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Representations by degenerate Daehee polynomials

  • Taekyun Kim , Dae San Kim EMAIL logo , Hyunseok Lee and Jongkyum Kwon EMAIL logo
Published/Copyright: April 11, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Daehee polynomials and more generally of the higher-order degenerate Daehee polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

Keywords: degenerate Daehee polynomial; higher-order degenerate Daehee polynomial; umbral calculus
MSC 2010: 05A19; 05A40; 11B68; 11B83

1 Introduction and preliminaries

The aim of this paper is to derive formulas (see Theorem 3.1) expressing any polynomial in terms of the degenerate Daehee polynomials (see (1.12)) with the help of umbral calculus and to illustrate our results with some examples (see Chapter 6). This can be generalized to the higher-order degenerate Bernoulli polynomials (see (1.13)). Indeed, we deduce formulas (see Theorems 4.1) for representing any polynomial in terms of the higher-order degenerate Daehee polynomials again by using umbral calculus. Letting λ → 0 , we obtain formulas (see Remarks 3.2 and 4.2) for expressing any polynomial in terms of the Daehee polynomials (see (1.10)) and of the higher-order Daehee polynomials (see (1.11)). These formulas are also illustrated in Chapter 5. The contribution of this paper is the derivation of such formulas that, we think, have many potential applications.

Let p ( x ) ∈ C [ x ] , with deg p ( x ) = n . Write p ( x ) = ∑ k = 0 n a k B k ( x ) , where B k ( x ) are the Bernoulli polynomials (see (1.3)). Then, it is known (see [1]) that

(1.1) a k = 1 k ! ∫ 0 1 p ( k ) ( x ) d x , for k = 0 , 1 , … , n .

The following identity (see [1,2]) is obtained by applying the formula in (1.1) to the polynomial p ( x ) = ∑ k = 1 n − 1 1 k ( n − k ) B k ( x ) B n − k ( x ) and after slight modification:

(1.2) ∑ k = 1 n − 1 1 2 k ( 2 n − 2 k ) B 2 k ( x ) B 2 n − 2 k ( x ) + 2 2 n − 1 B 1 ( x ) B 2 n − 1 ( x )

= 1 n ∑ k = 1 n 1 2 k 2 n 2 k B 2 k B 2 n − 2 k ( x ) + 1 n H 2 n − 1 B 2 n ( x ) + 2 2 n − 1 B 1 ( x ) B 2 n − 1 ,

where n ≥ 2 and H n = 1 + 1 2 + ⋯ + 1 n .

Letting x = 0 and x = 1 2 in (1.2), respectively, give a slight variant of Miki’s identity and the Faber-Pandharipande-Zagier (FPZ) identity. Here, it should be emphasized that the other proofs of Miki’s (see [3,4,5]) and FPZ identities (see [6,7]) are quite involved, while our proofs of Miki’s and FPZ identities follow from the simple formula in (1.1) involving only derivatives and integrals of the given polynomials.

Analogous formulas to (1.1) can be obtained for the representations by Euler, Frobenius-Euler, ordered Bell and Genocchi polynomials. Many interesting identities have been derived by using these formulas (see [1,8,9, 10,11,12, 13,14] and references therein). The list in the references is far from being exhaustive. However, the interested reader can easily find more related papers in the literature. Also, we should mention here that there are other ways of obtaining the same result as the one in (1.2). One of them is to use Fourier series expansion of the function obtained by extending by periodicity 1 of the polynomial function restricted to the interval [ 0 , 1 ) (see [2,15,16]).

The outline of this paper is as follows. In Section 1, we recall some necessary facts that are needed throughout this paper. In Section 2, we go over umbral calculus briefly. In Section 3, we derive formulas expressing any polynomial in terms of the degenerate Daehee polynomials. In Section 4, we derive formulas representing any polynomial in terms of the higher-order degenerate Daehee polynomials. In Section 5, we illustrate our results with examples of representation by the Daehee polynomials. In Section 6, we illustrate our results with examples of representation by the degenerate Daehee polynomials. Finally, we conclude our paper in Section 7.

The Bernoulli polynomials B n ( x ) are defined by

(1.3) t e t − 1 e x t = ∑ n = 0 ∞ B n ( x ) t n n ! .

When x = 0 , B n = B n ( 0 ) are called the Bernoulli numbers. We observe that B n ( x ) = ∑ j = 0 n n j B n − j x j , d d x B n ( x ) = n B n − 1 ( x ) , and B n ( x + 1 ) − B n ( x ) = n x n − 1 . The first few terms of B n are given by:

B 0 = 1 , B 1 = − 1 2 , B 2 = 1 6 , B 4 = − 1 30 , B 6 = 1 42 , B 8 = − 1 30 , B 10 = 5 66 , B 12 = − 691 2730 , … ; B 2 k + 1 = 0 , ( k ≥ 1 ) .

More generally, for any nonnegative integer r , the Bernoulli polynomials B n ( r ) ( x ) of order r are given by

(1.4) t e t − 1 r e x t = ∑ n = 0 ∞ B n ( r ) ( x ) t n n ! .

When x = 0 , B n ( r ) = B n ( r ) ( 0 ) are called the Bernoulli numbers of order r . We observe that B n ( r ) ( x ) = ∑ j = 0 n n j B n − j ( r ) x j , d d x B n ( r ) ( x ) = n B n − 1 ( r ) ( x ) , B n ( r ) ( x + 1 ) − B n ( r ) ( x ) = n B n − 1 ( r − 1 ) ( x ) .

The Euler polynomials E n ( x ) are defined by

(1.5) 2 e t + 1 e x t = ∑ n = 0 ∞ E n ( x ) t n n ! .

When x = 0 , E n = E n ( 0 ) are called the Euler numbers. We observe that E n ( x ) = ∑ j = 0 n n j E n − j x j , d d x E n ( x ) = n E n − 1 ( x ) , E n ( x + 1 ) + E n ( x ) = 2 x n . The first few terms of E n are given by:

E 0 = 1 , E 1 = − 1 2 , E 3 = 1 4 , E 5 = − 1 2 , E 7 = 17 8 , E 9 = − 31 2 , … ; E 2 k = 0 , ( k ≥ 1 ) .

