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A note on polyexponential and unipoly Bernoulli polynomials of the second kind

  • Minyoung Ma and Dongkyu Lim ORCID logo EMAIL logo
Published/Copyright: August 27, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are attached to an arithmetic function. We derive their explicit expressions and some identities involving poly-Bernoulli numbers of the second kind and unipoly Bernoulli numbers of the second kind.

Keywords: polyexponential function; unipoly function; poly-Bernoulli numbers of the second kind
MSC 2010: 05A19; 11B83; 34A34

1 Introduction

Kaneko [1] studied the poly-Bernoulli polynomials that are defined using the polylogarithm functions. The polyexponential functions were first studied by Hardy [2,3] and reconsidered by Kim and Kim [4,5] as an inverse type to the polylogarithm function. And recently, Kim et al. [6] studied the degenerate poly-Bernoulli polynomials and numbers arising from polyexponential functions, and they derived explicit identities involving them.

For k ∈ Z , the polylogarithm functions L i k ( x ) are defined by power series in x as

(1) L i k ( x ) = ∑ n = 1 ∞ x n n k = x + x 2 2 k + x 3 3 k ⋯ , ( ∣ x ∣ < 1 ) .

Note that

L i 1 ( x ) = ∑ n = 1 ∞ x n n = − log ( 1 − x ) .

The polyexponential function e ( x , a ∣ s ) is given by

e ( x , a ∣ s ) = ∑ n = 0 ∞ x n ( n + a ) s n ! , ( ℜ ( a ) > 0 ) .

And, an inverse type to the polylogarithm functions in (1), which are again called the modified polyexponential functions E i k ( x ) , is given by

E i k ( x ) = ∑ n = 1 ∞ x n n k ( n − 1 ) ! , ( k ∈ Z ) .

We note that e ( x , 1 ∣ k ) = 1 x E i k ( x ) and E i 1 ( x ) = e x − 1 .

For n ≥ 0 , the Stirling numbers of the first kind S ℓ ( n , ℓ ) are defined by

(2) ( x ) n = ∑ ℓ = 0 n S 1 ( n , ℓ ) x ℓ ,

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

From (2), we can easily see that

1 k ! ( log ( 1 + t ) ) k = ∑ n = k ∞ S 1 ( n , k ) t n n ! .

In the inversion expression to (2), for n ≥ 0 , the Stirling numbers of the second kind S 2 ( n , ℓ ) are defined by [7,8]

(3) x n = ∑ ℓ = 0 n S 2 ( n , ℓ ) ( x ) ℓ .

From (3), it is easily seen that

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

This paper is organized as follows. In Section 2, we study the poly-Bernoulli numbers of the second kind and derive explicit expressions and some identities involving them. See Theorems 2.1, 2.2, 2.4 and 2.7. In Section 3, we study the unipoly function attached to arithmetic functions, which are defined by Kim and Kim [4,5]. We also define unipoly Bernoulli numbers of the second kind and we investigate some identities of them. See Theorems 3.1, 3.2 and 3.6.

2 Poly-Bernoulli polynomials and numbers of the second kind

The Bernoulli polynomials of the second kind b n ( x ) are given by

t log ( 1 + t ) ( 1 + t ) x = ∑ n = 0 n b n ( x ) t n n ! ( see [ 9 , 10 ] ) .

When x = 0 , b n = b n ( 0 ) are called the Bernoulli numbers of the second kind.

For k ∈ Z , we introduce the poly-Bernoulli polynomials of the second kind b n ( k ) ( x ) by means of the modified polylogarithm function as follows:

(4) E i k ( log ( 1 + t ) ) log ( 1 + t ) ( 1 + t ) x = ∑ n = 0 n b n ( k ) ( x ) t n n ! .

When x = 0 , b n ( k ) = b n ( k ) ( 0 ) are called the poly-Bernoulli numbers of the second kind. For k = 1 , b n ( 1 ) ( x ) = b n ( x ) , ( n ≥ 0 ) are just Bernoulli polynomials of the second kind.

In the following theorems, we use the idea developed by Kim et al. in [11].

Theorem 2.1

For n ≥ 0 and k ∈ Z , we have

b n ( k ) ( x ) = ∑ m = 0 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) ℓ k − 1 b n − m ( x ) .

Proof

First, we note that

E i k ( log ( 1 + t ) ) = ∑ n = 1 ∞ ∑ ℓ = 1 n ( log ( 1 + t ) ) ℓ ( ℓ − 1 ) ! ℓ k .

