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Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers

  • Feng Qi ORCID logo , Muhammet Cihat Dağlı ORCID logo and Dongkyu Lim ORCID logo EMAIL logo
Published/Copyright: August 16, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

Keywords: Narumi number; Narumi polynomial; Cauchy number; degenerate Cauchy number; explicit formula; Bell polynomial of the second kind; Faà di Bruno formula; generating function
MSC 2010: 11B83; 11C08; 33B10

1 Preliminaries and motivations

In this section, we first recall several known notions and mention our motivations.

1.1 Stirling numbers

The Stirling numbers of the first kind s ( n , k ) for n ≥ k ≥ 0 can be generated [1,2] by

[ ln ( 1 + x ) ] k k ! = ∑ n = k ∞ s ( n , k ) x n n ! , ∣ x ∣ < 1

and can be computed by

∣ s ( n + 1 , k + 1 ) ∣ = n ! ∑ ℓ 1 = k n 1 ℓ 1 ∑ ℓ 2 = k − 1 ℓ 1 − 1 1 ℓ 2 ⋯ ∑ ℓ k − 1 = 2 ℓ k − 2 − 1 1 ℓ k − 1 ∑ ℓ k = 1 ℓ k − 1 − 1 1 ℓ k , n ≥ k ≥ 1 ,

which was derived in [3, Corollary 2.3] and can be reformulated as

∣ s ( n + 1 , k + 1 ) ∣ n ! = ∑ m = k n ∣ s ( m , k ) ∣ m ! , n ≥ k ≥ 1 .

The Stirling numbers of the second kind S ( n , k ) for n ≥ k ≥ 0 can be generated [1,2] by

(1) ( e x − 1 ) k k ! = ∑ n = k ∞ S ( n , k ) x n n !

and can be computed [4] by

S ( n , k ) = 1 k ! ∑ ℓ = 0 k ( − 1 ) k − ℓ k ℓ ℓ n .

In [5, p. 303, equation (1.2)], see also [6,7,8], the r -associate Stirling numbers of the second kind, denoted by S ( n , k ; r ) , were defined by

(2) e x − ∑ i = 0 r x i i ! k = ∑ i = r + 1 ∞ x i i ! k = k ! ∑ n = ( r + 1 ) k ∞ S ( n , k ; r ) x n n ! .

It is clear that S ( n , k ; 0 ) = S ( n , k ) .

Proposition 1.1

([5, p. 306, (3.11)] and [6, Theorem 3.1]) For k ≥ 1 , the 1-associate Stirling numbers of the second kind S ( n , k ; 1 ) satisfy S ( 0 , 0 ; 1 ) = 1 , S ( n , 0 ; 1 ) = 0 for n ≥ 1 , and

S ( n , k ; 1 ) = 1 k ! ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ⟨ n ⟩ m ( k − j ) n − m , n ≥ 2 k ≥ 2 ; 0 , 0 ≤ n < 2 k ,

where the falling factorial

⟨ x ⟩ n = ∏ k = 0 n − 1 ( x − k ) = x ( x − 1 ) ⋯ ( x − n + 1 ) , n ≥ 1 ; 1 , n = 0 .

1.2 Degenerate Narumi and Cauchy numbers and polynomials

The Narumi polynomials N n ( α ) ( x ) were defined by means of the generating function

(3) t ln ( 1 + t ) α ( 1 + t ) x = ∑ n = 0 ∞ N n ( α ) ( x ) t n n !

in [9, p. 127]. In particular, the quantities N n ( α ) ( 0 ) = N n ( α ) are called the Narumi numbers. In [10], several Sheffer sequences and many relations of several polynomials arising from umbral calculus were dealt with. In [11], the Narumi polynomials of the Barnes type were defined and many interesting identities in the light of umbral calculus were established.

When letting α = 1 and x = 0 in (3), the quantities N n ( α ) ( x ) become the Cauchy numbers of the first kind C n , which have been investigated in [3,12, 13,14] and closely related references therein.

In [15], Kim and two coauthors defined degenerate Cauchy numbers C n ( λ ) by

λ [ e [ ( 1 + t ) λ − 1 ] / λ − 1 ] ( 1 + t ) λ − 1 = ∑ n = 0 ∞ C n ( λ ) t n n !

and showed that the family of nonlinear differential equations

( 1 + t ) n [ ( 1 + t ) λ − 1 ] n F λ ( n ) ( t ) = F λ ( t ) ∑ i = 1 2 n a i ( n , λ ) ( 1 + t ) i λ + ∑ i = 1 2 n − 1 b i ( n , λ ) ( 1 + t ) i λ

for n ∈ N has the same solutions

(4) F λ ( t ) = e [ ( 1 + t ) λ − 1 ] / λ − 1 ( 1 + t ) λ − 1 ,

where a i ( n , λ ) for 1 ≤ i ≤ 2 n and b i ( n , λ ) for 1 ≤ i ≤ 2 n − 1 are uniquely determined by

(5) a 1 ( n , λ ) = − 1 λ ⟨ n − 1 − λ ⟩ n + 1 , a 2 ( n , λ ) = ⟨ n − 1 − 2 λ ⟩ n − 1 − 1 λ ∑ i = 0 n − 2 [ λ − ( λ + 1 ) ( n − i ) ] ⟨ n − i − 2 − λ ⟩ n − i ⟨ n − 1 − 2 λ ⟩ i , a i ( n , λ ) = [ ( i − 1 ) λ − ( λ + 1 ) n ] a i ( n − 1 , λ ) + a i − 2 ( n − 1 , λ ) + ( n − 1 − i λ ) a i ( n − 1 , λ )

for 3 ≤ i ≤ 2 n − 2 ,

(6) a 2 n − 1 ( n , λ ) = 1 2 n [ ( λ − 1 ) ( n − 1 ) − 2 ( λ + 1 ) ] , a 2 n ( n , λ ) = 1 , b 1 ( n , λ ) = ⟨ n − 1 − λ ⟩ n − 1 , b i ( n , λ ) = [ ( i − 1 ) λ − ( λ + 1 ) ( n − 1 ) ] b i − 1 ( n − 1 , λ ) + a i − 1 ( n − 1 , λ ) + ( n − 1 − i λ ) b i ( n − 1 , λ )

for 2 ≤ i ≤ 2 n − 3 , and

(7) b 2 n − 2 ( n , λ ) = ( λ − 1 ) n − 1 2 − 2 ( n − 1 ) − λ , b 2 n − 1 ( n , λ ) = 1 .

