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A study on a type of degenerate poly-Dedekind sums

  • Yuankui Ma , Lingling Luo , Taekyun Kim EMAIL logo , Hongze Li and Wenpeng Zhang EMAIL logo
Published/Copyright: January 10, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.

Keywords: degenerate poly-Dedekind sums; degenerate polylogarithm functions; degenerate poly-Bernoulli polynomials
MSC 2010: 11F20; 11B68; 11B83

1 Introduction and preliminaries

As is well known, the Euler polynomials and Bernoulli polynomials are given by

(1) 2 e t + 1 e x t = ∑ n = 0 ∞ E n ( x ) t n n ! , t e t − 1 e x t = ∑ n = 0 ∞ B n ( x ) t n n ! ( see [1,2] ) ,

when x = 0 , E n = E n ( 0 ) are the Euler numbers; B n = B n ( 0 ) are the Bernoulli numbers.

Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them

(2) S p ( h , m ) = ∑ μ = 1 m − 1 μ m B ¯ p h μ m ( see [3–9] ) ,

where B ¯ p ( x ) = B p ( x − [ x ] ) are the Bernoulli functions, where [ x ] denotes the greatest integer function not exceeding x .

As one generalization of the Apostol’s generalized Dedekind sums, the poly-Dedekind sums associated with the type 2 poly-Bernoulli functions were introduced (see [10,11] ). The poly-Dedekind sums associated with the poly-Bernoulli functions were recently introduced,

(3) S p ( k ) ( h , m ) = ∑ μ = 1 m − 1 μ m B ¯ p ( k ) h μ m ( see [12] ) ,

when k = 1 , we have B p ( 1 ) ( x ) = B p ( x ) , where B ¯ p ( k ) ( x ) = B p ( k ) ( x − [ x ] ) are the poly-Bernoulli functions, B p ( k ) ( x ) are the poly-Bernoulli polynomials given by

(4) L i k ( 1 − e − t ) 1 − e − t = ∑ p = 0 ∞ B p ( k ) ( x ) t p p ! ( see [12] ) ,

when x = 0 , B p ( k ) = B p ( k ) ( 0 ) are the poly-Bernoulli numbers.

Kim first introduced the Dedekind-type DC sums

(5) T p ( h , m ) = 2 ∑ μ = 0 m − 1 ( − 1 ) μ μ m E ¯ p h μ m , ( h , m ∈ N ) ( see [13,14] ) ,

where E ¯ p ( x ) = E p ( x − [ x ] ) are the Euler functions (see [3,13,14,19]).

Simsek (see [14]) found trigonometric representations of the Dedekind-type DC sums and their relations to Clausen functions, polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), and Hardy-Berndt sums.

The poly-Dedekind-type DC sums, which are obtained by replacing the Euler functions appearing in Dedekind-type DC sums by any poly-Euler functions of arbitrary indices, were given by

(6) T p ( k ) ( h , m ) = 2 ∑ μ = 1 m − 1 ( − 1 ) μ μ m E ¯ p ( k ) h μ m ( see [15] ) ,

where h , m , p ∈ N , E ¯ p ( k ) ( x ) are the poly-Euler functions, E ¯ p ( k ) ( x ) = E p ( k ) ( x − [ x ] ) , E p ( k ) ( x ) are the poly-Euler polynomials given by

(7) L i k ( 1 − e − 2 t ) t ( e t + 1 ) e x t = ∑ p = 0 ∞ E p ( k ) ( x ) t p p ! ( see [1,2,12] ) ,

when x = 0 , E p ( k ) = E p ( k ) ( 0 ) are the poly-Euler numbers.

In this article, as another generalization, we consider degenerate poly-Dedekind sums, which are obtained by replacing the first Bernoulli function appearing in Dedekind sums by degenerate poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for them.

