Next Article in Journal
Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making
Next Article in Special Issue
Fibonacci and Lucas Numbers of the Form 2a + 3b + 5c + 7d
Previous Article in Journal
A Systematic Review of the Use of Blockchain in Healthcare
Previous Article in Special Issue
On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# A Note on Modified Degenerate Gamma and Laplace Transformation

by
YunJae Kim
1,
Byung Moon Kim
2,
Lee-Chae Jang
3 and
Jongkyum Kwon
4,*
1
Department of Mathematics, Dong-A University, Busan 49315, Korea
2
Department of Mechanical System Engineering, Dongguk University, Gyungju-si, Gyeongsangbukdo 38066, Korea
3
Graduate School of Education, Konkuk University, Seoul 139-701, Korea
4
Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 471; https://doi.org/10.3390/sym10100471
Submission received: 8 September 2018 / Revised: 1 October 2018 / Accepted: 3 October 2018 / Published: 10 October 2018
(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

## Abstract

:
Kim-Kim studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. In this paper, we define modified degenerate gamma and modified degenerate Laplace transformation and investigate some properties and formulas related to them.

## 1. Introduction

It is well known that gamma function is defied by
$Γ ( s ) = ∫ 0 ∞ e − t t s − 1 d t , where s ∈ C with R e ( s ) > 0 ,$
(see [1,2]). From (1), we note that
$Γ ( s + 1 ) = s Γ ( s ) , and Γ ( n + 1 ) = n ! , where n ∈ N .$
Let $f ( t )$ be a function defined for $t ≥ 0$. Then, the integral
$L ( f ( t ) ) = ∫ 0 ∞ e − s t f ( t ) d t ,$
(see [1,2,3,4]), is said to be the Laplace transform of f, provided that the integral converges. For $λ ∈ ( 0 , ∞ )$. Kim-Kim [2] introduced the degenerate gamma function for the complex variable s with $0 < R e ( s ) < 1 λ$ as follows:
$Γ λ ( s ) = ∫ 0 ∞ ( 1 + λ t ) − 1 λ t s − 1 d t ,$
(see [2]) and degenerate Laplace transformation which was defined by
$L λ ( f ( t ) ) = ∫ 0 ∞ ( 1 + λ t ) − s λ f ( t ) d t ,$
(see [2,5]), if the integral converges. The authors obtained some properties and interesting formulas related to the degenerate gamma function. For examples, For $λ ∈ ( 0 , 1 )$ and $0 < R e ( s ) < 1 − λ λ$,
$Γ λ ( s + 1 ) = s ( 1 − λ ) s − 1 Γ λ 1 − λ ( s ) ,$
and $λ ∈ ( 0 , 1 k + s )$ with $k ∈ N$ and $0 < R e ( s ) < 1 − λ λ$,
