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On the type 2 poly-Bernoulli polynomials associated with umbral calculus

  • Taekyun Kim EMAIL logo , Dae San Kim , Dmitry V. Dolgy and Jin-Woo Park EMAIL logo
Published/Copyright: August 27, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

Keywords: type 2 poly-Bernoulli polynomials; umbral calculus; modified poly-exponential functions
MSC 2010: 11B83

1 Introduction

The poly-Bernoulli polynomials, which are defined with the help of polylogarithm functions, were studied by Kaneko in [1], while the type 2 poly-Bernoulli polynomials, which are defined with the help of modified polyexponential functions, were investigated very recently in [2]. We note that the modified polyexponential functions are inverse to the polylogarithm functions. Thus, it is very natural to replace the polylogarithms by the modified polyexponential functions in the definition of generating function of poly-Bernoulli polynomials. Indeed, the generating function of type 2 poly-Bernoulli polynomials is obtained in this way (see (1), (3)), and hence we may say that it arises in a natural manner.

Let k ≥ 1 be an integer, Ei k ( x ) the modified polyexponential function (see (1)), and let B n ( k ) be the type 2 poly-Bernoulli numbers (see (3)). In [2], the function η k ( s ) , for Re ( s ) > 0 , is defined as

η k ( s ) = 1 Γ ( s ) ∫ 0 ∞ t s − 1 e t − 1 Ei k ( log ( 1 + t ) ) d t .

It was shown that this function can be continued to an entire function on C and its values at non-positive integers are given by η k ( − m ) = ( − 1 ) m B m ( k ) , ( m ≥ 0 ) . In addition, for any integer k ≥ 2 , the generating function of the type 2 poly-Bernoulli numbers is given by

∑ n = 0 ∞ B n ( k ) x n n ! = 1 e x − 1 ∫ 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 .

The aim or motivation of this paper is to further derive some properties, recurrence relations, and identities related to the type 2 poly-Bernoulli polynomials by using umbral calculus techniques. Especially, those polynomials are represented in terms of some well-known special polynomials. In general, special polynomials and numbers can be studied by employing various different methods including combinatorial methods, generating functions, differential equations, umbral calculus techniques, p -adic analysis, and probability theory.

The outline of this paper is as follows. In Section 1, we give some necessary definitions and some basic facts about umbral calculus. As to definitions, we recall the definitions of polyexponential functions, type 2 poly-Bernoulli polynomials, higher-order Bernoulli polynomials, higher-order Cauchy polynomials, higher-order Euler polynomials, and Stirling numbers of the first and second kinds. As to umbral calculus, we give very basic facts such as Sheffer sequence, generating functions of Sheffer polynomials, and the formula for representing one Sheffer polynomial by another. For further details on umbral calculus, we let the reader refer to [3,4,5]. In Section 2, we find an explicit expression for the type 2 poly-Bernoulli polynomials involving Bernoulli numbers and Stirling numbers of the first kind, a recurrence relation for them, and an identity involving the type 2 poly-Bernoulli numbers and Stirling numbers of the first kind. In addition, we express the type 2 poly-Bernoulli polynomials as linear combinations of higher-order Cauchy polynomials, higher-order Euler polynomials, and of higher-order Frobenius-Euler polynomials.

It is one of our future projects to continue to work on various special polynomials and numbers by using umbral calculus, just as we did in the present paper.

Hardy introduced the polyexponential functions [6,7], while Kim-Kim considered the modified polyexponential functions which are given by

(1) Ei k ( x ) = ∑ n = 1 ∞ x n ( n − 1 ) ! n k , ( k ∈ Z ) ( see [ 1 , 8 , 9 ] ) .

From (1), we note that

(2) Ei 1 ( x ) = e x − 1 .

The type 2 poly-Bernoulli polynomials, which are defined by using the modified polyexponential functions, are given by

(3) Ei k ( log ( 1 + t ) ) e t − 1 e x t = ∑ n = 0 ∞ B n ( k ) ( x ) t n n ! ( see [1,10,11] ) .

For x = 0 , B n ( k ) = B n ( k ) ( 0 ) are called the type 2 poly-Bernoulli numbers.

For r ∈ N , the Bernoulli polynomials B n ( r ) ( x ) of order r are given by

(4) t e t − 1 r e x t = ∑ n = 0 ∞ B n ( r ) ( x ) t n n ! ( see [1,3–5,8–16] ) .