Since

lim λ → 0 ( 1 + t ) λ − 1 λ = ln ( 1 + t ) ,

we have

lim λ → 0 C n ( λ ) = C n , n ≥ 0 .

This should be what the word “degenerate” means.

In previous study [16], for simplifying and signifying those expressions in (5), (6), and (7), by virtue of the Faà di Bruno formula (12), the identity (13), and the closed-form expression (15), Qi and his two coauthors established the following two conclusions.

  1. For n ≥ 0 , degenerate Cauchy numbers C n ( λ ) and the Cauchy numbers C n can be explicitly computed by

    (8) C n ( λ ) = ∑ k = 0 n ( − 1 ) k ( k + 1 ) ! 1 λ k ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ⟨ ℓ λ ⟩ n

    and

    (9) C n = ∑ k = 0 n s ( n , k ) k + 1 ,

    respectively. See [16, Theorem 1].

  2. For n ∈ N , the function F λ ( t ) in (4) and its derivatives satisfy

    F λ ( n ) ( t ) = F λ ( t ) ( 1 + t ) n [ 1 − ( 1 + t ) λ ] n ∑ i = 1 2 n α i ( n , λ ) ( 1 + t ) i λ + ∑ i = 1 2 n − 1 β i ( n , λ ) ( 1 + t ) i λ

    with

    α i ( n , λ ) = ∑ k + m = i 1 ≤ k ≤ n 0 ≤ m ≤ n ( − 1 ) m A k ( n , λ ) ∑ ℓ = 0 min { n − m , k } λ ℓ ( k − ℓ ) ! n − ℓ m

    for 1 ≤ i ≤ 2 n and

    β i ( n , λ ) = ∑ k + m = i 1 ≤ k ≤ n 0 ≤ m ≤ n − 1 ( − 1 ) m + 1 A k ( n , λ ) ∑ ℓ = 0 min { k − 1 , n − m − 1 } λ ℓ ( k − ℓ ) ! n − ℓ − 1 m

    for 1 ≤ i ≤ 2 n − 1 , where

    A k ( n , λ ) = ( − 1 ) k λ k ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ∏ q = 0 n − 1 ( ℓ λ − q ) .

    See [16, Theorem 2].

From the explicit formula (8), we obtain the first seven values of degenerate Cauchy numbers C n ( λ ) for 0 ≤ n ≤ 6 as follows:

C 0 ( λ ) = 1 , C 1 ( λ ) = 1 2 , C 2 ( λ ) = 1 6 ( 3 λ − 1 ) , C 3 ( λ ) = 1 4 ( 2 λ 2 − 2 λ + 1 ) , C 4 ( λ ) = λ 3 2 − 2 λ 2 3 + λ − 19 30 , C 5 ( λ ) = 1 12 ( 6 λ 4 + 5 λ 2 − 36 λ + 27 ) , C 6 ( λ ) = λ 5 2 + 17 λ 4 6 − 10 λ 3 + 61 λ 2 12 + 12 λ − 863 84 .

In recent years, the authors also investigated other sequences of special numbers and polynomials in [17,18,19, 20,21,22, 23,24,25, 26,27,28] and closely related references therein.

1.3 Motivations

In this paper, we introduce degenerate Narumi polynomials N n ( α ) ( x , λ ) by

(10) λ [ e [ ( 1 + t ) λ − 1 ] / λ − 1 ] ( 1 + t ) λ − 1 α e x [ ( 1 + t ) λ − 1 ] / λ = ∑ n = 0 ∞ N n ( α ) ( x , λ ) t n n ! .

It is clear that

lim λ → 0 N n ( α ) ( x , λ ) = N n ( α ) ( x ) , N n ( 1 ) ( 0 , λ ) = C n ( λ ) .

When x = 0 , we call N n ( α ) ( 0 , λ ) degenerate Narumi numbers and denote them by the notation N n ( α ) ( λ ) , that is, taking x = 0 in (10),

(11) λ [ e [ ( 1 + t ) λ − 1 ] / λ − 1 ] ( 1 + t ) λ − 1 α = ∑ n = 0 ∞ N n ( α ) ( λ ) t n n ! .

It is easy to see that N 0 ( α ) ( x , λ ) = 1 and N 0 ( α ) ( λ ) = 1 .

In this paper, we mainly aim at establishing explicit formulas for degenerate Narumi numbers N n ( α ) ( λ ) and degenerate Narumi polynomials N n ( α ) ( x , λ ) . Consequently, we derive explicit formulas for the Cauchy numbers C n and degenerate Cauchy numbers C n ( λ ) .

2 Properties of second kind Bell polynomials

The Bell polynomials of the second kind B n , k ( x 1 , x 2 , … , x n − k + 1 ) for n ≥ k ≥ 0 were defined in [2, p. 134] by

B n , k ( x 1 , x 2 , … , x n − k + 1 ) = ∑ 1 ≤ i ≤ n − k + 1 ℓ i ∈ { 0 } ∪ N ∑ i = 1 n − k + 1 i ℓ i = n ∑ i = 1 n − k + 1 ℓ i = k n ! ∏ i = 1 n − k + 1 ℓ i ! ∏ i = 1 n − k + 1 x i i ! ℓ i .

For n ∈ N , the Faà di Bruno formula is described in [2, p. 139] in terms of the Bell polynomials of the second kind B n , k ( x 1 , x 2 , … , x n − k + 1 ) by

(12) d n d t n f ∘ h ( t ) = ∑ k = 1 n f ( k ) ( h ( t ) ) B n , k ( h ′ ( t ) , h ″ ( t ) , … , h ( n − k + 1 ) ( t ) ) .

In [2, p. 135], there is an identity

(13) B n , k ( a b x 1 , a b 2 x 2 , … , a b n − k + 1 x n − k + 1 ) = a k b n B n , k ( x 1 , x 2 , … , x n − k + 1 ) ,

where n ≥ k ≥ 0 and a , b , λ , α are any complex numbers. At the end of [2, p. 133], there is the formula

(14) 1 k ! ∑ m = 1 ∞ x m t m m ! k = ∑ n = k ∞ B n , k ( x 1 , x 2 , … , x n − k + 1 ) t n n ! , k ≥ 0 .

