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Erratum

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

by
Taekyun Kim
1,
Dae San Kim
2,
Han Young Kim
1 and
Jongkyum Kwon
3,*
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.
Symmetry 2019, 11(12), 1530; https://doi.org/10.3390/sym11121530
Submission received: 28 November 2019 / Accepted: 13 December 2019 / Published: 17 December 2019

## Corrigendum

The authors wish to make the following corrections to the published paper [1]:
Equations (31) and (32) must be replaced as follows:
$P [ Y = y | Y ≥ 0 ] = p ( y ) = e λ − 1 ( α ) α y ( 1 ) y , α y !$
by
$P [ Y = y | Y ≥ 0 ] = p ( y ) = e λ − 1 ( α ) α y ( 1 ) y , λ y ! .$
$P [ X = x | X > 0 ] = p ( x ) = 1 1 − e λ − 1 ( α ) e λ − 1 ( α ) ( 1 ) x , α α x x !$
by
$P [ X = x | X > 0 ] = p ( x ) = 1 1 − e λ − 1 ( α ) e λ − 1 ( α ) ( 1 ) x , λ α x x ! .$
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
$∑ y = 0 ∞ p ( y ) = e λ − 1 ( α ) ∑ y = 0 ∞ α y ( 1 ) y , λ y ! = 1 ,$
and
$∑ x = 1 ∞ p ( x ) = 1 e λ ( α ) − 1 ∑ x = 1 ∞ ( 1 ) x , λ α x x ! = 1 .$
In Equations (33) and (35), $( 1 ) x , α$ should be replaced by $( 1 ) x , λ$. We rewrite those equations as follows:
$E [ t X j ] = ∑ x = 1 ∞ P [ X j = x ] t x = 1 e λ ( α ) − 1 ∑ x = 1 ∞ ( 1 ) x , λ α x x ! t x = 1 e λ ( α ) − 1 ( e λ ( α t ) − 1 ) ,$
$E [ t Y ] = ∑ y = 0 ∞ P [ Y = y ] t y = e λ − 1 ( α ) ∑ y = 0 ∞ α y ( 1 ) y , λ y ! t y = e λ − 1 ( α ) e λ ( α t ) .$
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:
$P [ X = x | X ≥ r ] = p ( x ) = e λ − 1 ( α ) 1 − e λ − 1 ( α ) ∑ x = 0 r − 1 α x ( 1 ) x , λ x ! α x ( 1 ) x , λ x ! ,$
$E [ t X j ] = ∑ n = r ∞ P [ X j = n ] t n = ∑ n = r ∞ 1 e λ ( α ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j α n ( 1 ) n , λ n ! t n = 1 e λ ( α ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j e λ ( α t ) − ∑ j = 0 r − 1 ( 1 ) j , λ α j j ! t j = C λ ( λ , r ) e λ ( α t ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j t j ,$
where $C λ ( λ , r ) = 1 e λ ( α ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j$.
$∏ j = 1 k E [ t X j ] = C λ k ( λ , r ) e λ ( α t ) − ∑ j = 0 r − 1 ( 1 ) j , λ α j j ! t j k .$
In lines 5 and 6 from top on page 9, $( 1 ) x , α$ should be replaced by $( 1 ) x , λ$.
$E [ t X + Y ] = k ! C λ k ( λ , r ) 1 k ! e λ ( α t ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j t j k e λ − 1 ( α ) e λ ( α t ) = k ! C λ k ( λ , r ) e λ − 1 ( α ) 1 k ! e λ ( α t ) − ∑ j = 0 r − 1 ( 1 ) j , λ j ! α j t j k e λ ( α t ) = ∑ n = k r ∞ k ! C λ k ( λ , r ) e λ ( α ) S 2 , λ ( 1 ) ( n , k ∣ r ) α n n ! t n .$
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

1. 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] [Green Version]

## Share and Cite

MDPI and ACS Style

Kim, T.; Kim, D.S.; Kim, H.Y.; Kwon, J. Kim, T. et al. Degenerate Stirling Polynomials of the Second Kind and Some Applications. Symmetry, 2019, 11(8), 1046. Symmetry 2019, 11, 1530. https://doi.org/10.3390/sym11121530

AMA Style

Kim T, Kim DS, Kim HY, Kwon J. Kim, T. et al. Degenerate Stirling Polynomials of the Second Kind and Some Applications. Symmetry, 2019, 11(8), 1046. Symmetry. 2019; 11(12):1530. https://doi.org/10.3390/sym11121530

Chicago/Turabian Style

Kim, Taekyun, Dae San Kim, Han Young Kim, and Jongkyum Kwon. 2019. "Kim, T. et al. Degenerate Stirling Polynomials of the Second Kind and Some Applications. Symmetry, 2019, 11(8), 1046" Symmetry 11, no. 12: 1530. https://doi.org/10.3390/sym11121530

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