When x = 0 , B n ( r ) = B n ( r ) ( 0 ) , ( n ≥ 0 ) , are called the Bernoulli numbers of order r .

Note that B n ( 1 ) ( x ) = B n ( 1 ) ( x ) , which will be denoted by B n ( x ) , are the Bernoulli polynomials.

It is well known that the Cauchy polynomials of order r are defined by

(5) t log ( 1 + t ) r ( 1 + t ) x = ∑ n = 0 ∞ C n ( r ) ( x ) t n n ! , ( r ∈ N ) ( see [ 5 ] ) .

For r ∈ N , the Euler polynomials of order r are defined by

(6) 2 e t + 1 r e x t = ∑ n = 0 ∞ E n ( r ) ( x ) t n n ! ( see [ 5 ] ) .

For n ≥ 0 , the falling factorial sequence is defined by

( x ) 0 = 1 , ( x ) n = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) , ( n ≥ 1 ) .

Here we note that the Stirling numbers of the first kind are defined by

(7) ( x ) n = ∑ l = 0 n S 1 ( n , l ) x l , 1 m ! ( log ( 1 + t ) ) m = ∑ n = m ∞ S 1 ( n , m ) t n n ! , ( n ≥ 0 ) ( see [ 5 , 11 , 17 ] ) .

As an inversion formula of (7), the Stirling numbers of the second kind are defined by

(8) x n = ∑ l = 0 n S 2 ( n , l ) ( x ) l , 1 m ! ( e t − 1 ) m = ∑ n = m ∞ S 2 ( n , m ) t n n ! , ( n ≥ 0 ) ( see [ 5 , 11 , 13 , 15 ] ) .

Let C be the field of complex numbers and let

(9) F = f ( t ) = ∑ k = 0 ∞ a k t k k ! a k ∈ C

be the algebra of formal power series. For P = C [ x ] , let P ∗ denote the vector space of all linear functional on P . ⟨ L ∣ p ( x ) ⟩ denotes the action of the linear functional L on the polynomial p ( x ) , and it is well known that the vector space operations on P ∗ are defined by

⟨ L + M ∣ p ( x ) ⟩ = ⟨ L ∣ p ( x ) ⟩ + ⟨ M ∣ p ( x ) ⟩ , ⟨ c L ∣ p ( x ) ⟩ = c ⟨ L ∣ p ( x ) ⟩ ,

where c is a complex constant (see [3,4,5]).

For f ( t ) ∈ F , let ⟨ f ( t ) ∣ x n ⟩ = a n , ( n ≥ 0 ) . From (9), we note that

⟨ t k ∣ x n ⟩ = n ! δ n , k , ( n , k ≥ 0 ) ( see [ 4 , 12 , 13 ] ) ,

where δ n , k is the Kronecker symbol.

The order o ( f ( t ) ) of the power series f ( t ) ( ≠ 0 ) is the smallest integer k for which a k does not vanish. If o ( f ( t ) ) = 0 , then f ( t ) is said to be an invertible series; if o ( f ( t ) ) = 1 , then f ( t ) is called a delta series.

For f ( t ) , g ( t ) ∈ F , we have

(10) ⟨ f ( t ) g ( t ) ∣ p ( x ) ⟩ = ⟨ f ( t ) ∣ g ( t ) p ( x ) ⟩ = ⟨ g ( t ) ∣ f ( t ) p ( x ) ⟩ .

From (10), we note that

(11) f ( t ) = ∑ k = 0 ∞ ⟨ f ( t ) ∣ x k ⟩ t k k ! , p ( x ) = ∑ k = 0 ∞ ⟨ t k ∣ p ( x ) ⟩ x k k ! ( see [ 4 ] ) ,

where f ( t ) ∈ F and p ( x ) ∈ P .

Thus, by (11), we get

(12) p ( k ) ( 0 ) = ⟨ t k ∣ p ( x ) ⟩ = ⟨ 1 ∣ p ( k ) ( x ) ⟩ ( see [ 4 , 16 ] ) ,

where p ( k ) ( x ) = d k d x k p ( x ) .