By combining this with (4), we observe that

t log ( 1 + t ) ( 1 + t ) x 1 t E i k ( log ( 1 + t ) ) = ∑ j = 0 ∞ b j ( x ) t j j ! ∑ m = 0 ∞ ∑ ℓ = 1 m + 1 S 1 ( m + 1 , ℓ ) ( m + 1 ) ℓ k − 1 t m m ! = ∑ n = 0 ∞ ∑ m = 0 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) ℓ k − 1 b n − m ( x ) t n n ! .

Comparing this with (4) leads to the desired result.□

Let us take k = 1 in Theorem 2.1, then we yield

(5) b n ( x ) = ∑ m = 0 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) b n − m ( x ) .

For the case m = 0 in the right hand side of (5), we have the following identity:

(6) ∑ m = 1 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) b n − m ( x ) = 0 .

Taking x = 0 in (5), we get

b n = ∑ m = 0 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) ℓ k − 1 b n − m ,

and taking x = 0 in (6), we get

∑ m = 1 n ∑ ℓ = 1 m + 1 n m S 1 ( m + 1 , ℓ ) ( m + 1 ) ℓ k − 1 b n − m = 0 .

For the next theorem, we need the following well-known identity:

t log ( 1 + t ) r ( 1 + t ) x − 1 = ∑ n = 0 ∞ B n ( n − r + 1 ) ( x ) t n n ! ( cf . [ 4 , 5 ] ) .

Here, B n ( r ) ( x ) are the Bernoulli polynomials of order r , which are given by

(7) t e t − 1 r e x t = ∑ n = 0 ∞ B n ( r ) ( x ) t n n ! ( see [ 4 , 5 ] ) .

Theorem 2.2

For n ≥ 0 and k ∈ Z , we have

b n ( k ) = ∑ m = 0 n n m ∑ m 1 + m 2 + ⋯ + m k − 1 = m m m 1 , m 2 , … , m k − 1 b n − m × B m 1 ( m 1 ) ( 0 ) m 1 + 1 B m 2 ( m 2 ) ( 0 ) m 1 + m 2 + 1 ⋯ B m k − 1 ( m k − 1 ) ( 0 ) m 1 + m 2 + ⋯ + m k − 1 + 1 .

Proof

We observe that

(8) d d x E i k ( log ( 1 + x ) ) = 1 ( 1 + x ) log ( 1 + x ) E i k − 1 ( log ( 1 + x ) ) .

Thus from (8), for k ≥ 2 , we have

(9) E i k ( log ( 1 + x ) ) = ∫ 0 x 1 ( 1 + t ) log ( 1 + t ) E i k − 1 ( log ( 1 + t ) ) d t = ∫ 0 x 1 ( 1 + t ) log ( 1 + t ) ∫ 0 t 1 ( 1 + t ) log ( 1 + t ) ⋯ ∫ 0 t E i k − 1 log ( 1 + t ) ( 1 + t ) log ( 1 + t ) d t ⋯ d t = ∫ 0 x 1 ( 1 + t ) log ( 1 + t ) ∫ 0 t 1 ( 1 + t ) log ( 1 + t ) ⋯ ∫ 0 t ︸ ( k − 2 ) -times t ( 1 + t ) log ( 1 + t ) d t ⋯ d t .

By (4), (7) and (9), it follows that

∑ n = 0 ∞ b n ( k ) x n n ! = E i k ( log ( 1 + x ) ) log ( 1 + x ) = 1 log ( 1 + x ) ∫ 0 x 1 ( 1 + t ) log ( 1 + t ) ∫ 0 t 1 ( 1 + t ) log ( 1 + t ) ⋯ ∫ 0 t ︸ ( k − 2 ) -times t ( 1 + t ) log ( 1 + t ) d t ⋯ d t = x log ( 1 + x ) ∑ m = 0 ∞ ∑ m 1 + m 2 + ⋯ + m k − 1 = m m m 1 , m 2 , … , m k − 1 × B m 1 ( m 1 ) ( 0 ) m 1 + 1 B m 2 ( m 2 ) ( 0 ) m 1 + m 2 + 1 ⋯ B m k − 1 ( m k − 1 ) ( 0 ) m 1 + m 2 + ⋯ + m k − 1 + 1 x m m ! = ∑ n = 0 ∞ ∑ m = 0 n n m ∑ m 1 + m 2 + ⋯ + m k − 1 = m m m 1 , m 2 , … , m k − 1 b n − m × B m 1 ( m 1 ) ( 0 ) m 1 + 1 B m 2 ( m 2 ) ( 0 ) m 1 + m 2 + 1 ⋯ B m k − 1 ( m k − 1 ) ( 0 ) m 1 + m 2 + ⋯ + m k − 1 + 1 x n n ! .