The Genocchi polynomials G n ( x ) are defined by

(1.6) 2 t e t + 1 e x t = ∑ n = 0 ∞ G n ( x ) t n n ! .

When x = 0 , G n = G n ( 0 ) are called the Genocchi numbers. We observe that G n ( x ) = ∑ j = 0 n n j G n − j x j , d d x G n ( x ) = n G n − 1 ( x ) , G n ( x + 1 ) + G n ( x ) = 2 n x n − 1 , and deg G n ( x ) = n − 1 , for n ≥ 1 . The first few terms of G n are given by:

G 0 = 0 , G 1 = 1 , G 2 = − 1 , G 4 = 1 , G 6 = − 3 , G 8 = 17 , G 10 = − 155 G 12 = 2073 , … ; G 2 k + 1 = 0 , ( k ≥ 1 ) .

For any nonzero real number λ , the degenerate exponentials are given by

(1.7) e λ x ( t ) = ( 1 + λ t ) x λ = ∑ n = 0 ∞ ( x ) n , λ t n n ! , e λ ( t ) = e λ 1 ( t ) = ( 1 + λ t ) 1 λ = ∑ n = 0 ∞ ( 1 ) n , λ t n n ! .

Here, we recall that the λ -falling factorials are given by

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

Especially, ( x ) n = ( x ) n , 1 are called the falling factorials and hence given by

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

The compositional inverse of e λ ( t ) is called the degenerate logarithm and given by

log λ ( t ) = 1 λ ( t λ − 1 ) ,

which satisfies e λ ( log λ ( t ) ) = log λ ( e λ ( t ) ) = t .

Note here that lim λ → 0 e λ x ( t ) = e x t , lim λ → 0 log λ ( t ) = log ( t ) .

Recall that the Daehee polynomials D n ( x ) are given by

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

When x = 0 , D n = D n ( 0 ) are the Daehee numbers.

More generally, for any nonnegative integer r , the Daehee polynomials D n ( r ) ( x ) of order r are given by

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

When x = 0 , D n ( r ) = D n ( r ) ( 0 ) are the Daehee numbers of order r .

The degenerate Daehee polynomials D n , λ ( x ) are defined by

(1.12) log λ ( 1 + t ) t ( 1 + t ) x = ∑ n = 0 ∞ D n , λ ( x ) t n n ! ,

which are degenerate versions of the Daehee polynomials in (1.10). For x = 0 , D n , λ = D n , λ ( 0 ) are called the degenerate Daehee numbers and introduced in [7] (see also [14]).

More generally, for any nonnegative integer r , the degenerate Daehee polynomials D n , λ ( r ) ( x ) of order r are defined by

(1.13) log λ ( 1 + t ) t r ( 1 + t ) x = ∑ n = 0 ∞ D n , λ ( r ) ( x ) t n n ! ,

which are degenerate versions of the Daehe polynomials of order r in (1.11). We remark that D n , λ ( x ) → D n ( x ) , and D n , λ ( r ) ( x ) → D n ( r ) ( x ) , as λ tends to 0.

We recall some notations and facts about forward differences. Let f be any complex-valued function of the real variable x . Then, for any real number a , the forward difference Δ a is given by

(1.14) Δ a f ( x ) = f ( x + a ) − f ( x ) .

If a = 1 , then we let

(1.15) Δ f ( x ) = Δ 1 f ( x ) = f ( x + 1 ) − f ( x ) .

In general, the n th oder forward differences are given by

(1.16) Δ a n f ( x ) = ∑ i = 0 n n i ( − 1 ) n − i f ( x + i a ) .

For a = 1 , we have

(1.17) Δ n f ( x ) = ∑ i = 0 n n i ( − 1 ) n − i f ( x + i ) .

Finally, we recall that the Stirling numbers of the second kind S 2 ( n , k ) can be given by means of

(1.18) 1 k ! ( e t − 1 ) k = ∑ n = k ∞ S 2 ( n , k ) t n n ! .

2 Review of umbral calculus

Here, we will briefly go over very basic facts about umbral calculus. For more details on this, we recommend the reader to refer to [3, 20, 22]. Let C be the field of complex numbers. Then, ℱ denotes the algebra of formal power series in t over C , given by

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

and P = C [ x ] indicates the algebra of polynomials in x with coefficients in C .

Let P ∗ be the vector space of all linear functionals on P . If ⟨ L ∣ p ( x ) ⟩ denotes the action of the linear functional L on the polynomial p ( x ) , then 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 number.

For f ( t ) ∈ ℱ with f ( t ) = ∑ k = 0 ∞ a k t k k ! , we define the linear functional on P by

(2.1) ⟨ f ( t ) ∣ x k ⟩ = a k .

From (2.1), we note that

⟨ t k ∣ x n ⟩ = n ! δ n , k , ( n , k ≥ 0 ) ,

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

Some remarkable linear functionals are as follows:

(2.2) ⟨ e y t ∣ p ( x ) ⟩ = p ( y ) , ⟨ e y t − 1 ∣ p ( x ) ⟩ = p ( y ) − p ( 0 ) , e y t − 1 t p ( x ) = ∫ 0 y p ( u ) d u .

Let

(2.3) f L ( t ) = ∑ k = 0 ∞ ⟨ L ∣ x k ⟩ t k k ! .

Then, by (2.1) and (2.3), we obtain

⟨ f L ( t ) ∣ x n ⟩ = ⟨ L ∣ x n ⟩ .

That is, f L ( t ) = L . In addition, the map L ⟼ f L ( t ) is a vector space isomorphism from P ∗ onto ℱ .

Henceforth, ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on P . ℱ is called the umbral algebra and the umbral calculus is the study of umbral algebra. For each nonnegative integer k , the differential operator t k on P is defined by

(2.4) t k x n = ( n ) k x n − k , if k ≤ n , 0 , if k > n .

