbers of the first kind. In the second part, we proceed analogously by generalizing the univariate discrete \(q\)-Stirling distributions presented in Charalambides [@Charal2] to their multivariate counterparts, and derive the corresponding class of multiparameter signless \(q\)-Stirling numbers of the first kind.
Our approach provides a probabilistic framework for the study of these classes of multiparameter signless Stirling numbers of the first kind and their \(q\)-analogues, and leads naturally to the establishment of the
associated multivariate discrete distributions.
Consider a chamber of \(k\) consecutive lined cells initially containing \(m_j\) balls of color \(c_j\), where \(c_j\)
denotes the \(j\)-th shade of grey, for \(j=1,2,\ldots,k,\) and a group of \(n\) batches of \(s\) red balls. Suppose that
\(n\) balls are successively drawn one after the other from the first lined cell where after each trial the drawn ball is placed back in the first lined cell along with \(s\) red
balls.
Let \(A_{1,i}\) be the event of drawing a ball of color \(c_1\) at the \(i\)-th trial, \(i=1,2,\ldots,n\). Then, setting
\(\theta_1=m_1/s\), we get \[\begin{align}
p_{1,i}=P\left(A_{1,i}\right)=\frac{\theta_1}{\theta_{1}+i-1}, \,\, p'_{1,i}=P\left(A'_{1,i}\right)=\frac{i-1}{\theta_{1}+i-1}, i=1,2,\ldots,n,
\end{align}\] where \(A'_{1,i}\) denotes the complementary event of drawing a ball of color red at the \(i\)-th trial, \(i=1,2,\ldots,n\). After
\(n\) drawings (trials) we remove from the first cell \(n-x_1\) batches of \(s\) red balls, where \(x_1\) is the number of
balls of color \(c_1\) drawn and placed back to the first cell, \(x_1=1,2,\ldots,n\). Now, suppose that \(n-x_1\) balls are successively drawn one after the
other from the second lined cell where after each trial the drawn ball is placed back along with \(s\) red balls.
Let \(A_{2,i}\) be the event of drawing a ball of color \(c_2\) at the \(i\)-th trial, \(i=1,2,\ldots,n-x_1\). Then, setting
\(\theta_2=m_2/s\), we get \[\begin{align}
&&p_{2,i}=P\left(A_{2,i}\right)=\frac{\theta_2}{\theta_{2}+i-1},p'_{2,i}=P\left(A'_{2,i}\right)=\frac{i-1}{\theta_{2}+i-1},\\ \\ &&\,\,\,\,\,\,i=1,2,\ldots,n-x_1,\,\,\,x_1=1,2,\ldots,n.
\end{align}\] After \(n-x_1\) drawings (trials) we remove from the second cell \(n-x_1-x_2\) batches of \(s\) red balls, where \(x_2\) is the number of balls of color \(c_2\) drawn and placed back to the second cell, \(x_2=1,2,\ldots,n-x_1\).