Equating coefficients on the very ends of the above identity arrives at the required result.□

In particular, for k = 2 , we have the following:

Corollary 2.3

For n ≥ 0 , we have

b n ( 2 ) = ∑ ℓ = 0 n n ℓ B ℓ ( ℓ ) ( 0 ) ℓ + 1 b n − ℓ .

Theorem 2.4

For n ≥ 0 and k ∈ Z , we have

∑ m = 0 n b m ( k ) S 2 ( n , m ) = 1 ( n + 1 ) k .

Proof

Replacing t by e t − 1 in (4) and x = 0 , we have

(10) E i k ( t ) t = ∑ m = 0 ∞ b m ( k ) ( e t − 1 ) m m ! = ∑ m = 0 ∞ b m ( k ) ∑ n = m ∞ S 2 ( n , m ) t n n ! = ∑ n = 0 ∞ ∑ m = 0 n b m ( k ) S 2 ( n , m ) t n n ! .

On the other hand,

(11) E i k ( t ) t = 1 t ∑ ℓ = 1 ∞ t ℓ ( ℓ − 1 ) ! ℓ k = ∑ ℓ = 1 ∞ t ℓ − 1 ( ℓ − 1 ) ! ℓ k = ∑ ℓ = 0 ∞ t ℓ ℓ ! ( ℓ + 1 ) k .

Therefore, by comparing the coefficients of (10) and (11), we obtain

∑ m = 0 n b m ( k ) S 2 ( n , m ) = 1 ( n + 1 ) k . □

Taking k = 1 in Theorem 2.4, we obtain the following identity:

Corollary 2.5

For n ≥ 0 , we have

∑ m = 0 n b m S 2 ( n , m ) = 1 n + 1 .

Remark 2.6

In [8, p. 294], the Bernoulli numbers of the second kind are given by

(12) b n = ∫ 0 1 ( x ) n d x = ∑ m = 0 n S 1 ( n , m ) 1 m + 1 .

Thus, Corollary 2.5 can be regarded as the inversion formula of (12).

Recall from [12,13] that the Daehee numbers are defined by the generating function to be

log ( 1 + t ) t = ∑ n = 0 ∞ D n t n n ! .

Now, we express poly-Bernoulli numbers of the second kind related to Daehee numbers and Bernoulli numbers of the second kind.

Theorem 2.7

For n ≥ 0 and k ∈ Z , we have

∑ m = 0 n b m ( k ) S 2 ( n , m ) = ∑ m = 0 n ∑ j = 0 m n m m j D j b m − j 1 ( n − m + 1 ) k .

Proof

We observe that

E i k ( t ) t = log ( 1 + t ) t E i k ( t ) log ( 1 + t ) = log ( 1 + t ) t t log ( 1 + t ) ∑ ℓ = 1 ∞ t ℓ − 1 ( ℓ − 1 ) ! ℓ k = ∑ j = 0 ∞ D j t j j ! ∑ i = 0 ∞ b i t i i ! ∑ ℓ = 0 ∞ t ℓ ( ℓ + 1 ) k ℓ ! = ∑ m = 0 ∞ ∑ j = 0 m m j D j b m − j t m m ! ∑ ℓ = 0 ∞ t ℓ ( ℓ + 1 ) k ℓ ! = ∑ n = 0 ∞ ∑ m = 0 n ∑ j = 0 m n m m j D j b m − j 1 ( n − m + 1 ) k t n n ! .

Combining this identity with (10) results in the required identity.□

For the case k = 1 in Theorem 2.7 and Corollary 2.5, we have

Corollary 2.8

For n ≥ 0 , we have

∑ m = 0 n b m S 2 ( n , m ) = ∑ m = 0 n ∑ j = 0 m n m m j D j b m − j ∑ ℓ = 0 n − m b ℓ S 2 ( n − m , ℓ ) .

3 Unipoly Bernoulli polynomials and numbers of the second kind

Let p be any arithmetic function, which is real or complex valued function defined on the set of positive integer N . In [4], Kim and Kim defined the unipoly function attached to p by

(13) u k ( x ∣ p ) = ∑ n = 1 ∞ p ( n ) x n n k , ( k ∈ Z ) .

