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Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus

  • Taekyun Kim , Dae San Kim and Hye Kyung Kim EMAIL logo
Published/Copyright: October 3, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

Hyperharmonic numbers were introduced by Conway and Guy (The Book of Numbers, Copernicus, New York, 1996), whereas harmonic numbers have been studied since antiquity. Recently, the degenerate hyperharmonic and degenerate harmonic numbers were introduced as their respective degenerate versions. The aim of this article is to further investigate some properties and identities involving the degenerate hyperharmonic numbers and degenerate harmonic numbers. Especially, we derive some identities by making use of the transfer formula for associated sequences from umbral calculus.

Keywords: degenerate harmonic number; degenerate hyperharmonic number; umbral calculus
MSC 2010: 05A40; 11B73; 11B83

1 Introduction

Various degenerate versions of many special numbers and polynomials have been studied recently with their regained interests (see [1–6] and references therein). This investigation began with the pioneering work of Carlitz on the degenerate Bernoulli and degenerate Euler numbers in [7]. It is worthwhile to mention that these explorations for degenerate versions are not just limited to polynomials and numbers but also extended to transcendental functions, such as gamma functions (see [5]). It is also remarkable that the λ -umbral calculus and λ - q -umbral calculus were introduced as degenerate versions of the umbral calculus and the q -umbral calculus, respectively (see [1,8]).

The aim of this article is to further investigate some properties and identities involving the degenerate hyperharmonic numbers (see (7)) and degenerate harmonic numbers (see (6)). Especially, we derive some identities by making use of the transfer formula (see (22)) for associated sequences from umbral calculus.

The outline of this article is as follows. In Section 1, we recall the degenerate exponentials and the degenerate logarithms. We remind the reader of harmonic numbers and hyperharmonic numbers together with their explicit expression due to Conway and Guy (see [9]). Then, we recall their degenerate versions, namely, the degenerate harmonic numbers, and the degenerate hyperharmonic numbers together with their explicit expression (see [2,3]). We also mention the recently introduced degenerate Stirling numbers of the first kind and of the second kind. We also remind the reader of the degenerate Bell polynomials. Next, we briefly review some facts about umbral calculus, namely, the linear functionals and the linear differential operators on the vector space of all polynomials over C , Sheffer sequences, and the transfer formulas for associated sequences. In Section 2, we obtain explicit expressions for two formal power series whose coefficients involve the hyperharmonic numbers. Then, we derive an expression of some binomial in terms of the degenerate hyperharmonic numbers and the degenerate Stirling numbers of the first kind. In Section 3, we apply the transfer formula in order to derive some identities involving the degenerate hyperharmonic numbers in Theorems 1, 5, and 8. Also, we deduce an identity involving the degenerate Stirling numbers of both kinds. Finally, we conclude this article in Section 4. In the rest of this section, we recall what are needed throughout this study.

For integers n ≥ 0 , the harmonic numbers are defined by:

(1) H 0 = 0 , H n = 1 + 1 2 + ⋯ + 1 n ( see [10] ) .

For integers n and r ≥ 0 , we note that the hyperharmonic numbers are given by:

(2) H n ( r ) = 0 , if n = 0 , r ≥ 0 , 1 n , if r = 0 , n ≥ 1 , ∑ i = 1 n H i ( r − 1 ) , if n , r ≥ 1 ( see [9] ) .

From (2), we have

(3) H n ( r ) = n + r − 1 r − 1 ( H n + r − 1 − H r − 1 ) , ( r ≥ 1 ) ( see [9] ) .

For any nonzero λ ∈ R , the degenerate exponential functions are defined by:

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

where ( x ) 0 , λ = 1 , ( x ) n , λ = x ( x − λ ) ( x − 2 λ ) ⋯ ( x − ( n − 1 ) λ ) , ( n ≥ 1 ) (see [11]).

Let log λ t be the degenerate logarithm, which is the compositional inverse of e λ ( t ) so that it satisfies log λ ( e λ ( t ) ) = e λ ( log λ ( t ) ) = t .

In [2,3], the degenerate harmonic numbers are defined by:

(5) − 1 1 − t log λ ( 1 − t ) = ∑ n = 0 ∞ H n , λ t n ( see [ 2 ] ) .

Thus, by (5), we obtain

(6) H 0 , λ = 0 , H n , λ = ∑ k = 1 n 1 λ λ k ( − 1 ) k − 1 , ( n ∈ N ) ( see [2] ) .