In [29, Remark 1], there existed the formula

(15) B n , k 1 , 1 − λ , ( 1 − λ ) ( 1 − 2 λ ) , … , ∏ ℓ = 0 n − k ( 1 − ℓ λ ) = ( − 1 ) k k ! ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ∏ q = 0 n − 1 ( ℓ − q λ ) .

The explicit formula (15) has been applied and reviewed in [16, Lemma 3], [30, Lemma 2.6], [31, Section 2], [32, First proof of Theorem 2], [33, Lemma 2.2], [34, Remark 6.1], [35, Lemma 4], and [36, Section 1.3]. The explicit formula (15) is equivalent to

(16) B n , k ( ⟨ λ ⟩ 1 , ⟨ λ ⟩ 2 , … , ⟨ λ ⟩ n − k + 1 ) = ( − 1 ) k k ! ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ⟨ λ ℓ ⟩ n ,

which was presented in [37, Theorems 2.1 and 4.1]. In [38, Remark 7.5], the explicit formulas (15) and (16) were rearranged as

B n , k 1 , 1 − λ , ( 1 − λ ) ( 1 − 2 λ ) , … , ∏ ℓ = 0 n − k ( 1 − ℓ λ ) = ( − 1 ) k λ n − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ ℓ / λ − 1 n − 1

for λ ≠ 0 and

(17) B n , k ( ⟨ λ ⟩ 1 , ⟨ λ ⟩ 2 , … , ⟨ λ ⟩ n − k + 1 ) = ( − 1 ) k λ ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 ,

where n ∈ N and generalized binomial coefficient z w is defined by

z w = Γ ( z + 1 ) Γ ( w + 1 ) Γ ( z − w + 1 ) , z , w , z − w ∈ C ⧹ { − 1 , − 2 , … } ; 0 , z ∈ C ⧹ { − 1 , − 2 , … } , w , z − w ∈ { − 1 , − 2 , … } .

For establishing explicit formulas for degenerate Narumi polynomials N n ( α ) ( x , λ ) and for the Narumi polynomials N n ( α ) ( x ) , we derive two explicit formulas for special values of the Bell polynomials of the second kind

(18) B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 .

Lemma 2.1

Let a , b ∈ C .

  1. When n ≥ k ≥ 0 and n ∈ N , the Bell polynomials of the second kind B n , k ( x 1 , x 2 , … , x n − k + 1 ) satisfy

    (19) B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = ( − 1 ) k k ! n ! ( n + k ) ! ∑ ℓ = 1 k − 1 ( − 1 ) ℓ k ℓ ∑ q = 0 n − k n + k 2 ℓ + q b 2 ℓ + q a n + k − ( 2 ℓ + q ) × ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) 2 ℓ + q − m ⟨ 2 ℓ + q ⟩ m × ∑ j = 0 k − ℓ ( − 1 ) j k − ℓ j ∑ m = 0 j j m ( k − ℓ − j ) n + k − ( 2 ℓ + q ) − m ⟨ n + k − ( 2 ℓ + q ) ⟩ m + [ ( − 1 ) k b n + k + a n + k ] ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ( k − j ) n + k − m ⟨ n + k ⟩ m .

  2. When n ≥ k ≥ 0 , the Bell polynomials of the second kind B n , k ( x 1 , x 2 , … , x n − k + 1 ) satisfy

    (20) B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = 1 k ! ∑ r + s = k ∑ ℓ + m = n ( − 1 ) s k r n ℓ b r + ℓ r + ℓ r a s + m s + m s × ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) .

Proof

For b > a > 0 , let

(21) G a , b ( u ) = e b u − e a u u , u ≠ 0 ; b − a , u = 0 .

It is easy to see that

(22) G a , b ( k ) ( u ) = ∫ a b t k e t u d t , k ∈ { 0 } ∪ N

and

(23) G a , b ( k ) ( 0 ) = b k + 1 − a k + 1 k + 1 , k ∈ { 0 } ∪ N .

Employing the formula

B n , k x 2 2 , x 3 3 , … , x n − k + 2 n − k + 2 = n ! ( n + k ) ! B n + k , k ( 0 , x 2 , x 3 , … , x n + 1 )

in [2, p. 136], we acquire

B n , k ( G a , b ′ ( 0 ) , G a , b ″ ( 0 ) , … , G a , b ( n − k + 1 ) ( 0 ) ) = B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = n ! ( n + k ) ! B n + k , k ( 0 , b 2 − a 2 , b 3 − a 3 , … , b n + 1 − a n + 1 ) .

Making use of the formula (14) yields

∑ n = 0 ∞ B n + k , k ( x 1 , x 2 , … , x n + 1 ) k ! n ! ( n + k ) ! t n + k n ! = ∑ m = 1 ∞ x m t m m ! k , ∑ n = 0 ∞ B n + k , k ( x 1 , x 2 , … , x n + 1 ) n + k k t n + k n ! = ∑ m = 1 ∞ x m t m m ! k , B n + k , k ( x 1 , x 2 , … , x n + 1 ) = n + k k lim t → 0 d n d t n ∑ m = 0 ∞ x m + 1 t m ( m + 1 ) ! k .

Accordingly, considering (2), we obtain

B n + k , k ( 0 , b 2 − a 2 , b 3 − a 3 , … , b n + 1 − a n + 1 ) = n + k k lim t → 0 d n d t n ∑ m = 1 ∞ ( b m + 1 − a m + 1 ) t m ( m + 1 ) ! k = n + k k lim t → 0 d n d t n ( e b t − b t − 1 ) − ( e a t − a t − 1 ) t k = n + k k lim t → 0 d n d t n ( − 1 ) k t k ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ( e b t − b t − 1 ) ℓ ( e a t − a t − 1 ) k − ℓ = n + k k lim t → 0 d n d t n ( − 1 ) k t k ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ ℓ ! ∑ q = 2 ℓ ∞ S ( q , ℓ ; 1 ) b q t q q ! ( k − ℓ ) ! ∑ p = 2 ( k − ℓ ) ∞ S ( p , k − ℓ ; 1 ) a p t p p ! = ( − 1 ) k ( n + k ) ! n ! lim t → 0 d n d t n t k ∑ ℓ = 0 k ( − 1 ) ℓ ∑ q = 0 ∞ S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! t q ∑ p = 0 ∞ S ( 2 ( k − ℓ ) + p , k − ℓ ; 1 ) a 2 ( k − ℓ ) + p [ 2 ( k − ℓ ) + p ] ! t p = ( − 1 ) k ( n + k ) ! n ! lim t → 0 d n d t n ∑ ℓ = 0 k ( − 1 ) ℓ ∑ m = 0 ∞ ∑ q = 0 m S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! S ( 2 ( k − ℓ ) + m − q , k − ℓ ; 1 ) a 2 ( k − ℓ ) + m − q [ 2 ( k − ℓ ) + m − q ] ! t m + k = ( − 1 ) k ( n + k ) ! ∑ ℓ = 0 k ( − 1 ) ℓ ∑ q = 0 n − k S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! S ( n + k − 2 ℓ − q , k − ℓ ; 1 ) a n + k − 2 ℓ − q ( n + k − 2 ℓ − q ) ! .