For the case p ( n ) = 1 ,

u k ( x ∣ 1 ) = ∑ n = 1 ∞ x p n k = L i k ( x )

is the polylogarithm function in (1). And for k ≥ 2 , we have

d d x u k ( x ∣ p ) = 1 x u k − 1 ( x ∣ p ) ( see [ 4 , 14 ] )

and

u k ( x ∣ p ) = ∫ 0 x 1 t ∫ 0 t 1 t ⋯ ∫ 0 t ︸ ( k − 2 ) -times u 1 ( t ∣ p ) t d t ⋯ d t .

By using (13), we define the unipoly Bernoulli polynomials of the second kind as follows:

(14) 1 log ( 1 + t ) u k ( log ( 1 + t ) ∣ p ) ( 1 + t ) x = ∑ n = 0 ∞ b n , p ( k ) ( x ) t n n ! .

In the following theorem, we can connect the poly-Bernoulli polynomials of the second kind and the unipoly Bernoulli polynomials of the second kind, by taking p ( n ) = 1 Γ ( n ) , where Γ ( n ) = ( n − 1 ) ! is the well-known gamma function.

Theorem 3.1

If we take p ( n ) = 1 Γ ( n ) for n ≥ 0 and k ∈ Z , then we have

b n , p ( k ) ( x ) = b n ( k ) ( x ) .

Proof

From (14), we can derive

∑ n = 0 ∞ b n , p ( k ) ( x ) t n n ! = 1 log ( 1 + t ) u k log ( 1 + t ) 1 Γ ( n ) ( 1 + t ) x = 1 log ( 1 + t ) ∑ m = 1 ∞ ( log ( 1 + t ) ) m m k ( m − 1 ) ! ( 1 + t ) x = 1 log ( 1 + t ) E i k ( log ( 1 + t ) ) ( 1 + t ) x = ∑ n = 0 ∞ b n ( k ) ( x ) t n n ! .

Thus, by comparing the coefficients on both sides, we can obtain the result.□

Now, we have the explicit expression for the unipoly Bernoulli numbers of the second kind as follows:

Theorem 3.2

For n ≥ 0 and k ∈ Z , we have

b n , p ( k ) ( x ) = ∑ ℓ = 0 n ∑ m = 0 ℓ n ℓ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k S 1 ( ℓ + 1 , m + 1 ) ℓ + 1 b n − ℓ ( x ) .

Proof

This follows from the observation that

∑ n = 0 ∞ b n , p ( k ) ( x ) t n n ! = ( 1 + t ) x log ( 1 + t ) ∑ m = 1 ∞ p ( m ) m k ( log ( 1 + t ) ) m = ( 1 + t ) x log ( 1 + t ) ∑ m = 0 ∞ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k ( log ( 1 + t ) ) m + 1 ( m + 1 ) ! = ( 1 + t ) x log ( 1 + t ) ∑ m = 0 ∞ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k ∑ ℓ = m + 1 ∞ S 1 ( ℓ , m + 1 ) t ℓ ℓ ! = t ( 1 + t ) x log ( 1 + t ) ∑ m = 0 ∞ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k ∑ ℓ = m ∞ S 1 ( ℓ + 1 , m + 1 ) ℓ + 1 t ℓ ℓ ! = ∑ j = 0 ∞ b j ( x ) t j j ! ∑ ℓ = 0 ∞ ∑ m = 0 ℓ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k S 1 ( ℓ + 1 , m + 1 ) ℓ + 1 t ℓ ℓ ! = ∑ n = 0 ∞ ∑ ℓ = 0 n ∑ m = 0 ℓ n ℓ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k S 1 ( ℓ + 1 , m + 1 ) ℓ + 1 b n − ℓ ( x ) t n n ! . □

Taking x = 0 in the above theorem arrives at the following new results.

Corollary 3.3

For n ≥ 0 and k ∈ Z , we have

  1. b n , p ( k ) = ∑ m = 0 n p ( m + 1 ) m ! ( m + 1 ) k S 1 ( n , m ) ;

  2. b n , p ( k ) = ∑ ℓ = 0 n ∑ m = 0 ℓ n ℓ p ( m + 1 ) ( m + 1 ) ! ( m + 1 ) k S 1 ( ℓ + 1 , m + 1 ) ℓ + 1 b n − ℓ .