For the rest of this section, we recall some necessary facts that are needed throughout this article. For any λ ∈ R , the degenerate exponential function is defined by

(8) e λ x ( t ) = ∑ n = 0 ∞ ( x ) n , λ t n n ! , e λ ( t ) = e λ 1 ( t ) ( see [1,11,16–18] ) ,

where ( x ) 0 , λ = 1 , ( x ) n , λ = x ( x − λ ) ⋯ ( x − ( n − 1 ) λ ) , ( n ≥ 1 ) .

In [19], the degenerate Bernoulli polynomials are defined by

(9) t e λ ( t ) − 1 e λ x ( t ) = ∑ n = 0 ∞ β n , λ ( x ) t n n ! ,

when x = 0 , β n , λ = β n , λ ( 0 ) are the degenerate Bernoulli numbers.

We observe that

(10) ∑ l = 0 n − 1 e λ l ( t ) = t t ( e λ ( t ) − 1 ) ( e λ n ( t ) − 1 ) = 1 t ∑ j = 0 ∞ ( B j , λ ( n ) − B j , λ ) t j j ! = ∑ j = 0 ∞ B j + 1 , λ ( n ) − B j + 1 , λ j + 1 t j j ! .

On the other hand,

(11) ∑ l = 0 n − 1 e λ l ( t ) = ∑ l = 0 n − 1 e λ ∕ l ( l t ) = ∑ j = 0 ∞ ∑ l = 0 n − 1 l j ( 1 ) j , λ ∕ l t j j ! .

By (10) and (11), we have

(12) ∑ l = 0 n − 1 l j ( 1 ) j , λ ∕ l = B j + 1 , λ ( n ) − B j + 1 , λ j + 1 .

The type 2 degenerate Stirling numbers are defined by (see [20,21])

(13) 1 k ! ( e λ ( t ) − 1 ) k = ∑ n = k ∞ S 2 , λ ( n , k ) t n n ! , ( n ≥ 0 ) .

The degenerate polylogarithmic function of index k is defined by Kim and Kim to be (see [20])

(14) Li k , λ ( x ) = ∑ n = 1 ∞ ( − λ ) n − 1 ( 1 ) n , 1 ∕ λ ( n − 1 ) ! n k x n , ( k ∈ Z ) , ∣ x ∣ < 1 .

When k = 1 ,

(15) Li 1 , λ ( x ) = − log λ ( 1 − x ) = ∑ n = 1 ∞ ( − λ ) n − 1 ( 1 ) n , 1 ∕ λ n ! x n ,

where log λ ( x ) is the inverse to e λ ( x ) .

In [16], the degenerate poly-Bernoulli polynomials of index k are defined by

(16) Li k , λ ( 1 − e λ ( − t ) ) e λ ( t ) − 1 e λ x ( t ) = ∑ n = 0 ∞ B n , λ ( k ) ( x ) t n n ! .

Note that B n , λ ( 1 ) ( x ) = β n , λ ( x ) , when x = 0 , B n , λ ( k ) = B n , λ ( k ) ( 0 ) are the degenerate poly-Bernoulli numbers.

From (16), we note that

(17) B n , λ ( k ) ( x ) = ∑ l = 0 n n l B l , λ ( k ) ( x ) n − l , λ = ∑ l = 0 n n l B n − l , λ ( k ) ( x ) l , λ , ( n ≥ 0 ) .

2 Reciprocity relations for the degenerate poly-Dedekind sums

By replacing poly-Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices from the poly-Dedekind sums, the degenerate poly-Dedekind sums are given in the following,

(18) S p , λ ( k ) ( h , m ) = ∑ μ = 1 m − 1 μ m B ¯ p , λ ( k ) h μ m ,

where h , m , p ∈ N , B ¯ p , λ ( k ) ( x ) are the degenerate poly-Bernoulli functions, B ¯ p , λ ( k ) ( x ) = B p , λ ( k ) ( x − [ x ] ) .

Note that

(19) S p , λ ( 1 ) ( h , m ) = ∑ μ = 1 m − 1 μ m B ¯ p , λ ( 1 ) h μ m = ∑ μ = 1 m − 1 μ m β ¯ p , λ h μ m .