Consequently, we acquire

(24) B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = ( − 1 ) k n ! ∑ ℓ = 0 k ( − 1 ) ℓ ∑ q = 0 n − k S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! S ( n + k − 2 ℓ − q , k − ℓ ; 1 ) a n + k − 2 ℓ − q ( n + k − 2 ℓ − q ) ! .

Further applying Proposition 1.1 to (24) yields

B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = ( − 1 ) k n ! ∑ ℓ = 1 k − 1 ( − 1 ) ℓ ∑ q = 0 n − k S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! S ( n + k − 2 ℓ − q , k − ℓ ; 1 ) a n + k − 2 ℓ − q ( n + k − 2 ℓ − q ) !

+ ( − 1 ) k n ! ∑ q = 1 n − k S ( q , 0 ; 1 ) b q q ! S ( n + k − q , k ; 1 ) a n + k − q ( n + k − q ) ! + ( − 1 ) k n ! S ( 0 , 0 ; 1 ) b 0 0 ! S ( n + k , k ; 1 ) a n + k ( n + k ) ! + ( − 1 ) k n ! ( − 1 ) k ∑ q = 0 n − k − 1 S ( 2 k + q , k ; 1 ) b 2 k + q ( 2 k + q ) ! S ( n − k − q , 0 ; 1 ) a n − k − q ( n − k − q ) ! + ( − 1 ) k n ! ( − 1 ) k S ( n + k , k ; 1 ) b n + k ( n + k ) ! S ( 0 , 0 ; 1 ) a 0 0 ! = ( − 1 ) k n ! ∑ ℓ = 1 k − 1 ( − 1 ) ℓ ∑ q = 0 n − k S ( 2 ℓ + q , ℓ ; 1 ) b 2 ℓ + q ( 2 ℓ + q ) ! S ( n + k − 2 ℓ − q , k − ℓ ; 1 ) a n + k − 2 ℓ − q ( n + k − 2 ℓ − q ) ! + ( − 1 ) k n ! S ( n + k , k ; 1 ) a n + k ( n + k ) ! + n ! S ( n + k , k ; 1 ) b n + k ( n + k ) ! = ( − 1 ) k n ! ∑ ℓ = 1 k − 1 ( − 1 ) ℓ 1 ℓ ! 1 ( k − ℓ ) ! ∑ q = 0 n − k b 2 ℓ + q ( 2 ℓ + q ) ! a n + k − 2 ℓ − q ( n + k − 2 ℓ − q ) ! ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ⟨ 2 ℓ + q ⟩ m ( ℓ − j ) 2 ℓ + q − m × ∑ j = 0 k − ℓ ( − 1 ) j k − ℓ j ∑ m = 0 j j m ⟨ n + k − 2 ℓ − q ⟩ m ( k − ℓ − j ) n + k − 2 ℓ − q − m + [ b n + k + ( − 1 ) k a n + k ] n ! ( n + k ) ! 1 k ! ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ⟨ n + k ⟩ m ( k − j ) n + k − m ,

which can be rearranged as the explicit formula (19).

In [36, Section 1.8], it was given for n ≥ k ≥ 0 that

(25) B n , k 1 2 , 1 3 , … , 1 n − k + 2 = n ! ( n + k ) ! ∑ ℓ = 0 k ( − 1 ) k − ℓ n + k k − ℓ S ( n + ℓ , ℓ ) .

In [2, p. 136, equation [3n]], it was given that the Bell polynomials of the second kind B n , k satisfy

(26) B n , k ( x 1 + y 1 , x 2 + y 2 , … , x n − k + 1 + y n − k + 1 ) = ∑ r + s = k ∑ ℓ + m = n n ℓ B ℓ , r ( x 1 , x 2 , … , x ℓ − r + 1 ) B m , s ( y 1 , y 2 , … , y m − s + 1 ) .

Therefore, by virtue of the identities (26), (13), and (25) in sequence, we obtain

B n , k b 2 − a 2 2 , b 3 − a 3 3 , … , b n − k + 2 − a n − k + 2 n − k + 2 = ∑ r + s = k ∑ ℓ + m = n n ℓ B ℓ , r b 2 2 , b 3 3 , … , b ℓ − r + 2 ℓ − r + 2 B m , s − a 2 2 , − a 3 3 , … , − a m − s + 2 m − s + 2 = ∑ r + s = k ∑ ℓ + m = n n ℓ b ℓ + r B ℓ , r 1 2 , 1 3 , … , 1 ℓ − r + 2 ( − 1 ) s a m + s B m , s 1 2 , 1 3 , … , 1 m − s + 2 = ∑ r + s = k ∑ ℓ + m = n ( − 1 ) s n ℓ a m + s b ℓ + r ℓ ! ( ℓ + r ) ! ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) × m ! ( m + s ) ! ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) = 1 k ! ∑ r + s = k ∑ ℓ + m = n ( − 1 ) s k r n ℓ b r + ℓ r + ℓ r a s + m s + m s ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) × ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) .

The explicit formula (20) is thus proved. The proof of Lemma 2.1 is complete.□

Remark 2.1

The first few values of the Bell polynomials of the second kind in (18) for 1 ≤ n , k ≤ 4 can be computed by the explicit formula (19) and listed in Table 1.