From (12), we note that t k p ( x ) = p ( k ) ( x ) , ( k ≥ 0 ) . It is not difficult to show that

(13) e y t p ( x ) = p ( x + y ) , ⟨ e y t ∣ p ( x ) ⟩ = p ( y ) ( see [ 15 ] ) ,

where p ( x ) ∈ P .

Suppose that f ( t ) is a delta series and g ( t ) is an invertible series. Then there exists a unique sequence s n ( x ) of polynomials such that ⟨ g ( t ) ( f ( t ) ) k ∣ s n ( x ) ⟩ = n ! δ n , k , ( n , k ≥ 0 ) . The sequence s n ( x ) is called the Sheffer sequence for ( g ( t ) , f ( t ) ) , which is denoted by s n ( x ) ∼ ( g ( t ) , f ( t ) ) .

For s n ( x ) ∼ ( g ( t ) , f ( t ) ) , we have

(14) h ( t ) = ∑ k = 0 ∞ ⟨ h ( t ) ∣ s k ( x ) ⟩ k ! g ( t ) f ( t ) k

and

(15) p ( x ) = ∑ k = 0 ∞ ⟨ g ( t ) f ( t ) k ∣ p ( x ) ⟩ k ! s k ( x ) .

Thus, by (14), we easily get

(16) 1 g ( f ¯ ( t ) ) e x f ¯ ( t ) = ∑ n = 0 ∞ s n ( x ) t n n ! ,

where f ¯ ( t ) is the compositional inverse of f ( t ) with f ¯ ( f ( t ) ) = t , and

(17) f ( t ) s n ( x ) = n s n − 1 ( x ) , ( n ∈ N ) .

For s n ( x ) ∼ ( g ( t ) , t ) , we have

(18) s n + 1 ( x ) = x − g ′ ( t ) g ( t ) s n ( x ) ,

(19) s n ( x + y ) = ∑ k = 0 n n k s k ( x ) y n − k = ∑ k = 0 n n k s n − k ( x ) y k , ( n ≥ 0 ) ,

and

(20) s n ( x ) = 1 g ( t ) x n , ( n ≥ 0 ) ( see [ 15 ] ) .

We recall here that s n ( x ) is called the Appell sequence for g ( t ) if s n ( x ) ∼ ( g ( t ) , t ) . For example, the sequence B n ( k ) ( x ) of the type 2 poly-Bernoulli polynomials is an Appell sequence and hence has the properties stated in (18), (19), and (20). In particular, we have

(21) B n ( k ) ( x ) = ∑ j = 0 n B n − j ( k ) x j .

For s n ( x ) ∼ ( g ( t ) , f ( t ) ) , r n ( x ) ∼ ( h ( t ) , l ( t ) ) , it is known that

(22) s n ( x ) = ∑ m = 0 n A n , m r m ( x ) , ( n ≥ 0 ) ,

where

(23) A n , m = 1 m ! h ( f ¯ ( t ) ) g ( f ¯ ( t ) ) ( l ( f ¯ ( t ) ) ) m ∣ x n ( see [ 4 ] ) .

2 Some identities of type 2 poly-Bernoulli polynomials arising from umbral calculus

From (3), (4), and (16), we note that

(24) B n ( k ) ( x ) ∼ e t − 1 Ei k ( log ( 1 + t ) ) , t , B n ( r ) ( x ) ∼ e t − 1 t r , t .

By (17), we get

t B n ( k ) ( x ) = n B n − 1 ( k ) , t B n ( r ) ( x ) = n B n − 1 ( r ) , ( n ≥ 1 ) .

From (20) and (24), we have the next lemma.

Lemma 1

For n ≥ 0 , we have

Ei k ( log ( 1 + t ) ) e t − 1 x n = B n ( k ) ( x ) .

Now, we observe that

(25) Ei k ( log ( 1 + t ) ) = ∑ m = 1 ∞ ( log ( 1 + t ) ) m ( m − 1 ) ! m k = ∑ m = 1 ∞ 1 m k − 1 1 m ! ( log ( 1 + t ) ) m = ∑ m = 1 ∞ 1 m k − 1 ∑ l = m ∞ S 1 ( l , m ) t l l ! = ∑ l = 1 ∞ ∑ m = 1 l S 1 ( l , m ) m k − 1 t l l ! .