Note that lim λ → 0 H n , λ = H n , ( n ≥ 0 ) . For n , r ≥ 0 , the degenerate hyperharmonic numbers H n , λ ( r ) of order r are defined by:

(7) H 0 , λ ( r ) = 0 , ( r ≥ 0 ) , H n , λ ( 0 ) = 1 λ λ n ( − 1 ) n − 1 , ( n ≥ 1 ) , H n , λ ( r ) = ∑ k = 1 n H k , λ ( r − 1 ) , ( n , r ≥ 1 ) ( see [2] ) .

We observe from (6) and (7) that H n , λ ( 1 ) = H n , λ . From (7), we note that

(8) − log λ ( 1 − t ) ( 1 − t ) r = ∑ n = 1 ∞ H n , λ ( r ) t n

and

(9) H n , λ ( r + 1 ) = ( − 1 ) r λ − 1 r n + r n ( H n + r , λ − H r , λ ) , ( r ≥ 0 ) ( see [2] ) .

It is well known that Stirling numbers of the first kind are defined by:

(10) ( x ) n = ∑ k = 0 n S 1 ( n , k ) x k , ( n ≥ 0 ) ( see [12] ) ,

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

As the inversion formula of (10), the Stirling numbers of the second kind are given by:

(11) x n = ∑ k = 0 n S 2 ( n , k ) ( x ) k , ( n ≥ 0 ) ( see [7,10,13] ) .

Recently, the degenerate Stirling numbers of the first kind are defined by:

(12) ( x ) n = ∑ k = 0 n S 1 , λ ( n , k ) ( x ) k , λ , ( n ≥ 0 ) ( see [4] ) ,

and the unsigned degenerate Stirling numbers of the first kind are given by:

(13) n k λ = ( − 1 ) n − k S 1 , λ ( n , k ) , ( n , k ≥ 0 ) ( see [4] ) .

Also, the degenerate Stirling numbers of the second kind are defined by:

(14) ( x ) n , λ = ∑ k = 0 n S 2 , λ ( n , k ) ( x ) k , ( n ≥ 0 ) ( see [11] ) .

In [11], the degenerate Bell polynomials are given by:

(15) e x ( e λ ( t ) − 1 ) = ∑ n = 0 ∞ ϕ n , λ ( x ) t n n ! .

Thus, from (15), we obtain

(16) ϕ n , λ ( x ) = ∑ k = 0 n S 2 , λ ( n , k ) x k ( n ≥ 0 ) .

Note that lim λ → 0 ϕ n , λ ( x ) = ϕ n ( x ) , where ϕ n ( x ) are the ordinary Bell polynomials given by:

(17) e x ( e t − 1 ) = ∑ n = 0 ∞ ϕ n ( x ) t n n ! ( see [14] ) .

Let F be the algebra of all formal power series in t over C with:

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

Let P = C [ x ] be the algebra of all polynomials in x with coefficients in C , and let P * denote the vector space of all linear functionals on P . For f ( t ) = ∑ k = 0 ∞ a k t k k ! ∈ F , we define the linear functional on P by setting

(18) ⟨ f ( t ) ∣ x n ⟩ = a n for all n ≥ 0 ( see [14–17] ) .

Then, we see that ⟨ t k ∣ x n ⟩ = n ! δ n , k , where δ n , k is the Kronecker symbol. Let f L ( t ) = ∑ k = 0 ∞ ⟨ L ∣ x n ⟩ k ! t k . Then, we see that ⟨ f L ( t ) ∣ x n ⟩ = ⟨ L ∣ x n ⟩ . The map L ↦ f L ( t ) is a vector space isomorphism from P * onto F . Henceforth, F is thought of as both the algebra of formal power series in t and the vector space of all linear functionals on P (see [17]). F is called the umbral algebra, and the umbral calculus is the study of umbral algebra.

The order o ( f ) of the power series f ( t ) ( ≠ 0 ) is the smallest integer k for which a k is not zero. If o ( f ) = 1 , then f is called a delta series; if o ( f ) = 0 , then f ( t ) is called an invertible series (see [17]).