Extending (2.4) linearly, any power series

f ( t ) = ∑ k = 0 ∞ a k k ! t k ∈ ℱ

gives the differential operator on P defined by

(2.5) f ( t ) x n = ∑ k = 0 n n k a k x n − k , ( n ≥ 0 ) .

It should be observed that, for any formal power series f ( t ) and any polynomial p ( x ) , we have

(2.6) ⟨ f ( t ) ∣ p ( x ) ⟩ = ⟨ 1 ∣ f ( t ) p ( x ) ⟩ = f ( t ) p ( x ) ∣ x = 0 .

Here, we note that an element f ( t ) of ℱ is a formal power series, a linear functional, and a differential operator. Some notable differential operators are as follows:

(2.7) e y t p ( x ) = p ( x + y ) , ( e y t − 1 ) p ( x ) = p ( x + y ) − p ( x ) = Δ y p ( x ) , e y t − 1 t p ( x ) = ∫ x x + y p ( u ) d u .

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

For f ( t ) , g ( t ) ∈ ℱ with o ( f ( t ) ) = 1 and o ( g ( t ) ) = 0 , there exists a unique sequence s n ( x ) (deg s n ( x ) = n ) of polynomials such that

(2.8) ⟨ g ( t ) f ( t ) k ∣ s n ( x ) ⟩ = n ! δ n , k , ( n , k ≥ 0 ) .

The sequence s n ( x ) is said to be the Sheffer sequence for ( g ( t ) , f ( t ) ) , which is denoted by s n ( x ) ∼ ( g ( t ) , f ( t ) ) . We observe from (2.8) that

(2.9) s n ( x ) = 1 g ( t ) p n ( x ) ,

where p n ( x ) = g ( t ) s n ( x ) ∼ ( 1 , f ( t ) ) .

In particular, if s n ( x ) ∼ ( g ( t ) , t ) , then p n ( x ) = x n , and hence,

(2.10) s n ( x ) = 1 g ( t ) x n .

It is well known that s n ( x ) ∼ ( g ( t ) , f ( t ) ) if and only if

(2.11) 1 g ( f ¯ ( t ) ) e x f ¯ ( t ) = ∑ k = 0 ∞ s k ( x ) k ! t k ,

for all x ∈ C , where f ¯ ( t ) is the compositional inverse of f ( t ) such that f ¯ ( f ( t ) ) = f ( f ¯ ( t ) ) = t .

Equations (2.12)–(2.14) are equivalent to the fact that s n ( x ) is Sheffer for ( g ( t ) , f ( t ) ) , for some invertible g ( t ) :

(2.12) f ( t ) s n ( x ) = n s n − 1 ( x ) , ( n ≥ 0 ) ,

(2.13) s n ( x + y ) = ∑ j = 0 n n j s j ( x ) p n − j ( y ) ,

with p n ( x ) = g ( t ) s n ( x ) ,

(2.14) s n ( x ) = ∑ j = 0 n 1 j ! ⟨ g ( f ¯ ( t ) ) − 1 f ¯ ( t ) j ∣ x n ⟩ x j .

Let p n ( x ) , q n ( x ) = ∑ k = 0 n q n , k x k be sequences of polynomials. Then, the umbral composition of q n ( x ) with p n ( x ) is defined to be the sequence

(2.15) q n ( p ( x ) ) = ∑ k = 0 n q n , k p k ( x ) .

3 Representations by degenerate Daehee polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the degenerate Daehee polynomials.

From (1.7), (1.9), and (1.11), we first observe that

(3.1) D n , λ ( x ) ∼ g ( t ) = λ f ( t ) e λ t − 1 = λ ( e t − 1 ) e λ t − 1 , f ( t ) = e t − 1 ,

(3.2) ( x ) n ∼ ( 1 , f ( t ) = e t − 1 ) .

From (1.15), (2.7), (2.8), (2.12), (3.1), and (3.2), we note that

(3.3) f ( t ) D n , λ ( x ) = n D n − 1 , λ ( x ) = ( e t − 1 ) D n , λ ( x ) = Δ D n , λ ( x ) ,

(3.4) f ( t ) ( x ) n = n ( x ) n − 1 ,

(3.5) g ( t ) D n , λ ( x ) = ( x ) n .

Now, we assume that p ( x ) ∈ C [ x ] has degree n , and write p ( x ) = ∑ k = 0 n a k D k , λ ( x ) . Then, from (3.5), we have

(3.6) g ( t ) p ( x ) = ∑ k = 0 n a k g ( t ) D k , λ ( x ) = ∑ k = 0 n a k ( x ) k .

For k ≥ 0 , from (3.4) and (3.6), we obtain

(3.7) f ( t ) k g ( t ) p ( x ) = f ( t ) k ∑ l = 0 n a l ( x ) l , λ = ∑ l = k n l ( l − 1 ) ⋯ ( l − k + 1 ) a l ( x ) l − k , λ .

Letting x = 0 in (3.7), we finally obtain

(3.8) a k = 1 k ! f ( t ) k g ( t ) p ( x ) ∣ x = 0 = 1 k ! ⟨ g ( t ) f ( t ) k ∣ p ( x ) ⟩ , ( k ≥ 0 ) .

Now, we want to find more explicit expressions for (3.8). As λ t e λ t − 1 e x t = ∑ n = 0 ∞ λ n B n x λ t n n ! , we see from (2.10) that λ n B n x λ = λ t e λ t − 1 x n . To proceed further, we let p ( x ) = ∑ i = 0 n b i x i .

From (2.7), (2.15), and (3.1), noting that g ( t ) = e t − 1 t λ t e λ t − 1 , we have

(3.9) g ( t ) p ( x ) = e t − 1 t λ t e λ t − 1 p ( x ) = e t − 1 t ∑ i = 0 n b i λ t e λ t − 1 x i = e t − 1 t ∑ i = 0 n b i λ i B i x λ = e t − 1 t p λ B x λ = ∫ x x + 1 p λ B u λ d u ,

where p λ B x λ denotes the umbral composition of p ( x ) with λ i B i x λ , that is, it is given by p λ B x λ = ∑ i = 0 n b i λ i B i x λ .