Table 1

The first few values of the Bell polynomials of the second kind in (18) for 1 ≤ n , k ≤ 4

B n , k n = 1 n = 2 n = 3 n = 4
k = 1 1 2 ( b 2 − a 2 ) 1 3 ( b 3 − a 3 ) 1 4 ( b 4 − a 4 ) 1 5 ( b 5 − a 5 )
k = 2 0 1 4 ( a 2 − b 2 ) 2 1 2 ( a 5 − a 3 b 2 − a 2 b 3 + b 5 ) 1 6 ( 5 a 6 − 3 a 4 b 2 − 4 a 3 b 3 − 3 a 2 b 4 + 5 b 6 )
k = 3 0 0 1 8 ( b 2 − a 2 ) 3 − 1 2 ( a − b ) 3 ( a + b ) 2 ( a 2 + a b + b 2 )
k = 4 0 0 0 1 16 ( a 2 − b 2 ) 4

Remark 2.2

For new results and applications about the Bell polynomials of the second kind B n , k , please refer to the papers [6,7,18,19,25,28,36,39,40, 41,42] and closely related references therein.

3 Explicit formulas for degenerate Narumi and Cauchy numbers

In this section, we state and prove an explicit formula for degenerate Narumi numbers N n ( α ) ( λ ) , derive an explicit formula for degenerate Cauchy numbers C n ( λ ) , and list the first seven explicit expressions of degenerate Narumi numbers N n ( α ) ( λ ) for 0 ≤ n ≤ 6 .

Theorem 3.1

For n ∈ N , degenerate Narumi numbers N n ( α ) ( λ ) can be computed by

(27) N n ( α ) ( λ ) = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ⟨ α ⟩ ℓ ( k + ℓ ) ! ∑ j = 0 ℓ ( − 1 ) j k + ℓ ℓ − j S ( k + j , j ) .

Proof

Let F λ , α ( t ) denote the generating function of degenerate Narumi numbers N n ( α ) ( λ ) . If applying f ( u ) = e u − 1 u α and u = g ( t ) = ( 1 + t ) λ − 1 λ to the Faà di Bruno formula (12), then we can write

(28) d n d t n F λ , α ( t ) = ∑ k = 1 n d k d u k e u − 1 u α B n , k λ ( 1 + t ) λ − 1 λ , λ ( λ − 1 ) ( 1 + t ) λ − 2 λ , … , λ ( λ − 1 ) ⋯ [ λ − ( n − k ) ] ( 1 + t ) λ − ( n − k + 1 ) λ .

Utilizing the formula (25) and the Faà di Bruno formula (12), we arrive at

(29) lim u → 0 d k d u k e u − 1 u α = lim u → 0 ∑ i = 1 k ⟨ α ⟩ i v α − i B k , i e u − 1 u ′ , e u − 1 u ″ , … , e u − 1 u ( k − i + 1 ) = ∑ i = 1 k ⟨ α ⟩ i B k , i 1 2 , 1 3 , … , 1 k − i + 2 = ∑ i = 1 k ⟨ α ⟩ i k ! ( k + i ) ! ∑ j = 0 i ( − 1 ) i − j k + i i − j S ( k + j , j ) ,

where v = v ( u ) = e u − 1 u → 1 and

v ( ℓ ) ( u ) = e u − 1 u ( ℓ ) = ∑ j = 1 ∞ u j − 1 j ! ( ℓ ) = ∑ j = 0 ∞ u j ( j + 1 ) ! ( ℓ ) → 1 ℓ + 1

as u → 0 for ℓ ∈ N .

Employing (13) and (17), we acquire

(30) B n , k λ ( 1 + t ) λ − 1 λ , λ ( λ − 1 ) ( 1 + t ) λ − 2 λ , … , λ ( λ − 1 ) ⋯ ( λ − ( n − k ) ) ( 1 + t ) λ − ( n − k + 1 ) λ = ( 1 + t ) k λ − n λ k B n , k ( ⟨ λ ⟩ 1 , ⟨ λ ⟩ 2 , … , ⟨ λ ⟩ n − k + 1 ) = ( 1 + t ) k λ − n λ k ( − 1 ) k λ ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 → ( − 1 ) k λ k − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1

as t → 0 .

Taking t → 0 , which is equivalent to u → 0 , on both sides of (28) and making use of (29) and (30) give

lim t → 0 d n d t n F λ , α ( t ) = ∑ k = 1 n lim u → 0 d k d u k e u − 1 u α lim t → 0 B n , k λ ( 1 + t ) λ − 1 λ , λ ( λ − 1 ) ( 1 + t ) λ − 2 λ , … , λ ( λ − 1 ) ⋯ [ λ − ( n − k ) ] ( 1 + t ) λ − ( n − k + 1 ) λ = ∑ k = 1 n ∑ i = 1 k ⟨ α ⟩ i k ! ( k + i ) ! ∑ j = 0 i ( − 1 ) i − j k + i i − j S ( k + j , j ) ( − 1 ) k λ k − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 ∑ i = 1 k ⟨ α ⟩ i ( k + i ) ! ∑ j = 0 i ( − 1 ) i − j k + i i − j S ( k + j , j ) .

Considering the equation (11) leads to the formula (27). The proof of Theorem 3.1 is complete.□

Corollary 3.1

For n ∈ N , degenerate Cauchy numbers C n ( λ ) can be computed by

(31) C n ( λ ) = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k ( k + 1 ) ! 1 λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 .

Proof

This follows from setting α = 1 in the formula (27) and simplifying.□

Remark 3.1

From the explicit formula (27) in Theorem 3.1, we can obtain the first seven explicit expressions of degenerate Narumi numbers N n ( α ) ( λ ) for 0 ≤ n ≤ 6 as follows:

N 0 ( α ) ( λ ) = 1 , N 1 ( α ) ( λ ) = α 2 , N 2 ( α ) ( λ ) = 1 12 α ( 3 α + 6 λ − 5 ) , N 3 ( α ) ( λ ) = 1 8 α [ α 2 + α ( 6 λ − 5 ) + 4 λ 2 − 10 λ + 6 ] , N 4 ( α ) ( λ ) = 1 240 α [ 15 α 3 + 30 α 2 ( 6 λ − 5 ) + 5 α ( 84 λ 2 − 180 λ + 97 ) + 120 λ 3 − 580 λ 2 + 960 λ − 502 ] , N 5 ( α ) ( λ ) = 1 96 α [ 3 α 4 + 10 α 3 ( 6 λ − 5 ) + 5 α 2 ( 60 λ 2 − 120 λ + 61 ) + 2 α ( 180 λ 3 − 690 λ 2 + 910 λ − 401 ) + 8 ( 6 λ 4 − 45 λ 3 + 140 λ 2 − 196 λ + 95 ) ] , N 6 ( α ) ( λ ) = 1 4032 α [ 63 α 5 + 315 α 4 ( 6 λ − 5 ) + 315 α 3 ( 52 λ 2 − 100 λ + 49 ) + 7 α 2 ( 6480 λ 3 − 22320 λ 2 + 26370 λ − 10543 ) + 42 α ( 744 λ 4 − 4320 λ 3 + 9910 λ 2 − 10410 λ + 4075 ) + 8 ( 252 λ 5 − 2478 λ 4 + 11970 λ 3 − 31983 λ 2 + 41328 λ − 19087 ) ] .