Let f ( t ) ∈ F and p ( x ) ∈ P . Then, we have

(19) f ( t ) = ∑ k = 0 ∞ ⟨ f ( t ) ∣ x k ⟩ k ! t k and p ( x ) = ∑ k = 0 ∞ ⟨ t k ∣ p ( x ) ⟩ k ! x k ( see [14,17,18] ) .

From (19), we note that

(20) p ( k ) ( 0 ) = ⟨ t k ∣ p ( x ) ⟩ and ⟨ 1 ∣ p ( k ) ( x ) ⟩ = p ( k ) ( 0 ) ,

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

For each nonnegative integer k , the linear differential operator t k on P is defined by:

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

and hence, we have

(21) t k p ( x ) = p ( k ) ( x ) = d k p ( x ) d x k , ( k ≥ 0 ) ( see [14,17,19] ) .

For f ( t ) , g ( t ) ∈ F with o ( f ) = 1 and o ( g ) = 0 , we note that there exists a unique sequence s n ( x ) of polynomials such that ⟨ g ( t ) ( f ( t ) ) k ∣ s n ( x ) ⟩ = δ n , k , for n , k ≥ 0 (see [17]). 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 ) ) (see [17]).

For A n ( x ) ∼ ( 1 , f ( t ) ) and B n ( x ) ∼ ( 1 , g ( t ) ) , the transfer formula for the associated sequences is given by:

(22) B n ( x ) = x f ( t ) g ( t ) n x − 1 A n ( x ) , ( n ≥ 1 ) ( see [14,17] ) .

In this article, we investigate some properties of degenerate harmonic and hyperharmonic numbers that are derived from the generating functions of those numbers. From our investigation, we derive some new combinatorial identities involving degenerate harmonic and hyperharmonic numbers arising from transfer formula for the associated sequences.

2 Some properties of degenerate harmonic and degenerate hyperharmonic numbers

From (8), we note that

(23) ∑ n = 1 ∞ n H n , λ ( r ) t n − 1 = d d t − log λ ( 1 − t ) ( 1 − t ) r = ( − log λ ( 1 − t ) ) r ( 1 − t ) − r − 1 + 1 ( 1 − t ) r 1 1 − t ( λ log λ ( 1 − t ) + 1 ) = 1 ( 1 − t ) r + 1 ( ( λ − r ) log λ ( 1 − t ) + 1 ) .

Thus, by (23), we obtain

(24) ∑ n = 1 ∞ n H n , λ ( r ) t n = t ( 1 − t ) r + 1 ( 1 + ( λ − r ) log λ ( 1 − t ) ) .

Multiplying 1 1 − t on both sides of (24), we obtain

(25) t ( 1 − t ) r + 2 ( 1 + ( λ − r ) log λ ( 1 − t ) ) = 1 1 − t ∑ m = 1 ∞ m H m , λ ( r ) t m = ∑ m = 1 ∞ m H m , λ t m ∑ l = 0 ∞ t l = ∑ n = 1 ∞ ∑ m = 1 n m H m , λ ( r ) t n .

Thus, from (25), we have

(26) t ( 1 − t ) r + 2 ( 1 + ( λ − r ) log λ ( 1 − t ) ) = ∑ n = 1 ∞ ∑ m = 1 n m H m , λ ( r ) t n .

From (24) with r = 1 , we have

(27) ∑ k = 1 ∞ k H k , λ t k = t ( 1 − t ) 2 ( ( λ − 1 ) log λ ( 1 − t ) + 1 ) .

From (27), we note that

(28) ∑ n = 1 ∞ ( n + 1 ) H n , λ t n = t ( 1 − t ) 2 ( ( λ − 1 ) log λ ( 1 − t ) + 1 ) − 1 1 − t log λ ( 1 − t ) = t − log λ ( 1 − t ) ( 1 − λ t ) ( 1 − t ) 2 .

Hence, by (28), we obtain

(29) ∑ n = 1 ∞ ( n + 1 ) H n , λ t n = t − ( 1 − λ t ) log λ ( 1 − t ) ( 1 − t ) 2 .