From (16), we note that

(20) Li k , λ ( 1 − e λ ( − t ) ) e λ ( t ) − 1 = ∑ n = 0 ∞ B n , λ ( k ) t n n ! .

So we have

(21) Li k , λ ( 1 − e λ ( − t ) ) = ∑ l = 0 ∞ B l , λ ( k ) t l l ! ( e λ ( t ) − 1 ) = ∑ l = 0 ∞ B l , λ ( k ) t l l ! ∑ m = 0 ∞ ( 1 ) m , λ t m m ! − 1 = ∑ n = 1 ∞ ( B n , λ ( k ) ( 1 ) − B n , λ ( k ) ) t n n ! .

On the other hand, we also have

(22) Li k , λ ( 1 − e λ ( − t ) ) = ∑ m = 1 ∞ ( − λ ) m − 1 ( 1 ) m , 1 ∕ λ ( m − 1 ) ! m k ( 1 − e λ ( − t ) ) m = ∑ m = 1 ∞ ( − λ ) m − 1 ( 1 ) m , 1 ∕ λ m k − 1 ( − 1 ) m m ! ( e λ ( − t ) − 1 ) m = ∑ m = 1 ∞ ( − λ ) m − 1 ( 1 ) m , 1 ∕ λ ( − 1 ) m m k − 1 ∑ n = m ∞ S 2 , λ ( n , m ) ( − 1 ) n t n n ! = ∑ n = 1 ∞ ∑ m = 1 n λ m − 1 ( 1 ) m , 1 ∕ λ ( − 1 ) n − 1 m k − 1 S 2 , λ ( n , m ) t n n ! .

Therefore, by (21) and (22), we obtain the following theorem.

Theorem 1

For n ∈ N , we have

B n , λ ( k ) ( 1 ) − B n , λ ( k ) = ∑ m = 1 n λ m − 1 ( 1 ) m , 1 ∕ λ ( − 1 ) n − 1 m k − 1 S 2 , λ ( n , m ) .

Note that, for k = 1 , we have the following corollary.

Corollary 1

For n ∈ N , we have

∑ m = 1 n λ m − 1 ( 1 ) m , 1 ∕ λ ( − 1 ) n − 1 S 2 , λ ( n , m ) = δ n , 1 ,

where δ n , k is the Kronecker’s symbol.

For d ∈ N , we observe that

(23) ∑ n = 0 ∞ B n , λ ( k ) ( x ) t n n ! = Li k , λ ( 1 − e λ ( − t ) ) e λ ( t ) − 1 e λ x ( t ) = Li k , λ ( 1 − e λ ( − t ) ) ( e λ ∕ d ( d t ) − 1 ) ∑ i = 0 d − 1 e λ i + x ( t ) = Li k , λ ( 1 − e λ ( − t ) ) d t ∑ i = 0 d − 1 e λ ∕ d x + i d ( d t ) d t e λ ∕ d ( d t ) − 1 = ∑ j = 0 ∞ d j − 1 ∑ i = 0 d − 1 β j , λ ∕ d x + i d t j j ! 1 t ∑ l = 1 ∞ ( − λ ) l − 1 ( 1 ) l , 1 ∕ λ l k 1 ( l − 1 ) ! ( 1 − e λ ( − t ) ) l = ∑ j = 0 ∞ d j − 1 ∑ i = 0 d − 1 β j , λ ∕ d x + i d t j j ! 1 t ∑ l = 1 ∞ ( − 1 ) λ l − 1 ( 1 ) l , 1 ∕ λ l k − 1 1 l ! ( e λ ( − t ) − 1 ) l = ∑ j = 0 ∞ d j − 1 ∑ i = 0 d − 1 β j , λ ∕ d x + i d t j j ! 1 t ∑ m = 1 ∞ ∑ l = 1 m ( − 1 ) m − 1 λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( m , l ) l k − 1 t m m ! = ∑ j = 0 ∞ d j − 1 ∑ i = 0 d − 1 β j , λ ∕ d x + i d t j j ! ∑ m = 0 ∞ ∑ l = 1 m + 1 ( − 1 ) m λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( m + 1 , l ) l k − 1 ( m + 1 ) t m m ! = ∑ n = 0 ∞ ∑ j = 0 n ∑ i = 0 d − 1 ∑ l = 1 n − j + 1 n j d j − 1 β j , λ ∕ d x + i d ( − 1 ) n − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( n − j + 1 , l ) l k − 1 ( n − j + 1 ) t n n ! .