$Γ λ ( s + 1 ) = s ( s − 1 ) ⋯ ( s − ( k + 1 ) + 1 ) ( 1 − λ ) ( 1 − 2 λ ) ⋯ ( 1 − k λ ) ( 1 − ( k + 1 ) λ ) Γ λ 1 − ( k + 1 ) λ ( s − k ) ,$
and for $k ∈ N$ and $λ ∈ ( 0 , 1 k )$,
$Γ λ ( k ) = ( k − 1 ) ! ( 1 − λ ) ( 1 − 2 λ ) ⋯ ( 1 − k λ ) .$
The authors obtained some formulas related to the degenerate Laplace transformation. For examples,
$L λ ( 1 ) = 1 s − λ , if s > λ ,$
and
$L λ ( ( 1 + λ t ) − a λ ) = 1 s + a − λ , if s > − a + λ ,$
and
$L λ ( cos λ ( a t ) ) = s − λ ( s − λ ) 2 + a 2 ,$
and
$L λ ( sin λ ( a t ) ) = a ( s − λ ) 2 + a 2 ,$
where $cos λ ( t ) = 1 2 ( 1 + λ t ) i t λ + ( 1 + λ t ) − i t λ$ and $sin λ ( t ) = 1 2 i ( 1 + λ t ) i t λ − ( 1 + λ t ) − i t λ$.
Furthermore, the authors obtained that
$L λ ( t n ) = n ! ( s − λ ) ( s − 2 λ ) ⋯ ( s − n λ ) ( s − ( n + 1 ) λ ) ,$
for $n ∈ N$ and $s > ( n + 1 ) λ$, and
$L λ ( f ( n ) ( t ) ) = s ( s + λ ) ( s + 2 λ ) ⋯ ( s + ( n − 1 ) λ ) L λ ( ( 1 + λ t ) − n f ( t ) ) − ∑ i = 0 n − 1 f ( i ) ( 0 ) ∏ t = 1 n − i − 1 s + ( l − 1 ) λ .$
where $f , f ( 1 ) , ⋯ , f ( n − 1 )$ are continuous on $( 0 , ∞ )$ and are of degenerate exponential order and $f ( n ) ( t )$ is piecewise continuous on $( 0 , ∞ )$, and
$L λ ( ( log ( 1 + λ t ) ) n f ( t ) ) = ( − 1 ) n λ n d d s n L λ ( s ) ,$
for $n ∈ N$.
At first, L. Carlitz introduced the degenerate special polynomials (see [6,7]). The recently works which can be cited in this and researchers have studied the degenerate special polynomials and numbers (see [2,8,9,10,11,12,13,14,15,16,17,18,19]). Recently, the concept of degenerate gamma function and degenerate Laplace transformation was introduced by Kim-Kim [2]. They studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. We observe whether or not that holds. Thus, we consider the modified degenerate Laplace transform which are satisfied (16). The degenerate gamma and degenerate Laplace transformation applied to engineer’s mathematical toolbox as they make solving linear ODEs and related initial value problems. This paper consists of two sections. The first section contains the modified degenerate gamma function and investigate the properties of the modified gamma function. The second part of the paper provide the modified degenerate Laplace transformation and investigate interesting results of the modified degenerate Laplace transformation.
$L λ ( f ∗ g ) = L λ ( f ) L λ ( g )$