By (27), we obtain

(30) ∑ k = 1 ∞ k 2 H k , λ t k − 1 = d d t t ( λ − 1 ) log λ ( 1 − t ) + t ( 1 − t ) 2 = ( λ − 1 ) log λ ( 1 − t ) + 1 − t ( λ − 1 ) 1 1 − t ( λ log λ ( 1 − t ) + 1 ) ( 1 − t ) 2 + 2 t ( ( λ − 1 ) log λ ( 1 − t ) + 1 ) ( 1 − t ) 3 = ( 1 − t ) { ( λ − 1 ) log λ ( 1 − t ) + 1 } − t ( λ − 1 ) ( λ log λ ( 1 − t ) + 1 ) ( 1 − t ) 3 + 2 t ( ( λ − 1 ) log λ ( 1 − t ) + 1 ) ( 1 − t ) 3 = 1 + t ( 2 − λ ) + ( λ − 1 ) log λ ( 1 − t ) ( 1 + t ( 1 − λ ) ) ( 1 − t ) 3 .

Thus, by (30), we obtain

∑ n = 1 ∞ n 2 H n , λ t n = t { 1 + t ( 2 − λ ) + ( λ − 1 ) ( 1 + t ( 1 − λ ) ) log λ ( 1 − t ) } ( 1 − t ) 3 .

For r ≥ 0 , we have

(31) ∑ n = 0 ∞ n + r n z n = 1 ( 1 − z ) r + 1 = 1 ( 1 − z ) r ∑ k = 0 ∞ ( − 1 ) k , λ ( log λ ( 1 − z ) ) k k ! = 1 ( 1 − z ) r + ∑ k = 1 ∞ ( − 1 ) k , λ ( 1 − z ) r ( log λ ( 1 − z ) ) k k ! = 1 ( 1 − z ) r + ∑ k = 0 ∞ ( − 1 ) k + 1 , λ k + 1 ( log λ ( 1 − z ) ) k k ! log λ ( 1 − z ) ( 1 − z ) r = 1 ( 1 − z ) r − ∑ k = 0 ∞ ( − 1 ) k + 1 , λ k + 1 ∑ m = k ∞ S 1 , λ ( m , k ) ( − 1 ) m z m m ! ∑ l = 1 ∞ H l , λ ( r ) z l = 1 ( 1 − z ) r − ∑ n = 0 ∞ ∑ m = 0 n ∑ k = 0 m ( − 1 ) k + 1 , λ k + 1 S 1 , λ ( m , k ) m ! ( − 1 ) m H n − m , λ ( r ) z n = ∑ n = 0 ∞ r + n − 1 n − ∑ m = 0 n ∑ k = 0 m ( − 1 ) k + 1 , λ k + 1 S 1 , λ ( m , k ) m ! ( − 1 ) m H n − m , λ ( r ) z n .

Thus, by comparing the coefficients on the both sides of (31), we obtain

n + r n = n + r − 1 n − ∑ m = 0 n ∑ k = 0 m ( − 1 ) k + 1 , λ k + 1 S 1 , λ ( m , k ) m ! ( − 1 ) m H n − m , λ ( r ) .

That is,

(32) n + r − 1 n − 1 = ∑ m = 0 n ∑ k = 0 m ( − 1 ) k + 1 , λ k + 1 S 1 , λ ( m , k ) m ! ( − 1 ) m − 1 H n − m , λ ( r ) ,

where n is a positive integer.

Thus, we have shown the following theorem.

Theorem 1

For n ≥ 1 and r ≥ 0 , the following identity is valid:

n + r − 1 n − 1 = ∑ m = 0 n ∑ k = 0 m ( − 1 ) k + 1 , λ k + 1 S 1 , λ ( m , k ) m ! ( − 1 ) m − 1 H n − m , λ ( r ) .

3 Some identities of degenerate hyperharmonic numbers arising from umbral calculus

Assume that

(33) B n ( x ) ∼ ( 1 , t ( 1 − t ) r ) , ( r ≥ 0 ) .

Since x n ∼ ( 1 , t ) , from (22), we obtain

(34) B n ( x ) = x t t ( 1 − t ) r n x − 1 x n = x ( 1 − t ) − r n x n − 1 = x ∑ k = 0 ∞ r n + k − 1 k t k x n − 1 = x ∑ k = 0 n − 1 r n + k − 1 k ( n − 1 ) k x n − 1 − k = ∑ k = 0 n − 1 r n + k − 1 k ( n − 1 ) k x n − k = ∑ k = 1 n r n + n − k − 1 n − k ( n − 1 ) n − k x k .