Therefore, by (23), we obtain the following theorem.

Theorem 2

For k ∈ Z , d ∈ N , n ≥ 0 , we have

B n , λ ( k ) ( x ) = ∑ j = 0 n ∑ i = 0 d − 1 ∑ l = 1 n − j + 1 n j d j − 1 β j , λ ∕ d x + i d ( − 1 ) n − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( n − j + 1 , l ) l k − 1 ( n − j + 1 ) .

For m , h ∈ N , by (18) and Theorem 2, we obtain

(24) h m p S p , λ ( k ) ( h , m ) + m h p S p , λ ( k ) ( m , h ) = h m p ∑ μ = 0 m − 1 μ m B ¯ p , λ ( k ) h μ m + m h p ∑ ν = 0 h − 1 ν h B ¯ p , λ ( k ) m ν h = h m p ∑ μ = 0 m − 1 μ m ∑ j = 0 p h j − 1 p j ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 β j , λ ∕ h μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) + m h p ∑ ν = 0 h − 1 ν h ∑ j = 0 p m j − 1 p j ∑ μ = 0 m − 1 ∑ l = 1 p − j + 1 β j , λ ∕ m μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) = ∑ μ = 0 m − 1 μ m ∑ j = 0 p m p − j ( m h ) j p j ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 β j , λ ∕ h μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) + ∑ ν = 0 h − 1 ν h ∑ j = 0 p h p − j ( m h ) j p j ∑ μ = 0 m − 1 ∑ l = 1 p − j + 1 β j , λ ∕ m μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) = ∑ μ = 0 m − 1 ∑ j = 0 p ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 ( μ h ) ( m h ) − 1 m p − j ( m h ) j p j β j , λ ∕ h μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) + ∑ μ = 0 m − 1 ∑ j = 0 p ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 ( m ν ) ( m h ) − 1 h p − j ( m h ) j p j β j , λ ∕ m μ m + ν h ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) = ∑ μ = 0 m − 1 ∑ j = 0 p ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 p j ( m h ) j − 1 ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) × ( μ h ) m p − j β j , λ ∕ h μ m + ν h + ( m ν ) h p − j β j , λ ∕ m μ m + ν h .

Therefore, by (24), we obtain the following reciprocity theorem for the degenerate poly-Dedekind sums associated with degenerate poly-Bernoulli functions with index k .

Theorem 3

For m , h , p ∈ N , k ∈ Z , we have

h m p S p , λ ( k ) ( h , m ) + m h p S p , λ ( k ) ( m , h ) = ∑ μ = 0 m − 1 ∑ j = 0 p ∑ ν = 0 h − 1 ∑ l = 1 p − j + 1 p j ( m h ) j − 1 ( − 1 ) p − j λ l − 1 ( 1 ) l , 1 ∕ λ S 2 , λ ( p − j + 1 , l ) l k − 1 ( p − j + 1 ) × ( μ h ) m p − j β j , λ ∕ h μ m + ν h + ( m ν ) h p − j β j , λ ∕ m μ m + ν h .

When k = 1 , λ → 0 , by Corollary 1, we obtain the following reciprocity relation for the generalized Dedekind sums defined by Apostol.

Corollary 2

For m , h , p ∈ N , we have

h m p S p ( h , m ) + m h p S p ( m , h ) = ∑ μ = 0 m − 1 ∑ ν = 0 h − 1 ( m h ) p − 1 ( μ h + m ν ) B ¯ p μ m + ν h .