Remark 3.2

The explicit formula (31) in Corollary 3.1 is slightly different from the explicit formula (8).

4 Explicit formulas for degenerate Narumi polynomials and numbers

In this section, with the aid of the explicit formulas (19) and (20), we establish two explicit formulas for degenerate Narumi polynomials N n ( α ) ( x , λ ) , deduce a different formula for degenerate Narumi numbers N n ( α ) ( λ ) from the formula (27), and list the first five values of degenerate Narumi polynomials N n ( α ) ( x , λ ) for 0 ≤ n ≤ 4 .

Theorem 4.1

For n ∈ N , degenerate Narumi polynomials N n ( α ) ( x , λ ) can be computed by

(32) N n ( α ) ( x , λ ) = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 ∑ ℓ = 1 k ( − 1 ) ℓ 1 − 2 x α α − ℓ ⟨ α ⟩ ℓ ℓ ! ( k + ℓ ) ! × ∑ p = 1 ℓ − 1 ( − 1 ) p ℓ p ∑ q = 0 k − ℓ k + ℓ 2 p + q 1 − x α 2 p + q x α k + ℓ − ( 2 p + q ) × ∑ j = 0 p ( − 1 ) j p j ∑ m = 0 j j m ( p − j ) 2 p + q − m ⟨ 2 p + q ⟩ m × ∑ j = 0 ℓ − p ( − 1 ) j ℓ − p j ∑ m = 0 j j m ( ℓ − p − j ) k + ℓ − ( 2 p + q ) − m ⟨ k + ℓ − ( 2 p + q ) ⟩ m + 1 − x α k + ℓ + ( − 1 ) ℓ x α k + ℓ ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) k + ℓ − m ⟨ k + ℓ ⟩ m

and

(33) N n ( α ) ( x , λ ) = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k k ! 1 λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 × ∑ q = 1 k ⟨ α ⟩ q q ! 1 − 2 x α α − q ∑ r + s = q ∑ ℓ + m = k ( − 1 ) s q r k ℓ ( 1 − x / α ) r + ℓ r + ℓ r ( x / α ) s + m s + m s × ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) .

Proof

Let F λ , α , x ( t ) denote the generating function of degenerate Narumi polynomials N n ( α ) ( x , λ ) in (10). Applying f ( u ) = e u − 1 u α e x u and u = g ( t ) = ( 1 + t ) λ − 1 λ to the Faà di Bruno formula (12) yields

lim t → 0 d n d t n F λ , α , x ( t ) = lim t → 0 ∑ k = 1 n d k d u k e u − 1 u α e x u B n , k × λ ( 1 + t ) λ − 1 λ , λ ( λ − 1 ) ( 1 + t ) λ − 2 λ , … , λ ( λ − 1 ) ⋯ [ λ − ( n − k ) ] ( 1 + t ) λ − ( n − k + 1 ) λ .

Since

e u − 1 u α e x u = e u ( 1 − x / α ) − e u x / α u α = G x / α , 1 − x / α α ( u ) ,

where G a , b ( u ) is defined by (21), we obtain

(34) lim u → 0 d k d u k e u − 1 u α e x u = lim u → 0 d k d u k G x / α , 1 − x / α α ( u ) = lim u → 0 ∑ ℓ = 1 k d ℓ w α d w ℓ B k , ℓ ( G x / α , 1 − x / α ′ ( u ) , G x / α , 1 − x / α ′ ( u ) , … , G x / α , 1 − x / α ( k − ℓ + 1 ) ( u ) ) = ∑ ℓ = 1 k ⟨ α ⟩ ℓ 1 − 2 x α α − ℓ B k , ℓ ( G x / α , 1 − x / α ′ ( 0 ) , G x / α , 1 − x / α ′ ( 0 ) , … , G x / α , 1 − x / α ( k − ℓ + 1 ) ( 0 ) ) = ∑ ℓ = 1 k ⟨ α ⟩ ℓ 1 − 2 x α α − ℓ B k , ℓ ( 1 − x / α ) 2 − ( x / α ) 2 2 , … , ( 1 − x / α ) k − ℓ + 2 − ( x / α ) k − ℓ + 2 k − ℓ + 2 ,

where

w = w ( u ) = G x / α , 1 − x / α ( u ) → 1 − 2 x α , u → 0

and we used the formulas (22) and (23). Consequently, we acquire

(35) lim t → 0 d n d t n F λ , α , x ( t ) = ∑ k = 1 n ∑ ℓ = 1 k ⟨ α ⟩ ℓ 1 − 2 x α α − ℓ B k , ℓ ( 1 − x / α ) 2 − ( x / α ) 2 2 , ( 1 − x / α ) 2 − ( x / α ) 3 3 , … , ( 1 − x / α ) k − ℓ + 2 − ( x / α ) k − ℓ + 2 k − ℓ + 2 ( − 1 ) k λ k − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 ,

where we used the limit in (30). Further making use of the explicit formula (19) in Lemma 2.1 leads to

N n ( α ) ( x , λ ) = ∑ k = 1 n ( − 1 ) k λ k − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 × ∑ ℓ = 1 k ⟨ α ⟩ ℓ 1 − 2 x α α − ℓ ( − 1 ) ℓ ℓ ! k ! ( k + ℓ ) ! ∑ p = 1 ℓ − 1 ( − 1 ) p ℓ p ∑ q = 0 k − ℓ k + ℓ 2 p + q × 1 − x α 2 p + q ∑ j = 0 p ( − 1 ) j p j ∑ m = 0 j j m ( p − j ) 2 p + q − m ⟨ 2 p + q ⟩ m × x α k + ℓ − ( 2 p + q ) ∑ j = 0 ℓ − p ( − 1 ) j ℓ − p j ∑ m = 0 j j m ( ℓ − p − j ) k + ℓ − ( 2 p + q ) − m ⟨ k + ℓ − ( 2 p + q ) ⟩ m + ( 1 − x / α ) k + ℓ + ( − 1 ) ℓ ( x / α ) k + ℓ ℓ ! k ! ( k + ℓ ) ! ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) k + ℓ − m ⟨ k + ℓ ⟩ m ,

which can be simplified as (32).