By (15), we obtain

ϕ n , λ ( x ) = ∑ k = 0 n S 2 , λ ( n , k ) x k ∼ ( 1 , log λ ( 1 + t ) )

and

(35) ( − 1 ) n ϕ n , λ ( − x ) ∼ ( 1 , − log λ ( 1 − t ) ) .

On the other hand, by (22), (33), and (35), we obtain

(36) B n ( x ) = x − log λ ( 1 − t ) t ( 1 − t ) r n x − 1 ( − 1 ) n ϕ n , λ ( − x ) = x 1 t ∑ l = 1 ∞ H l , λ ( r ) t l n x − 1 ( − 1 ) n ϕ n , λ ( − x ) = x ∑ l = 0 ∞ H l + 1 , λ ( r ) t l n x − 1 ( − 1 ) n ϕ n , λ ( − x ) = x ∑ l = 0 ∞ H l + 1 , λ ( r ) t l n x − 1 ( − 1 ) n ∑ j = 0 n S 2 , λ ( n , j ) ( − 1 ) j x j = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j x ∑ l = 0 ∞ H l + 1 , λ ( r ) t l n x j − 1 = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j x ∑ l = 0 j − 1 ∑ l 1 + ⋯ + l n = l H l 1 + 1 , λ ( r ) ⋯ H l n + 1 , λ ( r ) ( j − 1 ) l x j − l − 1 = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j ∑ l = 1 j ∑ l 1 + ⋯ + l n = j − l H l 1 + 1 , λ ( r ) ⋯ H l n + 1 , λ ( r ) ( j − 1 ) j − l x l = ( − 1 ) n ∑ l = 1 n ∑ j = l n ∑ l 1 + ⋯ + l n = j − l S 2 , λ ( n , j ) ( − 1 ) j H l 1 + 1 , λ ( r ) ⋯ H l n + 1 , λ ( r ) ( j − 1 ) j − l x l ,

where n is a positive integer.

Therefore, by (34) and (36), we obtain the following theorem.

Theorem 2

For n ≥ 1 , r ≥ 1 , and 1 ≤ l ≤ n , we have

r n + n − l − 1 n − l ( n − 1 ) n − l = ( − 1 ) n ∑ j = l n ∑ l 1 + ⋯ + l n = j − l S 2 , λ ( n , j ) ( − 1 ) j H l 1 + 1 , λ ( r ) ⋯ H l n + 1 , λ ( r ) ( j − 1 ) j − l .

For n ≥ 1 , we have

log λ ( 1 + t ) t n = n ! t n 1 n ! ( log λ ( 1 + t ) ) n = n ! t n ∑ l = n ∞ S 1 , λ ( l , n ) t l l ! = n ! t n ∑ l = 0 ∞ S 1 , λ ( l + n , n ) l ! ( l + n ) ! t l + n l ! = ∑ l = 0 ∞ S 1 , λ ( l + n , n ) l + n l t l l ! .

For n ≥ 1 , by (33) and (35), we obtain

(37) B n ( x ) = x − log λ ( 1 − t ) t ( 1 − t ) r n x − 1 ( − 1 ) n ϕ n , λ ( − x ) = x log λ ( 1 − t ) − t n ( 1 − t ) − r n x − 1 ( − 1 ) n ϕ n , λ ( − x ) = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j x log λ ( 1 − t ) − t n ( 1 − t ) − r n x j − 1 = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j x log λ ( 1 − t ) − t n ∑ l = 0 j − 1 r n + l − 1 l ( j − 1 ) l x j − 1 − l = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j ∑ l = 0 j − 1 r n + l − 1 l ( j − 1 ) l x ∑ m = 0 j − 1 − l S 1 , λ ( m + n , n ) m + n m m ! ( − t ) m x j − 1 − l = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j ∑ l = 0 j − 1 r n + l − 1 l ( j − 1 ) l x ∑ m = 0 j − 1 − l S 1 , λ ( n + m , n ) m + n m m ! ( − 1 ) m ( j − 1 − l ) m t j − 1 − l − m = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ( − 1 ) j ∑ l = 0 j − 1 r n + l − 1 l ( j − 1 ) l ∑ k = 1 j − l S 1 , λ ( j − l − k + n , n ) ( − 1 ) j − l − k j − l − k + n n ( j − l − k ) ! ( j − 1 − l ) j − l − k x k = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ∑ l = 0 j − 1 r n + l − 1 l ( j − 1 ) l ∑ k = 1 j − l S 1 , λ ( j − l − k + n , n ) j − l − k + n n ( j − l − k ) ! ( j − 1 − l ) ! ( k − 1 ) ! ( − 1 ) l + k x k = ( − 1 ) n ∑ j = 1 n S 2 , λ ( n , j ) ∑ k = 1 j ∑ l = 0 j − k r n + l − 1 l ( j − 1 ) l j − 1 − l k − 1 j − l − k + n n S 1 , λ ( j − l − k + n , n ) ( − 1 ) l + k x k = ( − 1 ) n ∑ k = 1 n ∑ j = k n ∑ l = 0 j − k S 2 , λ ( n , j ) S 1 , λ ( j − l − k + n , n ) r n + l − 1 l ( j − 1 ) l j − 1 − l k − 1 j − l − k + n n ( − 1 ) k + l x k .