By (12) and (17), let h = 1 , we note that

(25) S p , λ ( k ) ( 1 , m ) = ∑ μ = 1 m − 1 μ m B ¯ p , λ ( k ) μ m = ∑ μ = 1 m − 1 μ m ∑ ν = 0 p p ν B ν , λ ( k ) μ m p − ν , λ = ∑ ν = 0 p p ν B ν , λ ( k ) ∑ μ = 1 m − 1 μ m p − ν + 1 ( 1 ) p − ν , m μ λ = ∑ ν = 0 p p ν B ν , λ ( k ) ∑ μ = 1 m − 1 μ m p − ν + 1 ( 1 ) p − ν + 1 , m μ λ + ( p − ν ) m μ λ ( 1 ) p − ν , m μ λ = ∑ ν = 0 p p ν B ν , λ ( k ) ∑ μ = 1 m − 1 μ m p − ν + 1 ( 1 ) p − ν + 1 , m μ λ + λ ∑ ν = 0 p p ν ( p − ν ) B ν , λ ( k ) ∑ μ = 1 m − 1 μ m p − ν ( 1 ) p − ν , m μ λ = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν ( B p + 2 − ν , λ ( m ) − B p + 2 − ν , λ ) + λ ∑ ν = 0 p p ν ( p − ν ) B ν , λ ( k ) 1 p + 1 − ν ( B p + 1 − ν , λ ( m ) − B p + 1 − ν , λ ) .

By (17), we have

(26) B p + 2 − ν , λ ( m ) − B p + 2 − ν , λ = ∑ i = 0 p + 2 − ν p + 2 − ν i B i , λ ( m ) p + 2 − ν − i , λ − B p + 2 − ν , λ = ∑ i = 0 p + 1 − ν p + 2 − ν i B i , λ ( m ) p + 2 − ν − i , λ .

In the same way as (26), we have

(27) B p + 1 − ν , λ ( m ) − B p + 1 − ν , λ = ∑ i = 0 p − ν p + 1 − ν i B i , λ ( m ) p + 1 − ν − i , λ .

By (25), (26), and (27), we obtain

(28) S p , λ ( k ) ( 1 , m ) = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν ( B p + 2 − ν , λ ( m ) − B p + 2 − ν , λ ) + λ ∑ ν = 0 p p ν ( p − ν ) B ν , λ ( k ) 1 p + 1 − ν ( B p + 1 − ν , λ ( m ) − B p + 1 − ν , λ ) = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν ∑ i = 0 p + 1 − ν p + 2 − ν i B i , λ ( m ) p + 2 − ν − i , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν ∑ i = 0 p − ν p + 1 − ν i B i , λ ( m ) p + 1 − ν − i , λ .

Now we assume that p ≥ 3 is an odd positive integer, so that B p , λ = 0 . Then we have

