On \(p, q\)-Poisson distributions and \(p, q\)-factorial moments (Q2811836)

In this paper, the authors define the \(p,q\)-analogues of some discrete \(q\)-distributions, called \(p,q\)-distributions, and introduce the \(p,q\)-factorial moments of some of these \(p,\) \(q\)-distributions.NEWLINENEWLINEClassically, a polynomial \(a_k(q)\) is a \(q\)-analogue of an integer \(a_k\) if its limit as \(q\) tends to 1 yields \(a_k\). Extending the classical \(q\)-analogue defintions such that for \(p\neq q\) NEWLINE\[NEWLINE [n]_{p,q}=\frac{p^n-q^n}{p-q},\quad [n]_{k,p,q}=\prod_{j=0}^{k-1}\frac{p^{n-j}-q^{n-j}}{p-q},\qquad[n]_{p,q}!=\prod_{j=1}^n[j]_{p,q} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \begin{bmatrix} n\\k \end{bmatrix}_{p,q}=\prod_{j=1}^k \frac{p^{n-j+1}-q^{n-j+1}}{p^j-q^j}, NEWLINE\]NEWLINE referred to as the \(p,q\)-real number of \(n\), the \(p,q\)-falling factorial of \(n\) of order \(k\), the \(p,q\)-factorial of \(n\), and the \(p,q\)-binomial coefficient, respectively, the authors show that when \(p=1\) we have NEWLINE\[NEWLINE [n]_{1,q}=[n]_{p},\quad [n]_{k,1,q}=[n]_{k,q},\quad [n]_{1,q}!=[n]_q!,\quad \begin{bmatrix} n\\k \end{bmatrix}_{1,q} =\begin{bmatrix} n\\k \end{bmatrix}_{q}, NEWLINE\]NEWLINE which is the requirement for an expressions to be a \(p,q\)-analogue. Moreover, it is shown that the \(p,q\)-binomial coefficient satisifies NEWLINE\[NEWLINE\begin{bmatrix} n\\k \end{bmatrix}_{p,q} =\frac{[n]_{k,p,q}}{[k]_{p,q}!}=\frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}, NEWLINE\]NEWLINE and negative \(p,q\)-binomial expansions are also considered.NEWLINENEWLINENEWLINEThe authors then define the type 1 and type 2 \(p,q\)-exponential functions \(\text{ê}_{p,q}(t)\) and \(e_{p,q}(t)\) such that NEWLINE\[NEWLINE \text{ê}_{p,q}(t)=\prod_{i=1}^\infty [(p^{i-1}+t(p-q)q^{i-1})(p^{i-1})^{-1}],\quad e_{p,q}(t)=\prod_{i=1}^\infty [(p^{i-1}-t(p-q)q^{i-1})^{-1}p^{i-1}], NEWLINE\]NEWLINE where \(\text{ê}_{p,q}(t)e_{p,q}(t)=1\), and go on to show (Theorem 2.5) that NEWLINE\[NEWLINE \text{ê}_{p,q}(t)=\text{ê}_{q/p}(tp)=\sum_{k=0}^\infty q^{\binom{k}{2}}\frac{(tp)^k}{[k]_{p,q}!},\quad e_{p,q}(t)=e_{q/p}(tp)=\sum_{k=0}^\infty p^{\binom{k}{2}}\frac{(tp)^k}{[k]_{p,q}!}, NEWLINE\]NEWLINE where \(e_{q/p}(t)\) is the standard \(q\)-exponential function.NEWLINENEWLINEDiscrete \(p,q\)-distributions for the binomial, negative binomial, Pascal and Poisson distributions are then considered, as well as \(p,q\)-analogues of the Stirling numbers and \(p,q\)-factorial moments.