From Lemma 1 and (25), we have

(26) B n ( k ) ( x ) = 1 n + 1 t e t − 1 Ei k ( log ( 1 + t ) ) x n + 1 = 1 n + 1 t e t − 1 ∑ l = 1 ∞ ∑ m = 1 l S 1 ( l , m ) m k − 1 t l l ! x n + 1 = 1 n + 1 ∑ j = 0 ∞ B j t j j ! ∑ l = 1 n + 1 ∑ m = 1 l S 1 ( l , m ) m k − 1 n + 1 l x n + 1 − l = 1 n + 1 ∑ l = 1 n + 1 ∑ m = 1 l ∑ j = 0 n + 1 − l S 1 ( l , m ) m k − 1 n + 1 l n + 1 − l j B j x n + 1 − l − j = 1 n + 1 ∑ l = 1 n + 1 ∑ m = 1 l ∑ j = 0 n + 1 − l S 1 ( l , m ) m k − 1 n + 1 l , j , n + 1 − l − j B j x n + 1 − l − j = 1 n + 1 ∑ j = 0 n ∑ l = 1 n + 1 − j ∑ m = 1 l S 1 ( l , m ) m k − 1 n + 1 l , j , n + 1 − l − j B n + 1 − l − j x j ,

where we used the trinomial coefficients n a , b , c = n ! a ! b ! c ! , with n = a + b + c .

Therefore, by (26), we obtain the following theorem.

Theorem 2

For n ≥ 0 , k ∈ Z , we have

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

By (18) and (24), we get

(27) B n + 1 ( k ) ( x ) = x − g ′ ( t ) g ( t ) B n ( k ) ( x ) , ( n ≥ 0 ) ,

where

g ( t ) = e t − 1 Ei k ( log ( 1 + t ) ) and g ′ ( t ) = d d t g ( t ) .

We observe that

(28) g ′ ( t ) g ( t ) = d d t ( log g ( t ) ) = d d t ( log ( e t − 1 ) − log Ei k ( log ( 1 + t ) ) ) = e t e t − 1 − 1 ( 1 + t ) log ( 1 + t ) Ei k − 1 ( log ( 1 + t ) ) Ei k ( log ( 1 + t ) ) .

As is known, the Cauchy numbers of the second kind are defined by

(29) t ( 1 + t ) log ( 1 + t ) = ∑ n = 0 ∞ c ^ n t n n ! .

From (27) and (28), we note that

(30) B n + 1 ( k ) ( x ) = x − g ′ ( t ) g ( t ) B n ( k ) ( x ) = x B n ( k ) ( x ) − e t e t − 1 − 1 ( 1 + t ) log ( 1 + t ) Ei k − 1 ( log ( 1 + t ) ) Ei k ( log ( 1 + t ) ) B n ( k ) ( x ) = x B n ( k ) ( x ) − e t e t − 1 Ei k ( log ( 1 + t ) ) e t − 1 x n + 1 ( 1 + t ) log ( 1 + t ) Ei k − 1 ( log ( 1 + t ) ) Ei k ( log ( 1 + t ) ) Ei k ( log ( 1 + t ) ) e t − 1 x n = x B n ( k ) ( x ) − t e t e t − 1 Ei k ( log ( 1 + t ) ) t ( e t − 1 ) x n + t ( 1 + t ) log ( 1 + t ) Ei k − 1 ( log ( 1 + t ) ) t ( e t − 1 ) x n = x B n ( k ) ( x ) − 1 n + 1 t e t e t − 1 Ei k ( log ( 1 + t ) ) e t − 1 x n + 1 + 1 n + 1 t ( 1 + t ) log ( 1 + t ) Ei k − 1 ( log ( 1 + t ) ) e t − 1 x n + 1 = x B n ( k ) ( x ) − e t n + 1 ∑ l = 0 ∞ ∑ j = 0 l l j B l − j B j ( k ) t l l ! x n + 1 + 1 n + 1 ∑ l = 0 ∞ ∑ j = 0 l l j c ^ n − j B j ( k − 1 ) t l l ! x n + 1 = x B n ( k ) ( x ) − e t n + 1 ∑ l = 0 n + 1 ∑ j = 0 l n + 1 j , l − j , n + 1 − l B l − j B j ( k ) x n + 1 − l + 1 n + 1 ∑ l = 0 n + 1 ∑ j = 0 l c ^ n − j B j ( k − 1 ) n + 1 j , l − j , n + 1 − l x n + 1 − l = x B n ( k ) ( x ) − 1 n + 1 ∑ l = 0 n + 1 ∑ j = 0 l n + 1 j , l − j , n + 1 − l B l − j B j ( k ) ( x + 1 ) n + 1 − l + 1 n + 1 ∑ l = 0 n + 1 ∑ j = 0 l n + 1 j , l − j , n + 1 − l c ^ n − j B j ( k − 1 ) x n + 1 − l .