Employing (20) in (35) arrives at

N n ( α ) ( x , λ ) = ∑ k = 1 n ( − 1 ) k λ k − 1 ( n − 1 ) ! k ! ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 × ∑ q = 1 k ⟨ α ⟩ q 1 − 2 x α α − q 1 q ! ∑ r + s = q ∑ ℓ + m = k ( − 1 ) s q r k ℓ ( 1 − x / α ) r + ℓ r + ℓ r ( x / α ) s + m s + m s × ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) ,

which can be rearranged as (33). The proof of Theorem 4.1 is complete.□

Remark 4.1

The first five values of degenerate Narumi polynomials N n ( α ) ( x , λ ) for 0 ≤ n ≤ 4 can be computed and listed as follows:

N 0 ( α ) ( x , λ ) = 1 , N 1 ( α ) ( x , λ ) = α 2 + x , N 2 ( α ) ( x , λ ) = α 2 4 + α λ 2 + x − 5 12 + x ( λ + x − 1 ) , N 3 ( α ) ( x , λ ) = 1 8 { α 3 + α 2 ( 6 λ + 6 x − 5 ) + 2 α [ 2 λ 2 + λ ( 12 x − 5 ) + 6 x 2 − 11 x + 3 ] + 8 x [ λ 2 + 3 λ ( x − 1 ) + x 2 − 3 x + 2 ] } , N 4 ( α ) ( x , λ ) = α 4 16 + α 3 8 ( 6 λ + 4 x − 5 ) + α 2 48 [ 84 λ 2 + 36 λ ( 6 x − 5 ) + 72 x 2 − 192 x + 97 ] + α λ 3 2 + λ 2 7 x − 29 12 + λ 9 x 2 − 33 x 2 + 4 + 2 x 3 − 17 x 2 2 + 19 x 2 − 251 120 + x [ λ 3 + λ 2 ( 7 x − 6 ) + λ ( 6 x 2 − 18 x + 11 ) + x 3 − 6 x 2 + 11 x − 6 ] .

Corollary 4.1

For n ∈ N , degenerate Narumi number N n ( α ) ( λ ) can be computed by

(36) N n ( α ) ( λ ) = ( n − 1 ) ! ∑ k = 1 n ( − 1 ) k λ k − 1 ∑ ℓ = 1 k ( − 1 ) ℓ ℓ k ℓ λ ℓ − 1 n − 1 × ∑ ℓ = 1 k ⟨ α ⟩ ℓ ℓ ! ( k + ℓ ) ! ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) k + ℓ − m ⟨ k + ℓ ⟩ m .

Proof

This follows from taking x → 0 in (32).□

Remark 4.2

The explicit formula (27) in Theorem 3.1 is different from the explicit formula (36) in Corollary 4.1.

Remark 4.3

If taking x → 0 in (33), then we deduce (27) readily.

5 Explicit formulas for Narumi polynomials and numbers

In this section, we present two explicit formulas for the Narumi polynomials N n ( α ) ( x ) and derive two explicit formulas for the Narumi numbers N n ( α ) .

Theorem 5.1

For n ∈ N , the Narumi polynomials N n ( α ) ( x ) have the explicit formulas

(37) N n ( α ) ( x ) = ∑ r = 0 n s ( n , r ) r ! ∑ k = 1 r ( − 1 ) k ⟨ α ⟩ k k ! ( r + k ) ! 1 − 2 x α α − k × ∑ ℓ = 1 k − 1 ( − 1 ) ℓ k ℓ ∑ q = 0 r − k r + k 2 ℓ + q 1 − x α 2 ℓ + q x α r + k − ( 2 ℓ + q ) × ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) 2 ℓ + q − m ⟨ 2 ℓ + q ⟩ m × ∑ j = 0 k − ℓ ( − 1 ) j k − ℓ j ∑ m = 0 j j m ( k − ℓ − j ) r + k − ( 2 ℓ + q ) − m ⟨ r + k − ( 2 ℓ + q ) ⟩ m + ( − 1 ) k 1 − x α r + k + x α r + k ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ( k − j ) r + k − m ⟨ r + k ⟩ m

and

(38) N n ( α ) ( x ) = ∑ q = 0 n s ( n , q ) ∑ k = 1 q ⟨ α ⟩ k k ! 1 − 2 x α α − k ∑ r + s = k ∑ ℓ + m = q ( − 1 ) s k r q ℓ ( 1 − x / α ) r + ℓ r + ℓ r ( x / α ) s + m s + m s × ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) .

Proof

Taking ln ( 1 + t ) = u in (3) yields

e u − 1 u α e x u = ∑ n = 0 ∞ N n ( α ) ( x ) ( e u − 1 ) n n ! ,

where, by (1),

∑ n = 0 ∞ N n ( α ) ( x ) ( e u − 1 ) n n ! = ∑ n = 0 ∞ N n ( α ) ( x ) ∑ i = n ∞ S ( i , n ) u i i ! = ∑ i = 0 ∞ ∑ n = 0 i S ( i , n ) N n ( α ) ( x ) u i i ! .

This implies that

lim u → 0 d i d u i e u − 1 u α e x u = ∑ n = 0 i S ( i , n ) N n ( α ) ( x ) , i ≥ 0 .

Making use of the result in (34) gives

∑ n = 0 i S ( i , n ) N n ( α ) ( x ) = ∑ ℓ = 1 i ⟨ α ⟩ ℓ 1 − 2 x α α − ℓ B i , ℓ ( 1 − x / α ) 2 − ( x / α ) 2 2 , … , ( 1 − x / α ) i − ℓ + 2 − ( x / α ) i − ℓ + 2 i − ℓ + 2

for i ∈ N . In [43, p. 171, Theorem 12.1], it is stated that if b α and a k are a collection of constants independent of n , then

a n = ∑ α = 0 n S ( n , α ) b α if and only if b n = ∑ k = 0 n s ( n , k ) a k .