Proof

  1. We observe that

    ∑ n = 0 ∞ b n , p ( k ) t n n ! = 1 log ( 1 + t ) ∑ m = 1 ∞ p ( m ) m k ( log ( 1 + t ) ) m = ∑ m = 0 ∞ p ( m + 1 ) m ! ( m + 1 ) k ( log ( 1 + t ) ) m m ! = ∑ m = 0 ∞ p ( m + 1 ) m ! ( m + 1 ) k ∑ n = m ∞ S 1 ( n , m ) t n n ! = ∑ n = 0 ∞ ∑ m = 0 n p ( m + 1 ) m ! ( m + 1 ) k S 1 ( n , m ) t n n ! .

    Thus, this completes the proof.

  2. If we take x = 0 in Theorem 3.2, we have the result.□

In particular, if we take p ( n ) = 1 Γ ( n ) in Corollary 3.3, then we have

Corollary 3.4

For n ≥ 0 and k ∈ Z , we have

  1. b n ( k ) = ∑ m = 0 n S 1 ( n , m ) ( m + 1 ) k ;

  2. b n ( k ) = ∑ ℓ = 0 n ∑ m = 0 ℓ n ℓ S 1 ( ℓ + 1 , m + 1 ) ( m + 1 ) k − 1 ( ℓ + 1 ) b n − ℓ .

Remark 3.5

Corollary 3.4 is the inversion formula of Theorem 2.4.

Theorem 3.6

For n ≥ 0 and k ∈ Z , we have

∑ m = 0 n b m , p ( k ) S 2 ( n , m ) = p ( n + 1 ) n ! ( n + 1 ) k .

Proof

Replacing t by e t − 1 in (14) gives

u k ( t ∣ p ) t = ∑ m = 0 n b m , p ( k ) ( e t − 1 ) m m ! = ∑ m = 0 n b m , p ( k ) ∑ n = m ∞ S 2 ( n , m ) t n n ! = ∑ n = 0 ∞ ∑ m = 0 n b m , p ( k ) S 2 ( n , m ) t n n ! .

Furthermore, we observe that

u k ( t ∣ p ) t = 1 t ∑ n = 1 ∞ p ( n ) n k t n = ∑ n = 1 ∞ p ( n ) n k t n − 1 = ∑ n = 0 ∞ p ( n + 1 ) n ! ( n + 1 ) k t n n ! .

The required result thus follows.□

If we take p ( n ) = 1 Γ ( n ) in Theorem 3.6, and taking into account Theorem 3.1, we reobtain Theorem 2.4.

Corollary 3.7

For n ≥ 0 and k ∈ Z , we have

∑ m = 0 n b m , 1 Γ ( k ) S 2 ( n , m ) = 1 ( n + 1 ) k .

4 Conclusion

The polyexponential functions were first studied by Hardy [2,15] and reconsidered by Kim and Kim [4,16] as an inverse type to the polylogarithm functions. Also, unipoly functions attached to any arithmetic functions were defined by Kim and Kim [4]. In this paper, we studied the poly-Bernoulli polynomials of the second kind and also the unipoly Bernoulli polynomials of the second kind. We derived explicit expressions and some identities involving those polynomials and numbers. In Section 2 we express in more detail the poly-Bernoulli polynomials of the second in terms of Bernoulli polynomials of the second kind and Stirling numbers of the first kind.

We deduce an expression of the poly-Bernoulli numbers of the second kind in terms of the Bernoulli numbers of the second kind and values of higher-order Bernoulli polynomials at zero. And, we derive an identity involving the poly-Bernoulli numbers of the second kind, Stirling numbers of the second kind and Bernoulli numbers of the second kind.

In Section 3, if we take p ( n ) = 1 Γ ( n ) , then we can see that our unipoly Bernoulli polynomials of the second kind are just the same poly-Bernoulli polynomials of the second. Also, we express the unipoly Bernoulli polynomials of the second kind in terms of Bernoulli polynomials of the second kind and Stirling numbers of the first kind. And, we deduce some expressions of the unipoly Bernoulli numbers of the second kind in terms of the Bernoulli numbers of the second kind and other special numbers.

Acknowledgements

The authors would like to thank the referees for their comments and suggestions that improved the original manuscript in its present form.

  1. Funding information: The work of D. Lim was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) NRF-2021R1C1C1010902.

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-08-30
Revised: 2021-03-16
Accepted: 2021-06-03
Published Online: 2021-08-27

© 2021 Minyoung Ma and Dongkyu Lim, published by De Gruyter

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

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  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
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  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
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  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
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  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0052
Keywords for this article
polyexponential function; unipoly function; poly-Bernoulli numbers of the second kind
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  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
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  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
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  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
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  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
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