(29) m p S p , λ ( k ) ( 1 , m ) = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν ∑ i = 0 p + 1 − ν p + 2 − ν i B i , λ m p ( m ) p + 2 − ν − i , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν ∑ i = 0 p − ν p + 1 − ν i B i , λ m p ( m ) p + 1 − ν − i , λ = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν m p ( m ) p + 2 − ν , λ + ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν ∑ i = 1 p + 1 − ν p + 2 − ν i B i , λ m p ( m ) p + 2 − ν − i , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν m p ( m ) p + 1 − ν , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν ∑ i = 1 p − ν p + 1 − ν i B i , λ m p ( m ) p + 1 − ν − i , λ = ∑ ν = 0 p p ν B ν , λ ( k ) 1 p + 2 − ν m p ( m ) p + 2 − ν , λ + ∑ i = 1 p + 1 ∑ ν = 0 p + 1 − i p ν p + 2 − ν i B ν , λ ( k ) p + 2 − ν B i , λ m p ( m ) p + 2 − ν − i , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν m p ( m ) p + 1 − ν , λ + λ ∑ i = 1 p ∑ ν = 0 p − i p ν p + 1 − ν i p − ν p + 1 − ν B ν , λ ( k ) B i , λ m p ( m ) p + 1 − ν − i , λ = ∑ ν = 0 p p ν B ν , λ ( k ) p + 2 − ν m p ( m ) p + 2 − ν , λ + ∑ i = 1 p − 1 ∑ ν = 0 p + 1 − i p ν p + 2 − ν i B ν , λ ( k ) p + 2 − ν B i , λ m p ( m ) p + 2 − ν − i , λ + p + 2 p + 1 B p + 1 , λ p + 2 m p ( m ) 1 , λ + ∑ ν = 0 1 p ν p + 1 − ν p B ν , λ ( k ) p + 2 − ν B p , λ m p ( m ) 2 − ν , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν m p ( m ) p + 1 − ν , λ + λ ∑ i = 1 p − 1 ∑ ν = 0 p − i p ν p + 1 − ν i p − ν p + 1 − ν B ν , λ ( k ) B i , λ m p ( m ) p + 1 − ν − i , λ + λ p + 1 p p p + 1 B p , λ m p ( m ) 1 , λ = ∑ ν = 0 p p ν B ν , λ ( k ) p + 2 − ν m p ( m ) p + 2 − ν , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν m p ( m ) p + 1 − ν , λ + ∑ i = 1 p − 1 ∑ ν = 0 p + 1 − i p ν p + 2 − ν i B ν , λ ( k ) p + 2 − ν B i , λ m p ( m ) p + 2 − ν − i , λ + λ ∑ i = 1 p − 1 ∑ ν = 0 p − i p ν p + 1 − ν i p − ν p + 1 − ν B ν , λ ( k ) B i , λ m p ( m ) p + 1 − ν − i , λ + B p + 1 , λ m p + 1 .

So, by (29), we obtain the following theorem.

Theorem 4

Let p ≥ 3 be an odd positive integer. Then we have

m p S p , λ ( k ) ( 1 , m ) = ∑ ν = 0 p p ν B ν , λ ( k ) p + 2 − ν m p ( m ) p + 2 − ν , λ + λ ∑ ν = 0 p p ν B ν , λ ( k ) p − ν p + 1 − ν m p ( m ) p + 1 − ν , λ + ∑ i = 1 p − 1 ∑ ν = 0 p + 1 − i p ν p + 2 − ν i B ν , λ ( k ) p + 2 − ν B i , λ m p ( m ) p + 2 − ν − i , λ + λ ∑ i = 1 p − 1 ∑ ν = 0 p − i p ν p + 1 − ν i p − ν p + 1 − ν B ν , λ ( k ) B i , λ m p ( m ) p + 1 − ν − i , λ + B p + 1 , λ m p + 1 .

3 Conclusion

As a further generalization of Dedekind sums, we further study the reciprocity relations of degenerate poly-Dedekind sums. In this article, we introduced the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli polynomials by degenerate poly-Bernoulli functions. We derived several explicit expressions and identities for the poly-Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind. We also investigated the poly-Bernoulli polynomials in terms of the degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind. Meanwhile, we obtained the reciprocity relations for the degenerate poly-Dedekind sums associated with degenerate poly-Bernoulli functions. It can be further explored in connection with modular forms, ζ function, and trigonometric sums, just as in the cases of Apostol-Dedekind sums.

  1. Funding information: This research was funded by the National Natural Science Foundation of China (No. 12271320, 12171311), Key Research and Development Program of Shaanxi (No. 2023-ZDLGY-02).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: All authors declare that there is no ethical problem in the production of this article.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2022-09-20
Revised: 2023-07-31
Accepted: 2023-09-28
Published Online: 2024-01-10

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
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  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
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  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
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  86. A study on a type of degenerate poly-Dedekind sums
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  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
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  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
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  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
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  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
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  108. New extensions related to Fejér-type inequalities for GA-convex functions
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  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
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  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2023-0121
Keywords for this article
degenerate poly-Dedekind sums; degenerate polylogarithm functions; degenerate poly-Bernoulli polynomials
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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