As regards the drawings from the \(k\)-th cell, there is a group of \(n-\sum_{j=0}^{k-1}x_j\) batches of \(s\) red balls. Suppose that we successively draw
\(n-\sum_{j=0}^{k-1}x_j\) balls from the \(k\)-th cell, where after each trial the drawn ball is placed back along with \(s\) red balls. Let \(A_{k,i}\) be the event of drawing a ball of color \(c_k\) at the \(i\)-th trial, \(i=1,2,\ldots,n-\sum_{j=0}^{k-1}x_j\). Then,
setting \(\theta_k=m_k/s\), we get \[\begin{align}
p_{k,i}=P\left(A_{k,i}\right)=\frac{\theta_k}{\theta_{k}+i-1},p'_{k,i}=P\left(A'_{k,i}\right)=\frac{i-1}{\theta_{k}+i-1},
\end{align}\] \(i=1,2,\ldots,n-\sum_{j=0}^{k-1}x_j,\) \(x_k=1,2,\ldots,n-\sum_{j=0}^{k-1}x_j\), \(\,k \geq 1\), where \(x_k\) are the number of balls of color \(c_k\) drawn and placed back to the \(k\)-th cell in \(n-\sum_{j=0}^{k-1}x_j\) trials,
with \(x_0=0\). So, \[\begin{align}
&&P\left( \prod_{j=1}^k A_{j,i_{j,1}}A_{j,i_{j,2}}\ldots A_{j,i_{j,x_j}}A'_{j,i_{j,x_j+1}}\ldots A'_{j,i_{j,n-\sum_{\nu=0}^{j-1}x_\nu}} \right)\\
&&=\prod_{j=1}^k P\left(A_{j,i_{j,1}}\right)
P\left(A_{j,i_{j,2}}\right)\ldots P\left(A_{j,i_{j,x_j}}\right) \\
&&\cdot P\left(A'_{j,i_{j,x_j+1}}\right) \ldots P\left(A'_{j,i_{j,n-\sum_{\nu=0}^{j-1}x_\nu}}\right)
\end{align}\] Therefore, the probability \(p\left(x_1,x_2,\ldots,x_k;n\right)\) of drawing \(x_j\) balls of color \(c_j\) from the \(j\)-th cell, \(j=1,2,\ldots,k\), \(k \geq 1\), under the described probability model, is given by the equation \[\begin{align}
&&p\left(x_1,x_2,\ldots,x_k;n\right)=\frac{\prod_{j=1}^k {\theta_j}^{x_j}}{\prod_{j=1}^k\left(\theta_j+n-\sum_{\nu=0}^{j-1}x_\nu-1\right)_{n-\sum_{\nu=0}^{j-1}x_\nu}}\\
&&\,\,\,\cdot \sum \left(i_{1,x_1+1}-1\right)\cdots \left(i_{1,n}-1\right)\left(i_{2,x_2+1}-1\right)\cdots \left(i_{2,n-x_1}-1\right)\\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdots\left(i_{k,x_k+1}-1\right)\cdots \left(i_{1,n-\sum_{j=0}^{k-1}x_j}-1\right), \sum_{j=1}^kx_j \leq n, k \geq 1, \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,x_j \geq1, j=1,2,\dots,k, x_0=0,
\end{align}\] where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{i_{j,x_j+1},i_{j,x_j+2},\ldots,i_{j,n-\sum_{\nu=0}^{j-1}x_\nu}\right
\}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu-1\) positive integers \(\left \{2,3,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu \right \},\) \(j=1,2,\ldots,k\), \(k \geq 1\). Thus, \[\begin{align}
&&p\left(x_1,x_2,\ldots,x_k;n\right)=\frac{\prod_{j=1}^k {\theta_j}^{x_j}}{\prod_{j=1}^k\left(\theta_j+n-\sum_{\nu=0}^{j-1}x_\nu-1\right)_{n-\sum_{\nu=0}^{j-1}x_\nu}}\\
&&\,\,\,\cdot \sum m_{1,1}\cdots m_{1,n-x_1}m_{2,1}\cdots m_{2,n-x_1-x_2}\cdots m_{k,1}\cdots m_{k,n-\sum_{\nu=1}^kx_\nu}, \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \sum_{j=1}^kx_j \leq n, k \geq 1, x_j \geq1, j=1,2,\dots,k, x_0=0,
\end{align}\] where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right
\}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu-1\) positive integers \(\left \{1,2,3,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \},\) \(j=1,2,\ldots,k\),
\(k \geq 1\).
Definition 1. Let \(|s\left(n,x_1,x_2,\ldots,x_k\right)|,\) \(\sum_{j=1}^kx_j \leq n,\) \(x_j\geq 1\), \(j=1,2\ldots,k,\) \(k \geq 1\), be the numbers given by the multiple sum \[\begin{align} |s\left(n,x_1,x_2,\ldots,x_k\right)| =\sum \prod_{j=1}^km_{j,1}\cdots m_{j,n-\sum_{\nu=1}^jx_\nu} , \end{align}\] where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right \}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu-1\) positive integers \(\left \{1,2,3,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \},\) \(j=1,2,\ldots,k\), \(k \geq 1\). The numbers \(|s\left(n,x_1,x_2,\ldots,x_k\right)|\) are called multiparameter signless Stirling numbers of the first kind.