We note from (3.5) and (3.9), in passing, that the following holds:

( x ) n = g ( t ) D n , λ ( x ) = ∫ x x + 1 D n , λ λ B u λ d u .

From (2.7) and (3.9), we deduce

(3.10) a k = 1 k ! f ( t ) k g ( t ) p ( x ) ∣ x = 0 = 1 k ! Δ k ∫ x x + 1 p λ B u λ d u x = 0 .

By making use of (1.17) and (3.10), an alternative expression of (3.10) is given by

(3.11) a k = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i ∫ i i + 1 p λ B u λ d u .

We obtain yet another expression from (1.18), (3.8), and (3.9), which is given by

(3.12) a k = 1 k ! ( e t − 1 ) k ∫ x x + 1 p λ B u λ d u x = 0 = ∑ l = k ∞ S 2 ( l , k ) t l l ! ∫ x x + 1 p λ B u λ d u x = 0 = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l ∫ x x + 1 p λ B u λ d u x = 0 ,

where we need to note that ∫ x x + 1 p λ B u λ d u has degree n .

Finally, from (3.10)–(3.12), and (3.8), we obtain the following theorem.

Theorem 3.1

Let p ( x ) ∈ C [ x ] , with deg p ( x ) = n . Then, we have p ( x ) = ∑ k = 0 n a k D k , λ ( x ) , where

a k = 1 k ! f ( t ) k g ( t ) p ( x ) ∣ x = 0 = 1 k ! Δ k ∫ x x + 1 p λ B u λ d u x = 0 = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i ∫ i i + 1 p λ B u λ d u = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l ∫ x x + 1 p λ B u λ d u x = 0 , for k = 0 , 1 , … , n ,

where g ( t ) = λ ( e t − 1 ) e λ t − 1 , f ( t ) = e t − 1 , and p λ B x λ denotes the umbral composition of p ( x ) with λ i B i x λ .

Remark 3.2

Let p ( x ) ∈ C [ x ] , with deg p ( x ) = n . Write p ( x ) = ∑ k = 0 n a k D k ( x ) . As λ tends to 0, g ( t ) → e t − 1 t , and p λ B x λ → p ( x ) . Thus, we obtain the following result.

a k = 1 k ! Δ k ∫ x x + 1 p ( u ) d u x = 0 = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i ∫ i i + 1 p ( u ) d u = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l ∫ x x + 1 p ( u ) d u x = 0 , for k = 0 , 1 , … , n .

4 Representations by higher-order degenerate Daehee polynomials

Our interest here is to derive formulas expressing any polynomial in terms of the higher-order degenerate Daehee polynomials.

With g ( t ) = λ f ( t ) e λ t − 1 = λ ( e t − 1 ) e λ t − 1 , f ( t ) = e t − 1 , from (1.11), we note that

(4.1) D n , λ ( r ) ( x ) ∼ ( g ( t ) r , f ( t ) ) ,

(4.2) ( x ) n ∼ ( 1 , f ( t ) ) .

From (1.15), (2.7), (2.8), (2.12), (4.1), and (4.2), we note that

(4.3) f ( t ) D n , λ ( r ) ( x ) = n D n − 1 , λ ( r ) ( x ) = ( e t − 1 ) D n , λ ( r ) ( x ) = Δ D n , λ ( r ) ( x ) ,

(4.4) f ( t ) ( x ) n = n ( x ) n − 1 ,

(4.5) g ( t ) r D n , λ ( r ) ( x ) = ( x ) n .

Now, we assume that p ( x ) ∈ C [ x ] has degree n , and write p ( x ) = ∑ k = 0 n a k D k , λ ( r ) ( x ) . Then, from (4.5), we have

(4.6) g ( t ) r p ( x ) = ∑ k = 0 n a k g ( t ) r D k , λ ( r ) ( x ) = ∑ k = 0 n a k ( x ) k .

For k ≥ 0 , from (4.4), we obtain

(4.7) f ( t ) k g ( t ) r p ( x ) = f ( t ) k ∑ l = 0 n a l ( x ) l = ∑ l = k n l ( l − 1 ) ⋯ ( l − k + 1 ) a l ( x ) l − k .

Letting x = 0 in (4.7), we finally obtain

(4.8) a k = 1 k ! f ( t ) k g ( t ) r p ( x ) ∣ x = 0 = 1 k ! ⟨ g ( t ) r f ( t ) k ∣ p ( x ) ⟩ , ( k ≥ 0 ) .

This also follows from the observation ⟨ g ( t ) r f ( t ) k ∣ D l , λ ( r ) ( x ) ⟩ = l ! δ l , k .

Now, we want to find more explicit expressions for (4.8). As λ t e λ t − 1 r e x t = ∑ n = 0 ∞ λ n B n ( r ) x λ t n n ! , we see from (2.10) that λ n B n ( r ) x λ = λ t e λ t − 1 r x n . To proceed further, we let p ( x ) = ∑ i = 0 n b i x i .

From (2.7), (2.15), and (4.1), noting that g ( t ) = e t − 1 t λ t e λ t − 1 , we have

(4.9) g ( t ) r p ( x ) = e t − 1 t r λ t e λ t − 1 r p ( x ) = e t − 1 t r ∑ i = 0 n b i λ t e λ t − 1 r x i = e t − 1 t r ∑ i = 0 n b i λ i B i ( r ) x λ = e t − 1 t r p λ B ( r ) x λ = I r p λ B ( r ) x λ ,

where p λ B ( r ) x λ denotes the umbral composition of p ( x ) with λ i B i ( r ) x λ , that is, it is given by p λ B ( r ) x λ = ∑ i = 0 n b i λ i B i ( r ) x λ , and I denotes the linear integral operator given by q ( x ) → ∫ x x + 1 q ( x ) d x .

We note from (4.5) and (4.9), in passing, that the following holds:

( x ) n = g ( t ) r D n , λ ( x ) = I r D n , λ λ B ( r ) x λ .