Therefore, by (34) and (37), we obtain the following theorem.

Theorem 3

For n ≥ 1 , r ≥ 1 , and 1 ≤ k ≤ n , we have

r n + n − k − 1 n − k n − 1 n − k ( n − k ) ! = ( − 1 ) n ∑ j = k n ∑ l = 0 j − k ( − 1 ) k + l r n + l − 1 l ( j − 1 ) l j − 1 − l k − 1 j − l − k + n n S 2 , λ ( n , j ) S 1 , λ ( j − k − l + n , n ) .

Consider the following associated sequence:

(38) q n ( x ) ∼ ( 1 , t ( 1 − t ) r + 2 ) .

From (22), for n ≥ 1 , we have

(39) q n ( x ) = x t t ( 1 − t ) r + 2 n x − 1 x n = x ( 1 − t ) − n ( r + 2 ) x n − 1 = x ∑ k = 0 n − 1 n ( r + 2 ) + k − 1 k t k x n − 1 = ∑ k = 0 n − 1 n ( r + 2 ) + k − 1 k ( n − 1 ) k x n − k = ∑ k = 1 n n ( r + 3 ) − k − 1 n − k ( n − 1 ) n − k x k .

Therefore, by (39), we obtain the following lemma.

Lemma 4

Let q n ( x ) ∼ ( 1 , t ( 1 − t ) r + 2 ) . Then, we have

q n ( x ) = ∑ k = 1 n n ( r + 3 ) − k − 1 n − k ( n − 1 ) n − k x k , ( n ≥ 1 ) .

Assume that

(40) p n ( x ) ∼ ( 1 , t ( 1 + ( λ − r ) log λ ( 1 − t ) ) ) .

By (22) and (40), for n ≥ 1 , we obtain

(41) p n ( x ) = x t t ( 1 + ( λ − r ) log λ ( 1 − t ) ) n x − 1 x n = x ∑ l = 0 ∞ n + l − 1 l ( λ − r ) l ( − 1 ) l ( log λ ( 1 − t ) ) l x n − 1 = x ∑ l = 0 ∞ n + l − 1 l ( − λ + r ) l l ! ∑ j = l n − 1 S 1 , λ ( j , l ) ( − 1 ) j j ! t j x n − 1 = ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ j = 0 n − l − 1 S 1 , λ ( j + l , l ) ( − 1 ) j + l ( j + l ) ! ( n − 1 ) j + l x n − j − l = ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ j = 0 n − l − 1 S 1 , λ ( j + l , l ) ( − 1 ) j + l n − 1 j + l x n − j − l = ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ k = 1 n − l S 1 , λ ( n − k , l ) ( − 1 ) n − k n − 1 n − k x k = ∑ k = 1 n ∑ l = 0 n − k ( λ − r ) l l ! n + l − 1 l n − 1 k − 1 ( − 1 ) n − k − l S 1 , λ ( n − k , l ) x k = ∑ k = 1 n ∑ l = 0 n − k ( λ − r ) ! l ! n + l − 1 l n − 1 k − 1 n − k l λ x k .

Therefore, by (41), we obtain the following lemma.

Lemma 5

Let p n ( x ) ∼ ( 1 , t ( 1 + ( λ − r ) log λ ( 1 − t ) ) ) . Then, we have

p n ( x ) = ∑ k = 1 n ∑ l = 0 n − k ( λ − r ) ! l ! n + l − 1 l n − 1 k − 1 n − k l λ x k , ( n ≥ 1 ) .