Accordingly, we derive

N i ( α ) ( x ) = ∑ n = 0 i s ( i , n ) ∑ k = 1 n ⟨ α ⟩ k 1 − 2 x α α − k B n , k ( 1 − x / α ) 2 − ( x / α ) 2 2 , … , ( 1 − x / α ) n − k + 2 − ( x / α ) n − k + 2 n − k + 2 = ∑ n = 0 i s ( i , n ) ∑ k = 1 n ⟨ α ⟩ k 1 − 2 x α α − k ( − 1 ) k k ! n ! ( n + k ) ! ∑ ℓ = 1 k − 1 ( − 1 ) ℓ k ℓ ∑ q = 0 n − k n + k 2 ℓ + q × 1 − x α 2 ℓ + q ∑ j = 0 ℓ ( − 1 ) j ℓ j ∑ m = 0 j j m ( ℓ − j ) 2 ℓ + q − m ⟨ 2 ℓ + q ⟩ m × x α n + k − ( 2 ℓ + q ) ∑ j = 0 k − ℓ ( − 1 ) j k − ℓ j ∑ m = 0 j j m ( k − ℓ − j ) n + k − ( 2 ℓ + q ) − m ⟨ n + k − ( 2 ℓ + q ) ⟩ m + n ! k ! ( n + k ) ! 1 − x α n + k + ( − 1 ) k x α n + k ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ( k − j ) n + k − m ⟨ n + k ⟩ m ,

where we used the formula (19) in Lemma 2.1. The explicit formula (37) is thus proved.

If we use the formula (20), then

N i ( α ) ( x ) = ∑ n = 0 i s ( i , n ) ∑ k = 1 n ⟨ α ⟩ k 1 − 2 x α α − k B n , k ( 1 − x / α ) 2 − ( x / α ) 2 2 , … , ( 1 − x / α ) n − k + 2 − ( x / α ) n − k + 2 n − k + 2 = ∑ n = 0 i s ( i , n ) ∑ k = 1 n ⟨ α ⟩ k 1 − 2 x α α − k 1 k ! ∑ r + s = k ∑ ℓ + m = n ( − 1 ) s k r n ℓ ( 1 − x / α ) r + ℓ r + ℓ r ( x / α ) s + m s + m s × ∑ j = 0 r ( − 1 ) r − j ℓ + r r − j S ( ℓ + j , j ) ∑ j = 0 s ( − 1 ) s − j m + s s − j S ( m + j , j ) ,

which can be rewritten as (38). The proof of Theorem 5.1 is complete.□

Corollary 5.1

For n ∈ N , the Narumi numbers N n ( α ) have the explicit formulas

(39) N n ( α ) = ∑ r = 0 n s ( n , r ) r ! ∑ k = 1 r ⟨ α ⟩ k k ! ( r + k ) ! ∑ j = 0 k ( − 1 ) j k j ∑ m = 0 j j m ( k − j ) r + k − m ⟨ r + k ⟩ m

and

(40) N n ( α ) = ∑ q = 0 n s ( n , q ) q ! ∑ k = 1 q ⟨ α ⟩ k ( k + q ) ! ∑ j = 0 k ( − 1 ) k − j q + k k − j S ( q + j , j ) .

Proof

These follow from taking x → 0 in (37) and (38), respectively.□

Remark 5.1

When taking α = 1 in (39) or (40), we derive the explicit formula (9) for the Cauchy numbers C n = N n ( 1 ) for n ∈ N .

Remark 5.2

The first six values of the Narumi polynomials N n ( α ) ( x ) for 0 ≤ n ≤ 5 can be computed by

N 0 ( α ) ( x ) = 1 , N 1 ( α ) ( x ) = α 2 + x , N 2 ( α ) ( x ) = α 2 4 + α x − 5 12 + ( x − 1 ) x , N 3 ( α ) ( x ) = 1 8 [ α 3 + α 2 ( 6 x − 5 ) + 2 α ( 6 x 2 − 11 x + 3 ) + 8 x ( x 2 − 3 x + 2 ) ] , N 4 ( α ) ( x ) = α 4 16 + 1 8 α 3 ( 4 x − 5 ) + α 2 3 x 2 2 − 4 x + 97 48 + α 2 x 3 − 17 x 2 2 + 19 x 2 − 251 120 + x ( x 3 − 6 x 2 + 11 x − 6 ) , N 5 ( α ) ( x ) = 1 96 [ 3 α 5 + 10 α 4 ( 3 x − 5 ) + 5 α 3 ( 24 x 2 − 84 x + 61 ) + 2 α 2 ( 120 x 3 − 660 x 2 + 1025 x − 401 ) + 4 α ( 60 x 4 − 460 x 3 + 1140 x 2 − 991 x + 190 ) + 96 x ( x 4 − 10 x 3 + 35 x 2 − 50 x + 24 ) ] .

The first nine values of the Narumi numbers N n ( α ) for 0 ≤ n ≤ 8 are

N 0 ( α ) = 1 , N 1 ( α ) = α 2 , N 2 ( α ) = 1 12 α ( 3 α − 5 ) , N 3 ( α ) = 1 8 α ( α 2 − 5 α + 6 ) , N 4 ( α ) = 1 240 α ( 15 α 3 − 150 α 2 + 485 α − 502 ) , N 5 ( α ) = 1 96 α ( 3 α 4 − 50 α 3 + 305 α 2 − 802 α + 760 ) , N 6 ( α ) = α 4032 ( 63 α 5 − 1575 α 4 + 15435 α 3 − 73801 α 2 + 171150 α − 152696 ) , N 7 ( α ) = α 1152 ( 9 α 6 − 315 α 5 + 4515 α 4 − 33817 α 3 + 139020 α 2 − 295748 α + 252336 ) , N 8 ( α ) = α 34560 ( 135 α 7 − 6300 α 6 + 124110 α 5 − 1334760 α 4 + 8437975 α 3 − 31231500 α 2 + 62333204 α − 51360816 ) .

, ,

Dedicated to Professor Ce-Wen Cao at School of Mathematics and Statistics, Zhengzhou University, China.

Acknowledgements

The authors thank anonymous referees for their careful corrections to, valuable comments on, and helpful suggestions to the original version of this paper.

  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. Author contributions: All authors contributed equally to the manuscript and read and approved the final version.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2021-05-17
Accepted: 2021-07-21
Published Online: 2021-08-16

© 2021 Feng Qi et al., published by De Gruyter

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

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https://doi.org/10.1515/math-2021-0079
Keywords for this article
Narumi number; Narumi polynomial; Cauchy number; degenerate Cauchy number; explicit formula; Bell polynomial of the second kind; Faà di Bruno formula; generating function
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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
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  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
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  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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