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

Kim, Taekyun; Kim, Dae San; Kim, Han Young; Kwon, Jongkyum

doi:10.3390/sym11121530

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 \geq 0] = p (y) = e_{λ}^{- 1} (α) \frac{α^{y} {(1)}_{y, α}}{y!}

(31)

by

P [Y = y | Y \geq 0] = p (y) = e_{λ}^{- 1} (α) \frac{α^{y} {(1)}_{y, λ}}{y!} .

P [X = x | X > 0] = p (x) = \frac{1}{1 - e_{λ}^{- 1} (α)} e_{λ}^{- 1} (α) \frac{{(1)}_{x, α} α^{x}}{x!}

(32)

by

P [X = x | X > 0] = p (x) = \frac{1}{1 - e_{λ}^{- 1} (α)} e_{λ}^{- 1} (α) \frac{{(1)}_{x, λ} α^{x}}{x!} .

In lines 8 and 10 from the top of page 7,

{(1)}_{x, α}

should be replaced by

{(1)}_{x, λ}

. We rewrite those equations as follows:

Note that

\sum_{y = 0}^{\infty} p (y) = e_{λ}^{- 1} (α) \sum_{y = 0}^{\infty} \frac{α^{y} {(1)}_{y, λ}}{y!} = 1,

and

\sum_{x = 1}^{\infty} p (x) = \frac{1}{e_{λ} (α) - 1} \sum_{x = 1}^{\infty} \frac{{(1)}_{x, λ} α^{x}}{x!} = 1 .

In Equations (33) and (35),

{(1)}_{x, α}

should be replaced by

{(1)}_{x, λ}

. We rewrite those equations as follows:

\begin{matrix} E [t^{X_{j}}] & = \sum_{x = 1}^{\infty} P [X_{j} = x] t^{x} \\ = \frac{1}{e_{λ} (α) - 1} \sum_{x = 1}^{\infty} \frac{{(1)}_{x, λ} α^{x}}{x!} t^{x} \\ = \frac{1}{e_{λ} (α) - 1} (e_{λ} (α t) - 1), \end{matrix}

(33)

\begin{matrix} E [t^{Y}] & = \sum_{y = 0}^{\infty} P [Y = y] t^{y} = e_{λ}^{- 1} (α) \sum_{y = 0}^{\infty} \frac{α^{y} {(1)}_{y, λ}}{y!} t^{y} \\ = e_{λ}^{- 1} (α) e_{λ} (α t) . \end{matrix}

(35)

In Equations (38), (40) and (41) on page 8–9,

{(1)}_{x, α}

should be replaced by

{(1)}_{x, λ}

. We rewrite those equations as follows:

P [X = x | X \geq r] = p (x) = \frac{e_{λ}^{- 1} (α)}{1 - e_{λ}^{- 1} (α) \sum_{x = 0}^{r - 1} \frac{α^{x} {(1)}_{x, λ}}{x!}} \frac{α^{x} {(1)}_{x, λ}}{x!},

(38)

\begin{matrix} E [t^{X_{j}}] & = \sum_{n = r}^{\infty} P [X_{j} = n] t^{n} \\ = \sum_{n = r}^{\infty} (\frac{1}{e_{λ} (α) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j}}) \frac{α^{n} {(1)}_{n, λ}}{n!} t^{n} \\ = (\frac{1}{e_{λ} (α) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j}}) (e_{λ} (α t) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ} α^{j}}{j!} t^{j}) \\ = C_{λ} (λ, r) (e_{λ} (α t) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j} t^{j}), \end{matrix}

(40)

where

C_{λ} (λ, r) = \frac{1}{e_{λ} (α) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j}}

.

\prod_{j = 1}^{k} E [t^{X_{j}}] = C_{λ}^{k} (λ, r) {(e_{λ} (α t) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ} α^{j}}{j!} t^{j})}^{k} .

(41)

In lines 5 and 6 from top on page 9,

{(1)}_{x, α}

should be replaced by

{(1)}_{x, λ}

.

\begin{matrix} E [t^{X + Y}] & = k! C_{λ}^{k} (λ, r) \frac{1}{k!} {(e_{λ} (α t) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j} t^{j})}^{k} e_{λ}^{- 1} (α) e_{λ} (α t) \\ = k! C_{λ}^{k} (λ, r) e_{λ}^{- 1} (α) \frac{1}{k!} {(e_{λ} (α t) - \sum_{j = 0}^{r - 1} \frac{{(1)}_{j, λ}}{j!} α^{j} t^{j})}^{k} e_{λ} (α t) \\ = \sum_{n = k r}^{\infty} \frac{k! C_{λ}^{k} (λ, r)}{e_{λ} (α)} S_{2, λ}^{(1)} (n, k ∣ r) \frac{α^{n}}{n!} t^{n} . \end{matrix}

In Equation (44) on page 9,

{(1)}_{x, α}

should be replaced by

{(1)}_{x, λ}

.

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

References

Kim, T.; Kim, D.S.; Kim, H.Y.; Kwon, J. Degenerate Stirling polynomials of the secind kind and some applications. Symmetry 2019, 11, 1046. [Google Scholar] [CrossRef] [Scilit]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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