Kim, T. et al. Degenerate Stirling Polynomials of the Second Kind and Some Applications. Symmetry, 2019, 11(8), 1046

Corrigendum

The authors wish to make the following corrections to the published paper [1]:

Equations (31) and (32) must be replaced as follows:

$$P[Y=y|Y\ge 0]=p\left(y\right)=e_{\lambda }^{-1}\left(\alpha \right)\frac{{\alpha }^y{\left(1\right)}_{y,\alpha }}{y!}$$

(31)

by

$$P[Y=y|Y\ge 0]=p\left(y\right)=e_{\lambda }^{-1}\left(\alpha \right)\frac{{\alpha }^y{\left(1\right)}_{y,\lambda }}{y!}.$$

$$P[X=x|X>0]=p\left(x\right)=\frac{1}{1-e_{\lambda }^{-1}\left(\alpha \right)}{e}_{\lambda }^{-1}\left(\alpha \right)\frac{{\left(1\right)}_{x,\alpha }{\alpha }^x}{x!}$$

(32)

by

$$P[X=x|X>0]=p\left(x\right)=\frac{1}{1-e_{\lambda }^{-1}\left(\alpha \right)}{e}_{\lambda }^{-1}\left(\alpha \right)\frac{{\left(1\right)}_{x,\lambda }{\alpha }^x}{x!}.$$

In lines 8 and 10 from the top of page 7, ${\left(1\right)}_{x,\alpha }$ should be replaced by ${\left(1\right)}_{x,\lambda }$. We rewrite those equations as follows:

Note that

$$\sum _{y=0}^{\infty }p\left(y\right)=e_{\lambda }^{-1}\left(\alpha \right)\sum _{y=0}^{\infty }\frac{{\alpha }^y{\left(1\right)}_{y,\lambda }}{y!}=1,$$

and

$$\sum _{x=1}^{\infty }p\left(x\right)=\frac{1}{e_{\lambda }\left(\alpha \right)-1}\sum _{x=1}^{\infty }\frac{{\left(1\right)}_{x,\lambda }{\alpha }^x}{x!}=1.$$

In Equations (33) and (35), ${\left(1\right)}_{x,\alpha }$ should be replaced by ${\left(1\right)}_{x,\lambda }$. We rewrite those equations as follows:

$$\begin{array}{cc}\hfill E\left[t^{X_j}\right]& =\sum _{x=1}^{\infty }P[X_j=x]t^x\hfill \\ \hfill & =\frac{1}{e_{\lambda }\left(\alpha \right)-1}\sum _{x=1}^{\infty }\frac{{\left(1\right)}_{x,\lambda }{\alpha }^x}{x!}{t}^x\hfill \\ \hfill & =\frac{1}{e_{\lambda }\left(\alpha \right)-1}(e_{\lambda }\left(\alpha t\right)-1),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill E\left[t^Y\right]& =\sum _{y=0}^{\infty }P[Y=y]t^y=e_{\lambda }^{-1}\left(\alpha \right)\sum _{y=0}^{\infty }\frac{{\alpha }^y{\left(1\right)}_{y,\lambda }}{y!}{t}^y\hfill \\ \hfill & =e_{\lambda }^{-1}\left(\alpha \right)e_{\lambda }\left(\alpha t\right).\hfill \end{array}$$

(35)

In Equations (38), (40) and (41) on page 8–9, ${\left(1\right)}_{x,\alpha }$ should be replaced by ${\left(1\right)}_{x,\lambda }$. We rewrite those equations as follows:

$$P[X=x|X\ge r]=p\left(x\right)=\frac{e_{\lambda }^{-1}\left(\alpha \right)}{1-e_{\lambda }^{-1}\left(\alpha \right){\sum }_{x=0}^{r-1}\frac{{\alpha }^x{\left(1\right)}_{x,\lambda }}{x!}}\frac{{\alpha }^x{\left(1\right)}_{x,\lambda }}{x!},$$

(38)

$$\begin{array}{cc}\hfill E\left[t^{X_j}\right]& =\sum _{n=r}^{\infty }P[X_j=n]t^n\hfill \\ \hfill & =\sum _{n=r}^{\infty }\left(\frac{1}{e_{\lambda }\left(\alpha \right)-{\sum }_{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j}\right)\frac{{\alpha }^n{\left(1\right)}_{n,\lambda }}{n!}{t}^n\hfill \\ \hfill & =\left(\frac{1}{e_{\lambda }\left(\alpha \right)-{\sum }_{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j}\right)\left(e_{\lambda }\left(\alpha t\right)-\sum _{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }{\alpha }^j}{j!}{t}^j\right)\hfill \\ \hfill & =C_{\lambda }(\lambda ,r)\left(e_{\lambda }\left(\alpha t\right)-\sum _{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j{t}^j\right),\hfill \end{array}$$

(40)

where $C_{\lambda }(\lambda ,r)=\frac{1}{e_{\lambda }\left(\alpha \right)-{\sum }_{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j}$.

$$\prod _{j=1}^{k}E\left[t^{X_j}\right]=C_{\lambda }^k(\lambda ,r){\left(e_{\lambda }\left(\alpha t\right)-\sum _{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }{\alpha }^j}{j!}{t}^j\right)}^k.$$

(41)

In lines 5 and 6 from top on page 9, ${\left(1\right)}_{x,\alpha }$ should be replaced by ${\left(1\right)}_{x,\lambda }$.

$$\begin{array}{cc}\hfill E\left[t^{X+Y}\right]& =k!C_{\lambda }^k(\lambda ,r)\frac{1}{k!}{\left(e_{\lambda }\left(\alpha t\right)-\sum _{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j{t}^j\right)}^k{e}_{\lambda }^{-1}\left(\alpha \right)e_{\lambda }\left(\alpha t\right)\hfill \\ \hfill & =k!C_{\lambda }^k(\lambda ,r)e_{\lambda }^{-1}\left(\alpha \right)\frac{1}{k!}{\left(e_{\lambda }\left(\alpha t\right)-\sum _{j=0}^{r-1}\frac{{\left(1\right)}_{j,\lambda }}{j!}{\alpha }^j{t}^j\right)}^k{e}_{\lambda }\left(\alpha t\right)\hfill \\ \hfill & =\sum _{n=kr}^{\infty }\frac{k!C_{\lambda }^k(\lambda ,r)}{e_{\lambda }\left(\alpha \right)}{S}_{2,\lambda }^{\left(1\right)}(n,k\mid r)\frac{{\alpha }^n}{n!}{t}^n.\hfill \end{array}$$

In Equation (44) on page 9, ${\left(1\right)}_{x,\alpha }$ should be replaced by ${\left(1\right)}_{x,\lambda }$.

The authors apologize for any convenience caused to the readers. The changes do not affect the results.