Remark 1. For \(k=1,\) the numbers \(|s(n,x_1)|,\) \(x_1=1,2,\ldots,n,\) according to the previous definition, are given by \[|s(n,x_1)|=\sum m_{1,1}\cdots m_{1,n-x_1}\] where the summation is extended over all \(\left(n-x_1\right)\)-combinations \(\left \{m_{1,1},m_{1,2},\ldots,m_{1,n-x_1}\right \}\) of the \(n-1\) positive integers \(\left \{1,2,3,\ldots,n-1\right \}\). Therefore, \(|s(n,x_1)|\) are the signless Stirling numbers of the first kind.
Next, we consider a chamber of \(k\) consecutive lined cells initially containing \(m_j\) balls of color \(c_{1:j}\) and \(b_j\) balls of color \(c_{2:j}\), \(j=1,2,\ldots,k\) and a group of \(n\) batches of \(s\) red
balls. Suppose that we apply the previous probability model of successive drawings described in Section 2.1. Let \(A_{j,i}\) be the event of drawing a ball of color \(c_{1:j},\) \(j=1,2,\ldots,k\), \(k \geq 1,\) at the \(i\)-th trial, \(i=1,2,\ldots,n-\sum_{\nu=0}^{j-1}x_j\). Then, setting \(\theta_j=m_j/s\), \(r_j=b_j/s\), we get \[\begin{align}
p_{j,i}=P\left(A_{j,i}\right)=\frac{\theta_j}{\theta_{j}+r_j+i}, \,\, p'_{j,i}=P\left(A'_{j,i}\right)=\frac{r_j+i}{\theta_{j}+r_j+i},
\end{align}\] \(i=1,2,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu,\) \(x_j=0,1,2,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu\), where \(x_j\) are the number of balls of
color \(c_{1:j}\) drawn and placed back to the \(j\)-th cell in \(n-\sum_{\nu=0}^{j-1}x_\nu\), trials \(j=1,2,\ldots,k\).
Working analogously as in Section 2.1, we derive the following expansion of Definition 1.
Definition 2. Let \(|s\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|,\) \(\sum_{j=1}^kx_j \leq n,\) \(x_j\geq 1\), \(j=1,2\ldots,k,\) \(k \geq 1\), be the numbers given by the multiple sum \[\begin{align}
&|s\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|\\
&=\sum \prod_{j=1}^k\left(r_1+m_{j,1}\right)\cdots \left(r_j+m_{j,n-\sum_{\nu=1}^jx_\nu}\right)
,
\end{align}\]
where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right \}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu-1\) positive integers \(\left \{1,2,3,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \},\) \(j=1,2,\ldots,k\), \(k
\geq 1\). The numbers \(|s\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|\) are called multiparameter signless noncentral Stirling numbers of the first kind.
Remark 2. For \(k=1,\) the numbers \(|s(n,x_1;r_1)|,\) \(x_1=1,2,\ldots,n,\) according to the previous definition, are given by \[|s(n,x_1;r_1)|=\sum \left(r_1+m_{1,1}\right)\cdots \left(r_1+ m_{1,n-x_1}\right)\] where the summation is extended over all \(\left(n-x_1\right)\)-combinations \(\left \{m_{1,1},m_{1,2},\ldots,m_{1,n-x_1}\right \}\) of the \(n-1\) positive integers \(\left \{1,2,3,\ldots,n-1\right \}\). Therefore, \(|s(n,x_1;r_1)|\) are the signless noncentral Stirling numbers of the first kind (see Charalambides [@Charal1]).