Therefore, we obtain the following theorem.

Theorem 3

For n ≥ 0 , k ∈ Z , we have

B n + 1 ( k ) ( x ) = x B n ( k ) ( x ) − 1 n + 1 ∑ l = 0 n + 1 ∑ j = 0 l n + 1 j , l − j , n + 1 − l B l − j B j ( k ) ( x + 1 ) n + 1 − l + 1 n + 1 ∑ l = 0 n + 1 ∑ j = 0 l n + 1 j , l − j , n + 1 − l c ^ n − j B j ( k − 1 ) x n + 1 − l .

Now, we compute ⟨ Ei k ( log ( 1 + t ) ) ∣ x n + 1 ⟩ , ( n ≥ 0 ) .

From (1), (3), and (10), we note that

(31) ⟨ Ei k ( log ( 1 + t ) ) ∣ x n + 1 ⟩ = Ei k ( log ( 1 + t ) ) e t − 1 ( e t − 1 ) x n + 1 = Ei k ( log ( 1 + t ) ) e t − 1 ( x + 1 ) n + 1 − x n + 1 = ∑ l = 0 n n + 1 l Ei k ( log ( 1 + t ) ) e t − 1 x l = ∑ l = 0 n n + 1 l B l ( k ) .

On the other hand,

(32) ⟨ Ei k ( log ( 1 + t ) ) ∣ x n + 1 ⟩ = ∑ m = 1 ∞ 1 m k − 1 ∑ l = m ∞ S 1 ( l , m ) l ! ⟨ t l ∣ x n + 1 ⟩ = ∑ l = 1 ∞ ∑ m = 1 l 1 m k − 1 S 1 ( l , m ) l ! ( n + 1 ) ! δ n + 1 , l = ∑ m = 1 n + 1 1 m k − 1 S 1 ( n + 1 , m ) .

Therefore, by (31) and (32), we obtain the following theorem.

Theorem 4

For n ≥ 0 and k ∈ Z , we have

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

Remark 5

Theorem 4 can be deduced also from Theorem 2. From (21) and Theorem 2, we see that

(33) n j B n − j ( k ) = ∑ l = 1 n + 1 − j ∑ m = 1 l 1 n + 1 S 1 ( l , m ) m k − 1 n + 1 l n + 1 − l j B n + 1 − l − j .

Replacing j by n − j , noting n + 1 j = n + 1 n + 1 − j n j , and summing over j in (33), we obtain

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

Now, Theorem 4 follows from (34) by noting that ∑ j = 0 n n + 1 j B j = δ n , 0 .

For the next result, we recall that, for any r ∈ N , the Cauchy polynomials C n ( r ) ( x ) of order r are given by

(35) t log ( 1 + t ) r ( 1 + t ) x = ∑ n = 0 ∞ C n ( r ) ( x ) t n n ! .

We consider the following two Sheffer sequences.

(36) B n ( k ) ( x ) ∼ e t − 1 Ei k ( log ( 1 + t ) ) , t , C n ( r ) ( x ) ∼ t e t − 1 r , e t − 1 .

From (22), (23), and (36), we have

(37) B n ( k ) ( x ) = ∑ m = 0 n A n , m C m ( r ) ( x ) ,

where, by making use of (8), we show

(38) A n , m = t e t − 1 r Ei k ( log ( 1 + t ) ) e t − 1 1 m ! ( e t − 1 ) m x n = ∑ l = m n n l S 2 ( l , m ) t e t − 1 r Ei k ( log ( 1 + t ) ) e t − 1 x n − l = ∑ l = m n n l S 2 ( l , m ) ∑ s = 0 ∞ B s ( r ) t s s ! B n − l ( k ) ( x ) = ∑ l = m n n l S 2 ( l , m ) ∑ s = 0 n − l B s ( r ) 1 s ! ⟨ 1 ∣ t s B n − l ( k ) ( x ) ⟩ = ∑ l = m n n l S 2 ( l , m ) ∑ s = 0 n − l n − l s B s ( r ) B n − l − s ( k ) .