From (2.7) and (4.9), we deduce

(4.10) a k = 1 k ! f ( t ) k g ( t ) r p ( x ) ∣ x = 0 = 1 k ! Δ k I r p λ B ( r ) x λ x = 0 .

By making use of (1.17) and (4.10), an alternative expression of (3.10) is given by

(4.11) a k = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i I r p λ B ( r ) x λ x = i .

We obtain yet another expression from (1.18), (4.8), and (4.9), which is given by

(4.12) a k = 1 k ! ( e t − 1 ) k I r p λ B ( r ) x λ x = 0

= ∑ l = k ∞ S 2 ( l , k ) t l l ! I r p λ B ( r ) x λ x = 0 = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l I r p λ B ( r ) x λ x = 0 ,

where we need to observe that I r p λ B ( r ) x λ has degree n .

Finally, from (4.10)–(4.12) and (4.8), we obtain the following theorem.

Theorem 4.1

Let p ( x ) ∈ C [ x ] , with deg p ( x ) = n . Then, we have p ( x ) = ∑ k = 0 n a k D k , λ ( r ) ( x ) , where

a k = 1 k ! f ( t ) k g ( t ) r p ( x ) ∣ x = 0 = 1 k ! Δ k I r p λ B ( r ) x λ x = 0 = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i I r p λ B ( r ) x λ x = i = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l I r p λ B ( r ) x λ x = 0 , for k = 0 , 1 , … , n ,

where g ( t ) = λ ( e t − 1 ) e λ t − 1 , f ( t ) = e t − 1 , p λ B ( r ) x λ indicates the umbral composition of p ( x ) with λ i B i ( r ) x λ , and I denotes the linear integral operator given by q ( x ) → ∫ x x + 1 q ( x ) d x .

We observe that I r p λ B ( r ) x λ x = i = ∫ i i + 1 I r − 1 p λ B ( r ) x λ d x .

Remark 4.2

Let p ( x ) ∈ C [ x ] , with deg p ( x ) = n . Write p ( x ) = ∑ k = 0 n a k D k ( r ) ( x ) . As λ tends to 0, g ( t ) → e t − 1 t , and p λ B ( r ) x λ → p ( x ) . Thus, we obtain the following result.

a k = 1 k ! Δ k ( I r p ( x ) ) x = 0 = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i I r p ( x ) x = i = ∑ l = k n S 2 ( l , k ) 1 l ! d d x l ( I r p ( x ) ) x = 0 , for k = 0 , 1 , … , n .

We note that I r p ( x ) ∣ x = i = ∫ i i + 1 I r − 1 p ( x ) d x .

5 Examples on representation by Daehee polynomials

Here, we illustrate our formulas in Remarks 3.2 and 4.2 with some examples.

(a) Let p ( x ) = B n ( x ) = ∑ k = 0 n a k D k ( x ) . Then, as B n ( x + 1 ) − B n ( x ) = n x n − 1 , ∫ x x + 1 B n ( u ) d u = x n , from Remark 3.2, we have

(5.1) a k = 1 k ! Δ k x n ∣ x = 0 = 1 k ! ∑ i = 0 k k i ( − 1 ) k − i i n = S 2 ( n , k ) ,

which are well known.

Thus, we obtain the following identity:

B n ( x ) = ∑ k = 0 n S 2 ( n , k ) D k ( x ) .

Next, we let p ( x ) = B n ( x ) = ∑ k = 0 n a k D k ( r ) ( x ) . Then, we first observe that

(5.2) I r B n ( x ) = 1 ( n + r ) r ∑ i = 0 r r i ( − 1 ) r − i B n + r ( x + i ) .

Now, by making use of Remark 4.2, we obtain

(5.3) a k = 1 k ! ( n + r ) r ∑ i = 0 r r i ( − 1 ) r − i Δ k B n + r ( x + i ) ∣ x = 0 = 1 ( n + r ) r ∑ l = k n ∑ i = 0 r ( − 1 ) r − i r i n + r l S 2 ( l , k ) B n + r − l ( i ) .

Thus, we have the following:

B n ( x ) = 1 ( n + r ) r ∑ k = 0 n 1 k ! ∑ i = 0 r r i ( − 1 ) r − i Δ k B n + r ( x + i ) ∣ x = 0 D k ( r ) ( x ) = 1 ( n + r ) r ∑ k = 0 n ∑ l = k n ∑ i = 0 r ( − 1 ) r − i r i n + r l S 2 ( l , k ) B n + r − l ( i ) D k ( r ) ( x ) .

(b) Here, we consider p ( x ) = ∑ k = 1 n − 1 1 k ( n − k ) B k ( x ) B n − k ( x ) , ( n ≥ 2 ) . For this, we first recall from [12] that

(5.4) p ( x ) = 2 n ∑ m = 0 n − 2 1 n − m n m B n − m B m ( x ) + 2 n H n − 1 B n ( x ) ,

where H n = 1 + 1 2 + ⋯ + 1 n is the harmonic number and a slight modification of (5.4) gives the identity in (1.2). Let p ( x ) = ∑ k = 0 n a k D k ( x ) . Then, we have

(5.5) a k = 2 n ∑ m = 0 n − 2 1 n − m n m B n − m ∑ l = k n S 2 ( l , k ) m l δ m , l + 2 n H n − 1 ∑ l = k n S 2 ( l , k ) n l δ n , l = 2 n ∑ m = k n − 2 1 n − m n m B n − m S 2 ( m , k ) + 2 n H n − 1 S 2 ( n , k ) ,

where we understand that the sum in (5.5) is zero for k = n − 1 or n . Thus, we obtain the following identity:

∑ k = 1 n − 1 1 k ( n − k ) B k ( x ) B n − k ( x ) = 2 n ∑ k = 0 n ∑ m = k n − 2 1 n − m n m B n − m S 2 ( m , k ) + H n − 1 S 2 ( n , k ) D k ( x ) .

