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

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

doi:10.3390/sym11121530

,

and

¹

Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea

²

Department of Mathematics, Sogang University, Seoul 121-742, Korea

³

Department of Mathematics Education and ERI, Gyeongsang National University, Jinju 52828, Korea

^*

Author to whom correspondence should be addressed.

Symmetry2019, 11(12), 1530;https://doi.org/10.3390/sym11121530

This article belongs to the Special Issue The 32th Congress of The Jangjeon Mathematical Society (ICJMS2019) will be Held at Far Eastern Federal Universit, Vladivostok Russia

Version Notes

Order Reprints

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]

© 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/).

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

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