Therefore, by (37) and (38), we obtain the following theorem.

Theorem 6

For n ≥ 0 , r ∈ N , and k ∈ Z , we have

B n ( k ) ( x ) = ∑ m = 0 n ∑ l = m n ∑ s = 0 n − l n l , s , n − l − s S 2 ( l , m ) B s ( r ) B n − l − s ( k ) C m ( r ) ( x ) ,

where C m ( r ) ( x ) are the Cauchy polynomials of order r .

For

B n ( k ) ( x ) ∼ e t − 1 Ei k ( log ( 1 + t ) ) , t , E n ( r ) ( x ) ∼ e t + 1 2 r , t , ( r ∈ N ) ,

we have

(39) B n ( k ) ( x ) = ∑ m = 0 n A n , m E m ( r ) ( x ) .

Here we note that

(40) A n , m = 1 m ! Ei k ( log ( 1 + t ) ) e t − 1 e t + 1 2 r t m x n = n m 2 r Ei k ( log ( 1 + t ) ) e t − 1 ( e t + 1 ) r x n − m = n m 2 r ∑ j = 0 r r j e j t Ei k ( log ( 1 + t ) ) e t − 1 x n − m = n m 2 r ∑ j = 0 r r j ⟨ e j t ∣ B n − m ( k ) ( x ) ⟩ = n m 2 r ∑ j = 0 r r j B n − m ( k ) ( j ) .

Therefore, by (39) and (40), we obtain the following theorem.

Theorem 7

For n ≥ 0 , r ∈ N , and k ∈ Z , we have

B n ( k ) ( x ) = 1 2 r ∑ m = 0 n n m ∑ j = 0 r r j B n − m ( k ) ( j ) E m ( r ) ( x ) .

Let λ ∈ C with λ ≠ 1 . For r ∈ N , the Frobenius-Euler polynomials of order r are defined by

(41) 1 − λ e t − λ r e x t = ∑ n = 0 ∞ H n ( r ) ( x ; λ ) t n n ! .

From (16) and (41), we note that

(42) H n ( r ) ( x ; λ ) ∼ e t − λ 1 − λ r , t .

Thus, by (22), (23), and (42), we get

(43) B n ( k ) ( x ) = ∑ m = 0 n A n , m H m ( r ) ( x ; λ ) ,

where

(44) A n , m = 1 m ! Ei k ( log ( 1 + t ) ) e t − 1 e t − λ 1 − λ r t m x n = n m ( 1 − λ ) r Ei k ( log ( 1 + t ) ) e t − 1 ( e t − λ ) r x n − m = n m ( 1 − λ ) r ∑ j = 0 r r j ( − λ ) r − j Ei k ( log ( 1 + t ) ) e t − 1 e j t x n − m = n m ( 1 − λ ) r ∑ j = 0 r r j ( − λ ) r − j t 1 Ei k ( log ( 1 + t ) ) e t − 1 ( x + j ) n − m = n m ( 1 − λ ) r ∑ j = 0 r r j ( − λ ) r − j B n − m ( k ) ( j ) .

Thus, by (43) and (44), we obtain the following theorem.

Theorem 8

For k ∈ Z , r ∈ N , and n ≥ 0 , we have

B m ( k ) ( x ) = 1 ( 1 − λ ) r ∑ m = 0 n n m ∑ j = 0 r r j ( − λ ) r − j B n − m ( k ) ( j ) H m ( r ) ( x ; λ ) .

Acknowledgments

The authors would like to thank the referees for their detailed comments and suggestions that helped improve the original manuscript in its present form. This research was supported by the Daegu University Research Grant, 2020.

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

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Received: 2020-12-08
Revised: 2021-06-12
Accepted: 2021-08-05
Published Online: 2021-08-27

© 2021 Taekyun Kim et al., 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. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  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
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  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
  117. Semiprimeness of semigroup algebras
  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-2021-0086
Keywords for this article
type 2 poly-Bernoulli polynomials; umbral calculus; modified poly-exponential functions
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  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
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  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
  117. Semiprimeness of semigroup algebras
  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
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