(c) In [12], it is shown that the following identity holds for n ≥ 2 :

(5.6) ∑ k = 1 n − 1 1 k ( n − k ) E k ( x ) E n − k ( x ) = − 4 n ∑ m = 0 n n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 B m ( x ) ,

where H n = 1 + 1 2 + ⋯ + 1 n is the harmonic number.

Write ∑ k = 1 n − 1 1 k ( n − k ) E k ( x ) E n − k ( x ) = ∑ k = 0 n a k D k ( x ) .

By proceeding similarly to (b), we see that

(5.7) a k = − 4 n ∑ m = 0 n n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 ∑ l = k n S 2 ( l , k ) m l δ m , l = − 4 n ∑ m = k n n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 S 2 ( m , k ) .

Thus, (5.7) implies the next identity:

∑ k = 1 n − 1 1 k ( n − k ) E k ( x ) E n − k ( x ) = − 4 n ∑ k = 0 n ∑ m = k n n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 S 2 ( m , k ) D k ( x ) .

(d) In [16], it is proved that the following identity is valid for n ≥ 2 :

(5.8) ∑ k = 1 n − 1 1 k ( n − k ) G k ( x ) G n − k ( x ) = − 4 n ∑ m = 0 n − 2 n m G n − m n − m B m ( x ) .

Again, by proceeding analogously to (b), we can show that

(5.9) a k = − 4 n ∑ l = k n − 2 S 2 ( l , k ) m l ∑ m = 0 n − 2 n m G n − m n − m δ m , l = − 4 n ∑ m = k n − 2 n m S 2 ( m , k ) G n − m n − m .

Therefore, we obtain the following identity:

∑ k = 1 n − 1 1 k ( n − k ) G k ( x ) G n − k ( x ) = − 4 n ∑ k = 0 n − 2 ∑ m = k n − 2 n m S 2 ( m , k ) G n − m n − m D k ( x ) .

(e) Nielsen [2,19] also represented products of two Euler polynomials in terms of Bernoulli polynomials as follows:

(5.10) E m ( x ) E n ( x ) = − 2 ∑ r = 1 m m r E r B m + n − r + 1 ( x ) m + n − r + 1 − 2 ∑ s = 1 n n s E s B m + n − s + 1 ( x ) m + n − s + 1 + 2 ( − 1 ) n + 1 m ! n ! ( m + n + 1 ) ! E m + n + 1 .

In the same way as (b), we can show that

(5.11) a k = 2 ( − 1 ) n + 1 m ! n ! ( m + n + 1 ) ! E m + n + 1 δ k , 0 − 2 ∑ r = 1 m m r E r m + n − r + 1 S 2 ( m + n − r + 1 , k ) − 2 ∑ s = 1 n n s E s m + n − s + 1 S 2 ( m + n − s + 1 , k ) .

Thus, we arrive at the next identity:

E m ( x ) E n ( x ) = 2 ( − 1 ) n + 1 m ! n ! ( m + n + 1 ) ! E m + n + 1 − 2 ∑ k = 1 m + n ∑ r = 1 m m r E r m + n − r + 1 S 2 ( m + n − r + 1 , k ) D k ( x ) − 2 ∑ k = 1 m + n ∑ s = 1 n n s E s m + n − s + 1 S 2 ( m + n − s + 1 , k ) D k ( x ) .

6 Examples on representation by degenerate Daehee polynomials

Here, we illustrate our formulas in Theorems 3.1 and 4.1.

(a) Let p ( x ) = B n ( x ) = ∑ k = 0 n a k D k , λ ( x ) . Then, as B n ( x ) = ∑ j = 0 n n j B n − j x j , we have

(6.1) ∫ x x + 1 B n λ B u λ d u = ∑ j = 0 n n j B n − j λ j + 1 1 j + 1 B j + 1 ( x + 1 λ ) − B j + 1 x λ .

Thus, for 1 ≤ l ≤ n ,

(6.2) d d x l ∫ x x + 1 B n λ B u λ d u = ∑ j = 0 n n j B n − j λ j d d x l − 1 B j x + 1 λ − B j x λ = ∑ j = l n n j B n − j λ j − l + 1 ( j ) l − 1 B j − l + 1 x + 1 λ − B j − l + 1 x λ .

Now, from Theorem 3.1, (6.1), and (6.2), we obtain

(6.3) a k = 1 k ! ∑ j = 0 n n j λ j + 1 j + 1 B n − j Δ k B j + 1 x + 1 λ − B j + 1 x λ x = 0 = 1 k ! ∑ i = 0 k ∑ j = 0 n ( − 1 ) k − i k i n j λ j + 1 j + 1 B n − j B j + 1 i + 1 λ − B j + 1 i λ = ∑ l = k n ∑ j = l n S 2 ( l , k ) 1 l ! n j λ j − l + 1 ( j ) l − 1 B n − j B j − l + 1 1 λ − B j − l + 1 ,

where we understand that ( j ) − 1 = 1 j + 1 .

Hence, from (6.3), we obtain the following identity:

B n ( x ) = ∑ k = 0 n 1 k ! ∑ j = 0 n n j λ j + 1 j + 1 B n − j Δ k B j + 1 x + 1 λ − B j + 1 x λ x = 0 D k , λ ( x ) = ∑ k = 0 n 1 k ! ∑ i = 0 n ∑ j = 0 n ( − 1 ) k − i k i n j λ j + 1 j + 1 B n − j B j + 1 i + 1 λ − B j + 1 i λ D k , λ ( x ) = ∑ k = 0 n ∑ l = k n ∑ j = l n S 2 ( l , k ) 1 l ! n j λ j − l + 1 ( j ) l − 1 B n − j B j − l + 1 1 λ − B j − l + 1 D k , λ ( x ) .

Next, we let p ( x ) = B n ( x ) = ∑ k = 0 n a k D k , λ ( r ) ( x ) . Then, we first observe that

(6.4) I r B n λ B ( r ) x λ = I r ∑ j = 0 n n j B n − j λ j B j ( r ) x λ = ∑ j = 0 n ∑ i = 0 r ( − 1 ) r − i n j r i λ j + r ( j + r ) r B n − j B j + r ( r ) x + i λ .