## 2. Modified Degenerate Gamma Function

In this section, we will define modified degenerate gamma functions which are different to degenerate gamma functions. For each $λ ∈ ( 0 , ∞ )$, we define modified degenerate gamma function for the complex variable s with $0 < R e ( s )$ as follows:
$Γ λ ∗ ( s ) = ∫ 0 ∞ ( 1 + λ ) − t λ t s − 1 d t .$
Let $λ ∈ ( 0 , 1 )$. Then, for $0 < R e ( s )$, we have
$Γ λ ∗ ( s + 1 ) = ∫ 0 ∞ ( 1 + λ ) − t λ t s d t = 1 ( log ( 1 + λ ) − 1 λ ( 1 + λ ) − t λ t s ∣ 0 ∞ + λ log ( 1 + λ ) ∫ 0 ∞ s ( 1 + λ ) − t λ t s − 1 d t = λ log ( 1 + λ ) s Γ λ ∗ ( s ) .$
Therefore, by (18), we obtain the following theorem.
Theorem 1.
Let $λ ∈ ( 0 , 1 )$. Then, for $0 < R e ( s )$, we have
$Γ λ ∗ ( s + 1 ) = λ s log ( 1 + λ ) Γ λ ∗ ( s ) .$
Then, for $0 < R e ( s )$ and $λ ∈ ( 0 , 1 )$, repeatly we calculate
$Γ λ ∗ ( s + 1 ) = λ s log ( 1 + λ ) Γ λ ∗ ( s ) = λ 2 ( s − 1 ) ( log ( 1 + λ ) ) 2 Γ λ ∗ ( s − 1 ) .$
Thus, continuing this process, for $0 < R e ( s )$ and $λ ∈ ( 0 , 1 )$, we have
$Γ λ ∗ ( s + 1 ) = λ k ( s − 1 ) ⋯ ( s − k + 1 ) ( log ( 1 + λ ) ) k Γ λ ∗ ( s − k ) .$
Therefore, by (21), we obtain the following theorem.
Theorem 2.
Let $λ ∈ ( 0 , 1 )$. Then, for $0 < R e ( s )$, we have
$Γ λ ∗ ( s + 1 ) = λ k ( s − 1 ) ⋯ ( s − k + 1 ) ( log ( 1 + λ ) ) k Γ λ ∗ ( s − k ) .$
Let us take $s = k + 1$. Then, by Theorem 2, we get
$Γ λ ∗ ( k + 2 ) = λ k + 1 k ⋯ 2 ( log ( 1 + λ ) ) k + 1 Γ λ ∗ ( 1 ) = λ k + 1 k ! ( log ( 1 + λ ) ) k + 1 Γ λ ∗ ( 1 )$
and
$Γ λ ∗ ( 1 ) = ∫ 0 ∞ ( 1 + λ ) − t λ d t = − λ ( log ( 1 + λ ) ( 1 + λ ) − t λ ∣ 0 ∞ = λ ( log ( 1 + λ ) .$
Therefore, by (23) and (24), we obtain the following theorem.
Theorem 3.
For $k ∈ N$ and $λ ∈ ( 0 , 1 )$, we have
$Γ λ ∗ ( k + 1 ) = λ k + 1 k ! ( log ( 1 + λ ) ) k + 1 .$