The multivariate discrete \(q\)-distributions, \(0<q<1\), are based on stochastic models of sequences of \(n\) independent Bernoulli trials with
chain-composite failures (or successes), where the odds of success of a certain kind at a trial is assumed to vary geometrically, with rate \(q\), with the number of previous trials or with the number of previous successes
or both with the number of previous trials and successes (see Charalambides [@Charal5]).
Next, we consider a sequence of independent Bernoulli trials with chain-composite failures, where the probability of success of the \(j\)-th kind at the \(i\)-th trial is given by
\[\begin{align} \label{sucess} p_{j,i}=\frac{1}{[r_j+i]_q},\,\, 0\leq r_j<\infty,\,\, j=1,2,\ldots,k, i=1,2,\ldots
\end{align}\tag{1}\] Let \(A_{j,i}\) be the event of success of the \(j\)-th kind at the \(i\)-th trial with probability of success given by
Eq. 1 for \(j=1,\ldots,k\), \(k \geq 1\), \(i=1,\ldots,\) and consider a permutation \((i_{j,1},\ldots,i_{j,x_j},i_{j,x_{j+1}},\ldots, i_{j,n-\sum_{\nu=0}^{j-1}x_\nu})\) of \(\{1,2,\ldots, n-\sum_{\nu=0}^{j-1}x_{\nu} \}\). Then, we get \[\begin{align}
&&P\left( \prod_{j=1}^k A_{j,i_{j,1}}\ldots A_{j,i_{j,x_j}}A'_{j,i_{j,x_j+1}}\cdots A'_{j,i_{j,n-\sum_{\nu=0}^{j-1}x_\nu}} \right)\\
&&=\prod_{j=1}^k P\left(A_{j,i_{j,1}}\right)
\ldots P\left(A_{j,i_{j,x_j}}\right)P\left(A'_{j,i_{j,x_j+1}}\right) \cdots P\left(A'_{j,i_{j,n-\sum_{\nu=0}^{j-1}x_\nu}}\right)
\end{align}\] and the following theorem holds.
Theorem 3. Let \(X_j\) be the number of successes of the \(j\)-th kind in a sequence of \(n\) independent Bernoulli trials with chain-composite failures, where the probability of success of the \(j\)-th kind at the \(i\)-th trial is given by Eq. 1 , for \(j=1,2,\ldots,k\). Then the probability function of the r.v. \(\left (X_1,X_2,\ldots,X_k \right)\) is given by \[\begin{align} P\left (X_1=x_1,X_2=x_2,\ldots,X_k=x_k\right)&=& \dfrac{q^{nk-\sum_{j=1}^k(k-j+1)x_j}}{\prod_{j=1}^k[r_j+n-\sum_{\nu=0}^{j-1}x_\nu]_{{n-\sum_{\nu=0}^{j-1}x_\nu},q}} \\ && \cdot \sum \prod_{j=1}^k [r_j+m_{j,1}]_q[r_j+m_{j,2}]_q\cdots[r_j+m_{j,n-\sum_{\nu=1}^jx_\nu}]_q, x_j=0,1,\ldots,n, \end{align}\] with \(\sum_{j=1}^kx_j \leq n\), \(x_0=0\), where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right \}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu\) nonnegative integers \(\left \{0,1,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \},\) \(j=1,2,\ldots,k\), \(k \geq 1\).
Definition 3. Let \(|s_q\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|,\) \(\sum_{j=1}^kx_j \leq n,\) \(x_j\geq 1\), \(0\leq r_j<\infty\), \(j=1,2\ldots,k,\) \(k \geq 1\), be the numbers given by the multiple sum \[\begin{align} &&|s_q\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|=q^{nk-\sum_{j=1}^k(k-j+1)x_j}\\ &&\cdot \sum \prod_{j=1}^k [r_j+m_{j,1}]_q[r_j+m_{j,2}]_q\cdots[r_j+m_{j,n-\sum_{\nu=1}^jx_\nu}]_q, \end{align}\] where the multiple sum is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right \}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu\) nonnegative integers \(\left \{0,1,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \}\) , \(j=1,2,\ldots,k\), \(k \geq 1\). The numbers \(|s_q\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|\) are called multiparameter signless noncentral \(q\)-Stirling numbers of the first kind.