So, for l with j + r ≥ l , we obtain

(6.5) d d x l I r B n λ B ( r ) x λ = ∑ j = 0 n ∑ i = 0 r ( − 1 ) r − i n j r i λ j + r ( j + r ) r B n − j d d x l B j + r ( r ) x + i λ = ∑ j = 0 n ∑ i = 0 r ( − 1 ) r − i n j r i λ j + r − l ( j + r ) r B n − j ( j + r ) l B j + r − l ( r ) x + i λ .

Thus, from Theorem 4.1, (6.4), and (6.5), we have

B n ( x ) = ∑ k = 0 n 1 k ! ∑ j = 0 n ∑ i = 0 r ( − 1 ) r − i n j r i λ j + r ( j + r ) r B n − j Δ k B j + r ( r ) x + i λ x = 0 D k , λ ( r ) ( x ) = ∑ k = 0 n ∑ l = k n S 2 ( l , k ) 1 l ! ∑ j = max { 0 , l − r } n ∑ i = 0 r ( − 1 ) r − i n j r i λ j + r − l ( j + r ) r B n − j ( j + r ) l B j + r − l ( r ) i λ D k , λ ( r ) ( x ) .

(b) Let p ( x ) = ∑ k = 1 n − 1 1 k ( n − k ) B k ( x ) B n − k ( x ) , for n ≥ 2 . As we stated earlier, it was shown in [12] that

(6.6) p ( x ) = 2 n ∑ m = 0 n − 2 1 n − m n m B n − m B m ( x ) + 2 n H n − 1 B n ( x ) ,

where H n = 1 + 1 2 + ⋯ + 1 n is the Harmonic number.

Write p ( x ) = ∑ k = 0 n a k D k ( x ) . Then, from Theorem 4.1 and (6.6), we have

(6.7) a k = 2 n ∑ l = k n − 2 ∑ m = l n − 2 ∑ j = l m S 2 ( l , k ) 1 l ! 1 n − m n m m j B n − m B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 + 2 n H n − 1 ∑ l = k n ∑ j = l n S 2 ( l , k ) 1 l ! n j B n − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 .

Thus, from (6.7), we obtain

∑ k = 1 n − 1 1 k ( n − k ) B k ( x ) B n − k ( x ) = 2 n ∑ k = 0 n ∑ l = k n − 2 ∑ m = l n − 2 ∑ j = l m S 2 ( l , k ) 1 l ! 1 n − m n m m j B n − m B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 + H n − 1 ∑ l = k n ∑ j = l n S 2 ( l , k ) 1 l ! n j B n − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 D k , λ ( x ) ,

where we understand that the triple sum in the parentheses is zero for k = n − 1 or k = n , and ( j ) − 1 = 1 j + 1 .

(c) Let p ( x ) = ∑ k = 1 n − 1 1 k ( n − k ) E k ( x ) E n − k ( x ) , for n ≥ 2 . Then, as we saw earlier, it was proved in [12] that

(6.8) p ( x ) = − 4 n ∑ m = 0 n n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 B m ( x ) .

Write p ( x ) = ∑ k = 0 n − 2 a k D k , λ ( x ) . Then, from Theorem 4.1 and (6.8), we can show that

(6.9) a k = − 4 n ∑ l = k n ∑ m = l n ∑ j = l m S 2 ( l , k ) 1 l ! n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 m j B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 ,

where we understand that ( j ) − 1 = 1 j + 1 .

Hence, from (6.9), we have

∑ k = 1 n − 1 1 k ( n − k ) E k ( x ) E n − k ( x ) = − 4 n ∑ k = 0 n ∑ l = k n ∑ m = l n ∑ j = l m S 2 ( l , k ) 1 l ! n m ( H n − 1 − H n − m ) n − m + 1 E n − m + 1 m j B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 D k , λ ( x ) .

(d) Here, we consider p ( x ) = ∑ k = 1 n − 1 1 k ( n − k ) G k ( x ) G n − k ( x ) , for n ≥ 2 . As we mentioned earlier, it was shown in [16] that

(6.10) p ( x ) = − 4 n ∑ m = 0 n − 2 n m G n − m n − m B m ( x ) .

Write p ( x ) = ∑ k = 0 n − 2 a k D k , λ ( x ) . Then, from Theorem 4.1 and (6.10), we obtain that

(6.11) a k = − 4 n ∑ l = k n − 2 ∑ m = l n − 2 ∑ j = l m S 2 ( l , k ) 1 l ! n m G n − m n − m m j B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 ,

where we understand that ( j ) − 1 = 1 j + 1 .

Thus, from (6.11), we obtain

∑ k = 1 n − 1 1 k ( n − k ) G k ( x ) G n − k ( x ) = − 4 n ∑ k = 0 n − 2 ∑ l = k n − 2 ∑ m = l n − 2 ∑ j = l m S 2 ( l , k ) 1 l ! n m G n − m n − m m j B m − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 D k , λ ( x ) .

(e) As we mentioned earlier, it was shown (see [17,18]) that, for positive integers m and n , with m + n ≥ 2 , we have

(6.12) B m ( x ) B n ( x ) = ∑ r m 2 r n + n 2 r m B 2 r B m + n − 2 r ( x ) m + n − 2 r + ( − 1 ) m + 1 B m + n m + n m .

Then, from Theorem 4.1 and (6.12), we can show that

(6.13) a k = ( − 1 ) m + 1 B m + n m + n m δ k , 0 + ∑ r ∑ l = k m + n ∑ j = l m + n − 2 r S 2 ( l , k ) 1 l ! m 2 r n + n 2 r m × B 2 r m + n − 2 r m + n − 2 r j B m + n − 2 r − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 .