## 3. Modified Degenerate Laplace Transformation

In this section, we will define modified Laplace transformation which are different to degenerate Laplace transformation. Let $λ ∈ ( 0 , ∞ )$ and let $f ( t )$ be a function defined for $t ≥ 0$. Then the integral
$L λ ∗ ( f ( t ) ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t ) d t .$
is said to be the modified degenerate Laplace transformation of f if the integral converges which is also defined by $L λ ∗ ( f ( t ) ) = F λ ( s )$.
From (26), we get
$L λ ∗ ( α f ( t ) + β g ( t ) ) = α L λ ∗ ( f ( t ) ) + β L λ ∗ ( g ( t ) ) ,$
where α and β are constant real numbers.
First, we observe that for $n ∈ N$,
$L λ ∗ ( t n ) = ∫ 0 ∞ ( 1 + λ s ) − t λ t n d t = − λ log ( 1 + λ s ) ( 1 + λ s ) − t λ t n ∣ 0 ∞ + λ n log ( 1 + λ s ) ∫ 0 ∞ ( 1 + λ s ) − t λ t n − 1 d t = λ n log ( 1 + λ s ) L λ ∗ ( t n − 1 ) = λ n log ( 1 + λ s ) − λ log ( 1 + λ s ) ( 1 + λ s ) − t λ t n − 1 ∣ 0 ∞ + λ ( n − 1 ) log ( 1 + λ s ) ∫ 0 ∞ ( 1 + λ s ) − t λ t n − 2 d t = λ log ( 1 + λ s ) 2 n ( n − 1 ) L λ ∗ ( t n − 2 ) = ⋯ = λ log ( 1 + λ s ) n n ! L λ ∗ ( 1 ) = λ log ( 1 + λ s ) n + 1 n ! .$
Therefore, by (28), we obtain the following theorem.
Theorem 4.
For $k ∈ N$ and $λ ∈ ( 0 , 1 )$, we have
$L λ ∗ ( t n ) = λ log ( 1 + λ s ) n + 1 n ! .$
Secondly, we note that if f is a periodic function with a period T.
$L λ ∗ ( f ( t ) ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t ) d t = ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t + ∫ T ∞ ( 1 + λ s ) − t λ f ( t ) d t = ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t + ∫ 0 ∞ ( 1 + λ s ) − t + T λ f ( t + T ) d t = ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t + ( 1 + λ s ) − T λ ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t ) d t$
By (30), we get
$1 − ( 1 + λ s ) − T λ L λ ∗ ( f ( t ) ) = ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t .$
Thus, by (31), we get
$L λ ∗ ( f ( t ) ) = 1 1 − ( 1 + λ s ) − T λ ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t .$
We recall that the degenerate Bernoulli numbers are introduced as
$t ( 1 + λ ) − t λ = ∑ n = 0 ∞ B n , λ t n n ! ,$
Thus, by (32) and (33), we have
$1 1 − ( 1 + λ S ) − T λ = − 1 T S S T ( 1 + λ s ) − T S λ S − 1 = − 1 T S ∑ n = 0 ∞ B n , λ S ( − 1 ) n S n T n n ! .$
Therefore, by (33) and (34), we obtain the following theorem.
Theorem 5.
If f is a function defined $t ≥ 0$ and $L λ ∗ ( f ( t ) )$ exists, then we have
$L λ ∗ ( f ( t ) ) = − 1 T S ∑ n = 0 ∞ B n , λ S ( − 1 ) n S n ∫ 0 T ( 1 + λ s ) − t λ f ( t ) d t T n n ! = − 1 T S ∑ n = 0 ∞ B n , λ S ( − 1 ) n S n L λ ∗ ( U ( t − T ) f ( t ) ) ,$
where $U ( t − a ) = 0 , for 0 ≥ t ≥ a , 1 , for t ≤ a .$ is the Heviside function.
Thirdly, we observe the modified degenerate Laplace transformation of $f ( t − a ) U ( t − a )$ as follows:
$L λ ∗ ( f ( t − a ) U ( t − a ) ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t − a ) U ( t − a ) d t = ∫ a ∞ ( 1 + λ s ) − t λ f ( t − a ) d t = ∫ 0 ∞ ( 1 + λ s ) − t + a λ f ( t ) d t = ( 1 + λ s ) − a λ ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t ) d t = ( 1 + λ s ) − a λ L λ ∗ ( f ( t ) ) .$
Therefore, by (36), we obtain the following theorem.
Theorem 6.
For $λ ∈ ( 0 , 1 )$ and $a ∈ ( 0 , ∞ )$ we have
$L λ ∗ ( f ( t − a ) U ( t − a ) ) = ( 1 + λ s ) − a λ L λ ∗ ( f ( t ) ) ,$
where $U ( t − a )$ is the Heviside function.
Fourthly, we observe the modified degenerate Laplace transformation of the convolution $f ∗ g$ of two function f, g as follows:
$L λ ∗ ( f ) L λ ∗ ( g ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ( t ) d t ∫ 0 ∞ ( 1 + λ s ) − τ λ g ( τ ) d τ = ∫ 0 ∞ ∫ 0 ∞ ( 1 + λ s ) − t + τ λ f ( t ) g ( τ ) d t d τ = ∫ 0 ∞ f ( t ) ∫ τ ∞ ( 1 + λ s ) − μ λ g ( μ − τ ) d μ d τ = ∫ 0 ∞ ∫ τ ∞ f ( t ) ( 1 + λ s ) − μ λ g ( μ − τ ) d μ d τ = ∫ 0 ∞ f ∗ g ( 1 + λ s ) − μ λ d μ = L λ ( f ∗ g ) .$
Therefore, by (38), we obtain the following theorem.
Theorem 7.
For $λ ∈ ( 0 , 1 ]$, we have
$L λ ∗ ( f ∗ g ) = L λ ∗ ( f ) L λ ∗ ( g ) .$
We note that
$L λ ∗ ( 1 ) = ∫ 0 ∞ ( 1 + λ s ) − t λ 1 d t = − λ log ( 1 + λ s ) ( 1 + λ s ) − t λ ∣ 0 ∞ = λ log ( 1 + λ s ) .$
By (40), we have
$L λ ∗ ( f ∗ 1 ) = L λ ∗ ( f ) L λ ∗ ( 1 ) = L λ ∗ ( f ) λ log ( 1 + λ s ) .$
Therefore, by (41), we obtain the following theorem.
Theorem 8.
For $λ ∈ ( 0 , 1 ]$, we have
$L λ ∗ − 1 ( L λ ∗ ( f ) λ log ( 1 + λ s ) ) = f ∗ 1 ( t ) = ∫ 0 t f ( t ) d t .$
Fifthly, we observe that the modified degenerate Laplace transformation of derivative of f which is $f ( t ) = 0 ( ( 1 + λ s ) − t λ )$, where $f ( t ) = 0 ( u ( t ) )$ means
$L λ ∗ ( f ′ ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ′ d t = ( 1 + λ s ) − t λ f ( t ) ∣ 0 ∞ + ∫ 0 ∞ log ( 1 + λ s ) λ ( 1 + λ s ) − t λ f ( t ) d t = − f ( 0 ) + log ( 1 + λ s ) λ L λ ∗ ( f ) .$
and
$L λ ∗ ( f ( 2 ) ) = ∫ 0 ∞ ( 1 + λ s ) − t λ f ( 2 ) d t = ( 1 + λ s ) − t λ f ′ ( t ) ∣ 0 ∞ + log ( 1 + λ s ) λ ∫ 0 ∞ ( 1 + λ s ) − t λ f ′ ( t ) d t = − f ( 0 ) + log ( 1 + λ s ) λ − f ( 0 ) + log ( 1 + λ s ) λ L λ ∗ ( f ) = log ( 1 + λ s ) λ 2 L λ ∗ ( f ) − log ( 1 + λ s ) λ f ( 0 ) − f ′ ( 0 ) .$
By using mathematical induction, we obtain the following theorem.
Theorem 9.
For $λ ∈ ( 0 , 1 ]$, we have
$L λ ∗ ( f ( n ) ) = log ( 1 + λ s ) λ n L λ ∗ ( f ) − ∑ i = 0 n − 1 log ( 1 + λ s ) λ n − 1 − i f ( i ) ( 0 ) .$
Finally, we observe
$d F λ ∗ d s = ∫ 0 ∞ λ 1 + λ s ( − t λ ) ( 1 + λ s ) − t λ f ( t ) d t = − 1 1 + λ s ∫ 0 ∞ ( 1 + λ s ) − t λ t f ( t ) d t = − 1 1 + λ s L λ ∗ ( t f ( t ) ) .$
By (46), we obtain the following theorem.
Theorem 10.
For $λ ∈ ( 0 , 1 ]$ and $0 < R e ( s )$, we have
$d F λ ∗ d s = − 1 1 + λ s L λ ∗ ( t f ( t ) ) .$