Remark 4. For \(k=1,\) the numbers \(|s_q(n,x_1;r_1)|,\) \(x_1=1,2,\ldots,n,\) \(n=1,2,\ldots,\) \(0 \leq r_1 <\infty,\) according to the previous definition, are given by \[|s_q(n,x_1;r_1)|=q^{n-x_1}\sum [r_1+m_{1,1}]_q[r_1+m_{1,2}]_q\cdots[r_1+m_{1,n-x_1}]_q,\] where the summation is extended over all \(\left(n-x_1\right)\)-combinations \(\left \{m_{1,1},m_{1,2},\ldots,m_{1,n-x_1}\right \}\) of the \(n\) nonnegative integers \(\left \{0,1,\ldots,n-1\right \}\). Therefore, \(|s_q(n,x_1;r_1)|\) are the signless noncentral \(q\)-Stirling numbers of the first kind (see Charalambides [@Charal2]).
Remark 5. The multiparameter noncentral \(q\)-Stirling numbers of the first kind are defined by the multiple sum as follows \[\begin{align} &&s_q\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)\nonumber\\ &&= \sum \prod_{j=1}^k [-1]_q^{n-\sum_{\nu=1}^{j}x_\nu} [r_j+m_{j,1}]_q[r_j+m_{j,2}]_q\cdots[r_j+m_{j,n-\sum_{\nu=1}^jx_\nu}]_q, \nonumber \end{align}\] where the summation is extended over all \(\left(n-\sum_{\nu=1}^j x_\nu\right)\)-combinations \(\left \{m_{j,1},m_{j,2},\ldots,m_{j,n-\sum_{\nu=1}^{j}x_\nu}\right \}\) of the \(n-\sum_{\nu=0}^{j-1}x_\nu\) nonnegative integers \(\left \{0,1,\ldots,n-\sum_{\nu=0}^{j-1}x_\nu -1\right \},\) \(j=1,2,\ldots,k\), \(k \geq 1\).
Note that if we change suitably the probability of success the following theorem also holds.
Theorem 6. Let \(X_j\) be the number of successes of the \(j\)-th kind in a a sequence of \(n\) independent Bernoulli trials with chain-composite failures, where the probability of success of the \(j\)-th kind at the \(i\)-th trial, is given by \[\begin{align} \label{sucess2} p_{j,i}=q^{r_j+i-1},\,\, 0\leq r_j<\infty,\,\, j=1,2,\ldots,k, i=1,2,\ldots, \,0<q<1. \end{align}\qquad{(1)}\] Then the probability function of the r.v. \(\left (X_1,X_2,\ldots,X_k \right)\) is given by \[\begin{align} P\left (X_1=x_1,X_2=x_2,\ldots,X_k=x_k\right) &=&q^{\sum_{j=1}^k {n-\sum_{\nu=0}^{j-1}x_\nu \choose 2}+\sum_{j=1}^k \left(n-\sum_{\nu=0}^{j-1} x_\nu \right) r_j}\\&&\,\,\,\,\cdot (1-q)^{\sum_{j=1}^k \left(n-\sum_{\nu=1}^{j}x_\nu\right)} |s_{q^{-1}}\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)| \end{align}\] with \(x_j=0,1,\ldots,n,\) \(\sum_{j=1}^kx_j \leq n\), \(x_0=0\), where \[\begin{align} |s_{q^{-1}}\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right)|&=&(-1)^{\sum_{j=1}^k \left(n-\sum_{\nu=1}^{j}x_\nu\right)} \\&&\,\,\,\,\,\,\,\cdot q^{-{\sum_{j=1}^k \left(n-\sum_{\nu=1}^{j}x_\nu\right)}}s_{q^{-1}}\left(n,x_1,x_2,\ldots,x_k;r_1,r_2,\ldots,r_k\right). \end{align}\]
[@*]