Thus, form (6.13), we obtain

B m ( x ) B n ( x ) = ( − 1 ) m + 1 B m + n m + n m + ∑ k = 0 m + n ∑ r ∑ l = k m + n − 2 r ∑ j = l m + n − 2 r S 2 ( l , k ) 1 l ! m 2 r n + n 2 r m × B 2 r m + n − 2 r m + n − 2 r j B m + n − 2 r − j λ j − l + 1 ( j ) l − 1 B j − l + 1 1 λ − B j − l + 1 D k , λ ( x ) ,

where ( j ) − 1 = 1 j + 1 .

7 Conclusion

In this paper, we were interested in representing any polynomial in terms of the degenerate Daehee polynomials and of the higher-order degenerate Daehee polynomials. We were able to derive formulas for such representations with the help of umbral calculus. We showed that, by letting λ tends to zero, they give formulas for representations by the Daehee polynomials and by the higher-order Daehee polynomials. Further, we illustrated the formulas with some examples.

As we mentioned in Section 1, both Faber-Pandharipande-Zagier (FPZ) identity and a variant of Miki’s identity follow from the one identity (see (1.2)) that can be derived from the formula (see (1.1)) involving only derivatives and integrals of the given polynomial, while all the other proofs are quite involved. We recall here that the FPZ identity was a conjectural relation between Hodge integrals in Gromov-Witten’s theory. It should be stressed that our method is very useful and powerful, even though it is elementary.

It is one of our future research projects to continue to find formulas representing polynomials in terms of some specific special polynomials and to apply those in discovering some interesting identities.

Acknowledgments

The authors would like to thank Jangjeon Institute for Mathematical Science for the support of this research.

  1. Funding information: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (No. 2020R1F1A1A01071564).

  2. Conflict of interest: The authors declare no conflict of interest.

References

[1] D. S. Kim, T. Kim, S.-H. Lee, and Y.-H. Kim, Some identities for the product of two Bernoulli and Euler polynomials, Adv. Differential Equations 2012 (2012), 95. 10.1186/1687-1847-2012-95Search in Google Scholar

[2] T. Kim, D. S. Kim, G.-W. Jang, and J. Kwon, Fourier series of sums of products of Genocchi functions and their applications, J. Nonlinear Sci. Appl. 10 (2017), 1683–1694. 10.22436/jnsa.010.04.31Search in Google Scholar

[3] I. M. Gessel, On Mikias identities for Bernoulli numbers, J. Number Theory 110 (2005), no. 1, 75–82. 10.1016/j.jnt.2003.08.010Search in Google Scholar

[4] H. Miki, A relation between Bernoulli numbers, J. Number Theory 10 (1978), no. 3, 297–302. 10.1016/0022-314X(78)90026-4Search in Google Scholar

[5] K. Shiratani and S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A 36 (1982), no. 1, 73–83. 10.2206/kyushumfs.36.73Search in Google Scholar

[6] G. V. Dunne and C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys. 7 (2013), no. 2, 225–249. 10.4310/CNTP.2013.v7.n2.a1Search in Google Scholar

[7] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. 10.1007/s002229900028Search in Google Scholar

[8] L.-C. Jang, W. Kim, H.-I. Kwon, and T. Kim, On degenerate Daehee polynomials and numbers of the third kind, J. Comput. Appl. Math. 364 (2020), 112343. 10.1016/j.cam.2019.112343Search in Google Scholar

[9] D. S. Kim and T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci. 2012 (2012), 463659. 10.1155/2012/463659Search in Google Scholar

[10] D. S. Kim and T. Kim, A note on higher-order Bernoulli polynomials, J. Inequal. Appl. 2013 (2013), 111. 10.1186/1029-242X-2013-111Search in Google Scholar

[11] T. Kim and D. S. Kim, On λ-Bell polynomials associated with umbral calculus, Russ. J. Math. Phys. 24 (2017), no. 1, 69–78. 10.1134/S1061920817010058Search in Google Scholar

[12] D. S. Kim, T. Kim, D. V. Dolgy, and S.-H. Rim, Higher-order Bernoulli, Euler and Hermite polynomials, Adv. Differential Equations 2013 (2013), 103. 10.1186/1687-1847-2013-103Search in Google Scholar

[13] D. S. Kim, T. Kim, and T. Mansour, Euler basis and the product of several Bernoulli and Euler polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 24 (2014), no. 4, 535–547. Search in Google Scholar

[14] T. Kim, D. S. Kim, H. Y. Kim, and J. Kwon, Some results on degenerate Daehee and Bernoulli numbers and polynomials, Adv. Differential Equations 2020 (2020), 311. 10.1186/s13662-020-02778-8Search in Google Scholar

[15] T. Kim, D. S. Kim, G.-W. Jang, and J. Kwon, Series of sums of products of higher-order Bernoulli functions, J. Inequal. Appl. 2017 (2017), 221. 10.1186/s13660-017-1494-9Search in Google Scholar PubMed PubMed Central

[16] T. Kim, D. S. Kim, G.-W. Jang, and J. Kwon, Fourier series of sums of products of higher-order Euler functions, J. Comput. Anal. Appl. 27 (2019), no. 2, 345–360. Search in Google Scholar

[17] N. Nielsen, Traité élémentaire des nombres de Bernoulli, Paris, 1923. Search in Google Scholar

[18] L. Carlitz, The product of two Eulerian polynomials, Math. Mag. 36 (1963), no. 1, 37–41. 10.1080/0025570X.1963.11975384Search in Google Scholar

[19] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math. 15 (1979), 51–88. Search in Google Scholar

[20] R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 433–438. Search in Google Scholar

[21] S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich Publishers], New York, 1984. Search in Google Scholar

[22] Y. Simsek, Special numbers and polynomials including their generating functions in umbral analysis methods, Axioms 7 (2018), no. 2, 22. 10.3390/axioms7020022Search in Google Scholar
Received: 2021-09-23
Accepted: 2022-01-11
Published Online: 2022-04-11

© 2022 Taekyun Kim et al., published by De Gruyter

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

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  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
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  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
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  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0013
Keywords for this article
degenerate Daehee polynomial; higher-order degenerate Daehee polynomial; umbral calculus
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