## 4. Conclusions

Kim-Kim ([9]) defined a degenerate gamma function and a degenerate Laplace transformation. The motivation of this paper is to define modified degenerate gamma functions and modified degenerate Laplace transformations which are different to degenerate gamma function and degenerate Laplace transformation and to obtain more useful results which are Theorems 7 and 8 for the modified degenerate Laplace transformation. We do not obtain these result from the degenerate Laplace transformation. Also, we investigated some results which are Theorems 1 and 3 for modified degenerate gamma functions. Furthermore, Theorems 6 and 9 are some interesting properties which are applied to differential equations in engineering mathematics.

## Author Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

## Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).

## Conflicts of Interest

The authors declare that they have no competing interests.

## References

1. Kreyszig, E.; Kreyszig, H.; Norminton, E.J. Advanced Engineering Mathematics; John Wiley & Sons Inc.: New Jersey, NJ, USA, 2011. [Google Scholar]
2. Kim, T.; Kim, D.S. Degenerate Laplace transform and degenerate Gamma function. Russ. J. Math. Phys. 2017, 24, 241–248. [Google Scholar] [CrossRef]
3. Chung, W.S.; Kim, T.; Kwon, H.I. On the q-analog of the Laplace transform. Russ. J. Math. Phys. 2014, 21, 156–168. [Google Scholar] [CrossRef]
4. Spiegel, M.R. Laplace Transforms (Schaum’s Outlines); McGraw Hill: New York, NY, USA, 1965. [Google Scholar]
5. Upadhyaya, L.M. On the degenerate Laplace transform. Int. J. Eng. Sci. Res. 2018, 6, 198–209. [Google Scholar] [CrossRef]
6. Carlitz, L. A degenerate Staudt-Clausen Theorem. Arch. Math. (Basel) 1956, 7, 28–33. [Google Scholar] [CrossRef]
7. Carlitz, L. Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 1979, 15, 51–88. [Google Scholar]
8. Dolgy, D.V.; Kim, T.; Seo, J.-J. On the symmetric identities of modified degenerate Bernoulli polynomials. Proc. Jangjeon Math. Soc. 2016, 19, 301–308. [Google Scholar]
9. Kim, D.S.; Kim, T. Some identities of degenerate Euler polynomials arising from p-adic fermionic integral on ℤp. Integral Transf. Spec. Funct. 2015, 26, 295–302. [Google Scholar] [CrossRef]
10. Kim, T. Degenerate Euler Zeta function. Russ. J. Math. Phys. 2015, 22, 469–472. [Google Scholar] [CrossRef]
11. Kim, T. On the degenerate q-Bernoulli polynomials. Bull. Korean Math. Soc. 2016, 53, 1149–1156. [Google Scholar] [CrossRef]
12. Kim, T.; Kim, D.S.; Dolgy, D.V. Degenerate q-Euler polynomials. Adv. Difference Equ. 2015, 2015, 13662. [Google Scholar] [CrossRef]
13. Kim, T.; Dolgy, D.V.; Kim, D.S. Symmetric identities for degenerate generalized Bernoulli polynomials. J. Nonlinear Sci. Appl. 2016, 9, 677–683. [Google Scholar] [CrossRef] [Green Version]
14. Kim, T.; Kim, D.S.; Seo, J.J. Differential equations associated with degenerate Bell polynomials. Int. J. Pure Appl. Math. 2016, 108, 551–559. [Google Scholar]
15. Kwon, H.I.; Kim, T.; Seo, J.-J. A note on degenerate Changhee numbers and polynomials. Proc. Jangjeon Math. Soc. 2015, 18, 295–305. [Google Scholar]
16. Kim, T.; Jang, G.-W. A note on degenerate gamma function and degenerate Stirling number of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 2018, 28, 207–214. [Google Scholar]
17. Kim, T.; Yao, Y.; Kim, D.S.; Jang, G.-W. Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 2018, 25, 44–58. [Google Scholar] [CrossRef]
18. Kim, T.; Kim, D.S. A new approach to Catalan numbers using differential equations. Russ. J. Math. Phys. 2017, 24, 465–475. [Google Scholar] [CrossRef] [Green Version]
19. Jang, G.-W.; Kim, T.; Kwon, H.I. On the extension of degenerate Stirling polynomials of the second kind and degenerate Bell polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 2018, 28, 305–316. [Google Scholar]

## Share and Cite

MDPI and ACS Style

Kim, Y.; Kim, B.M.; Jang, L.-C.; Kwon, J. A Note on Modified Degenerate Gamma and Laplace Transformation. Symmetry 2018, 10, 471. https://doi.org/10.3390/sym10100471

AMA Style

Kim Y, Kim BM, Jang L-C, Kwon J. A Note on Modified Degenerate Gamma and Laplace Transformation. Symmetry. 2018; 10(10):471. https://doi.org/10.3390/sym10100471

Chicago/Turabian Style

Kim, YunJae, Byung Moon Kim, Lee-Chae Jang, and Jongkyum Kwon. 2018. "A Note on Modified Degenerate Gamma and Laplace Transformation" Symmetry 10, no. 10: 471. https://doi.org/10.3390/sym10100471

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.