For n ≥ 1 , by (22), (26), (38), and (40), we obtain

(42) q n ( x ) = x t ( 1 + ( λ − r ) log λ ( 1 − t ) ) t ( 1 − t ) r + 2 n x − 1 p n ( x ) = x ∑ j = 1 ∞ ∑ m = 1 j m H m , λ ( r ) t j − 1 n ∑ a = 1 n ∑ l = 0 n − a ( λ − r ) l l ! n + l − 1 l n − 1 a − 1 n − a l λ x a − 1 = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ x ∑ j = 0 ∞ ∑ m = 1 j + 1 m H m , λ ( r ) t j n x a − 1 = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ x ∑ j = 0 ∞ ∑ j 1 + ⋯ + j n = j ∏ i = 1 n ∑ m i = 1 j i + 1 m i H m i , λ ( r ) t j x a − 1 = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ x ∑ j = 0 a − 1 ∑ j 1 + ⋯ + j n = j ∏ i = 1 n ∑ m i = 1 j i + 1 m i H m i , λ ( r ) ( a − 1 ) j x a − 1 − j = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ∑ k = 1 a ∑ j 1 + ⋯ + j n = a − k ∏ i = 1 n ∑ m i = 1 j i + 1 m i H m i , λ ( r ) ( a − 1 ) a − k x k = ∑ k = 1 n ∑ a = k n ∑ l = 0 n − a ∑ j 1 + ⋯ + j n = a − k ∏ i = 1 n ∑ m i = 1 j i + 1 m i H m i , λ ( r ) l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ( a − 1 ) n − k x k .

Therefore, by Theorem 3 and (42), we obtain the following theorem.

Theorem 6

For n , r ≥ 1 , and 1 ≤ k ≤ n , we have

n ( r + 3 ) − k − 1 n − k n − 1 n − k = 1 ( n − k ) ! ∑ a = k n ∑ l = 0 n − a ∑ j 1 + ⋯ + j n = a − k ∑ m 1 = 1 j 1 + 1 ⋯ ∑ m n = 1 j n + 1 m 1 ⋯ m n H m 1 , λ ( r ) ⋯ H m n , λ ( r ) × l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ( a − 1 ) n − k .

We recall from (24) that:

t ( 1 − t ) r + 1 ( 1 + ( λ − r ) log λ ( 1 − t ) ) = ∑ n = 1 ∞ n + 1 n , λ ( r ) t n .

Let us consider the following associated sequence:

(43) Q n ( x ) ∼ ( 1 , t ( 1 − t ) r + 1 ) .

Thus, by (22) and (43), we obtain

(44) Q n ( x ) = x t t ( 1 − t ) r + 1 n x − 1 x n = x ( 1 − t ) − ( r + 1 ) n x n − 1 = x ∑ k = 0 n − 1 ( r + 1 ) n + k − 1 k t k x n − 1 = ∑ k = 0 n − 1 n ( r + 1 ) + k − 1 k ( n − 1 ) k x n − k = ∑ k = 1 n n ( r + 2 ) − k − 1 n − k ( n − 1 ) n − k x k .

Therefore, by (44), we obtain the following lemma.

Lemma 7

Let Q n ( x ) ∼ ( 1 , t ( 1 − t ) r + 1 ) . Then, we have

Q n ( x ) = ∑ k = 1 n n ( r + 2 ) − k − 1 n − k ( n − 1 ) n − k x k , ( n ≥ 1 ) .

Assume that

(45) R n ( x ) ∼ ( 1 , t ( 1 + ( λ − r ) log λ ( 1 − t ) ) ) .

By (22) and (45), for n ≥ 1 , we obtain

(46) R n ( x ) = x t t ( 1 + ( λ − r ) log λ ( 1 − t ) ) n x − 1 x n = x ∑ l = 0 ∞ n + l − 1 l ( r − λ ) l ( log λ ( 1 − t ) ) l x n − 1 = x ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ k = 0 n − 1 − l ( − 1 ) k + l S 1 , λ ( k + l , l ) t k + l ( k + l ) ! x n − 1 = x ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ k = 0 n − 1 − l ( − 1 ) k + l S 1 , λ ( k + l , l ) ( n − 1 ) k + l ( k + l ) ! x n − 1 − k − l = ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ k = 0 n − 1 − l ( − 1 ) k + l S 1 , λ ( k + l , l ) n − 1 k + l x n − k − l = ∑ l = 0 n − 1 n + l − 1 l ( r − λ ) l l ! ∑ k = 0 n − l ( − 1 ) n − k S 1 , λ ( n − k , l ) n − 1 n − k x k = ∑ k = 1 n ∑ l = 0 n − k n + l − 1 l ( r − λ ) l l ! ( − 1 ) n − k S 1 , λ ( n − k , l ) n − 1 k − 1 x k = ∑ k = 1 n ∑ l = 0 n − k n + l − 1 l n − 1 k − 1 ( λ − r ) l l ! n − k l λ x k .

Therefore, by (46), we obtain the following lemma.

Theorem 8

Let R n ( x ) ∼ ( 1 , t ( 1 + ( λ − r ) log λ ( 1 − t ) ) ) . Then, we have

R n ( x ) = ∑ k = 1 n ∑ l = 0 n − k n + l − 1 l n − 1 k − 1 n − k l λ ( λ − r ) l l ! x k , ( n ≥ 1 ) .

From (22), (23), (43), and (45), we note that

(47) Q n ( x ) = x t ( 1 + ( λ − r ) log λ ( 1 − t ) ) t ( 1 − t ) r + 1 n x − 1 R n ( x ) = x ∑ j = 1 ∞ j H j , λ ( r ) t j − 1 n x − 1 ∑ a = 1 n ∑ l = 0 n − a l ! n + l − 1 l n − 1 a − 1 n − a l λ ( λ − r ) l x a = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ x ∑ j = 0 ∞ ( j + 1 ) H j + 1 , λ ( r ) t j n x a − 1 = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ × x ∑ j = 0 ∞ ∑ j 1 + ⋯ + j n = j ( j 1 + 1 ) ⋯ ( j n + 1 ) H j 1 + 1 , λ ( r ) ⋯ H j n + 1 , λ ( r ) t j x a − 1

= ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ∑ j = 0 a − 1 × ∑ j 1 + ⋯ + j n = j ( j 1 + 1 ) ⋯ ( j n + 1 ) H j 1 + 1 , λ ( r ) ⋯ H j n + 1 , λ ( r ) ( a − 1 ) j x a − j = ∑ a = 1 n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ × ∑ k = 1 a ∑ j 1 + ⋯ + j n = a − k ( j 1 + 1 ) ⋯ ( j n + 1 ) H j 1 + 1 , λ ( r ) ⋯ H j n + 1 , λ ( r ) ( a − 1 ) a − k x k = ∑ k = 1 n ∑ a = k n ∑ l = 0 n − a l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ∑ j 1 + ⋯ + j n = a − k ( j 1 + 1 ) ⋯ ( j n + 1 ) H j 1 + 1 , λ ( r ) ⋯ H j n + 1 , λ ( r ) ( a − 1 ) a − k x k .

Therefore, by Lemma 6 and (47), we obtain the following theorem.

Theorem 9

For n , r ≥ 1 , and 1 ≤ k ≤ n , we have

n ( r + 2 ) − k − 1 n − k n − 1 n − k = 1 ( n − k ) ! ∑ a = k n ∑ l = 0 n − a ∑ j 1 + ⋯ + j n = a − k ( j 1 + 1 ) ⋯ ( j n + 1 ) H j 1 + 1 , λ ( r ) ⋯ H j n + 1 , λ ( r ) × l ! ( λ − r ) l n + l − 1 l n − 1 a − 1 n − a l λ ( a − 1 ) a − k .

4 Conclusion

Many different tools have been used in the explorations for degenerate versions of some special numbers and polynomials. In this article, we used umbral calculus in order to study the degenerate harmonic and degenerate hyperharmonic numbers. Some properties and identities relating to those numbers were derived from the transfer formula from umbral calculus.

We would like to continue to investigate various degenerate versions of certain special numbers and polynomials, including their applications to physics, science, and engineering as well as to mathematics.

Acknowledgments

We would like to thank the Jangjeon Institute for Mathematical Sciences for supporting this research.

  1. Funding information: This article was supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).

  2. Conflict of interest: The authors declare that there is no conflict of interest.

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Received: 2023-03-07
Accepted: 2023-08-27
Published Online: 2023-10-03

© 2023 the author(s), 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-2023-0124
Keywords for this article
degenerate harmonic number; degenerate hyperharmonic number; umbral calculus
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
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  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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