Some identities related to degenerate Stirling numbers of the second kind

Taekyun Kim; Dae San Kim; Hye Kyung Kim

doi:10.1515/dema-2022-0170

Open Access Published by De Gruyter Open Access November 16, 2022

Some identities related to degenerate Stirling numbers of the second kind

Taekyun Kim , Dae San Kim and Hye Kyung Kim

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2022-0170

Abstract

The degenerate Stirling numbers of the second kind were introduced as a degenerate version of the ordinary Stirling numbers of the second kind. They appear very frequently when one studies various degenerate versions of some special numbers and polynomials. The aim of this article is to further study some identities and properties related to the degenerate Stirling numbers of the second kind, in connection with the degenerate Bell polynomials, the degenerate Fubini polynomials, the degenerate Bernoulli polynomials, and the degenerate Euler polynomials.

Keywords: degenerate Stirling numbers of the second kind; degenerate Fubini polynomials; degenerate Bell polynomials

MSC 2010: 11B68; 11B73; 11B83

1 Introduction and preliminaries

Carlitz [1,2] initiated study of the degenerate Bernoulli and Euler polynomials and numbers, which are degenerate versions of the ordinary Bernoulli and Euler polynomials and numbers. In recent years, studying degenerate versions of some special numbers and polynomials has regained the interests of some mathematicians. Some fascinating results were obtained not only in combinatorics and arithmetic but also in applications to differential equations, identities of symmetry, and probability theory (see [3,4,5, 6,7,8, 9,10,11, 12,13,14] and references therein). It is remarkable that studying degenerate versions is not only limited to polynomials but also extended to transcendental functions. Indeed, the degenerate gamma functions were introduced in connection with degenerate Laplace transforms (see [7]). It is also noteworthy that the degenerate umbral calculus is introduced as a degenerate version of the classical umbral calculus, where the central role of the Sheffer polynomials is replaced by the degenerate Sheffer polynomials (see [6]). Degenerate versions of special numbers and polynomials have been explored by various methods, including combinatorial methods, generating functions, umbral calculus techniques, p -adic analysis, differential equations, special functions, probability theory, and analytic number theory.

The Stirling number of the second S 2 ( n , k ) is the number of ways to partition a set of n objects into k nonempty subsets. The degenerate Stirling numbers of the second kind S 2 , λ ( n , k ) (see (5), (11)) are a degenerate version of the ordinary Stirling numbers of the second kind and appear very frequently when we study various degenerate versions of some special numbers and polynomials. The aim of this article is to further study some identities and properties related to the degenerate Stirling numbers of the second kind, in connection with the degenerate Bell polynomials, the degenerate Fubini polynomials, the degenerate Bernoulli polynomials, and the degenerate Euler polynomials.

The outline of this article is as follows. In Section 1, we recall the degenerate exponentials, the degenerate logarithms, the degenerate Stirling numbers of the first kind, and the degenerate Stirling numbers of the second kind. Also, we remind the reader of the degenerate Bernoulli polynomials, the degenerate Fubini polynomials, and the degenerate Bell polynomials. Section 2 outlines the main results of this article. In Theorem 2, we state an explicit value of a finite sum involving the degenerate falling factorials. In Theorem 3, we mention an identity involving the degenerate falling factorials and the degenerate Stirling numbers of the second kind. We express the degenerate Bell polynomials and the degenerate Fubini polynomials in terms of the differential operator x d d x m , λ , respectively, in Theorems 4 and 5. In Theorem 6, we express the degenerate Bernoulli numbers as finite sums involving the degenerate Stirling numbers of the second kind. The consecutive sum of the values of the degenerate falling factorials is expressed as a finite sum involving the degenerate Stirling numbers of the second kind in Theorem 8. In the rest of this section, we recall the facts that are needed throughout this article.

For any λ ∈ R , the degenerate exponentials are defined by

(1) e λ x ( t ) = ( 1 + λ t ) x λ = ∑ n = 0 ∞ ( x ) n , λ n ! t n ( see [5–14] ) ,

where the degenerate falling factorials are given by

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

In particular, for x = 1 , we use the notation

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

Let log λ t be the compositional inverse of e λ ( t ) satisfying log λ ( e λ ( t ) ) = e λ ( log λ ( t ) ) = t .

They are called the degenerate logarithms and are given by

(3) log λ ( 1 + t ) = ∑ n = 1 ∞ ( 1 ) n , 1 ∕ λ λ n − 1 n ! t n ( see [5] ) .

In [8], the degenerate Stirling numbers of the first kind are defined by

(4) ( x ) n = ∑ k = 0 n S 1 , λ ( n , k ) ( x ) k , λ , ( n ≥ 0 ) ,

where ( x ) 0 = 1 , ( x ) n = x ( x − 1 ) ⋯ ( x − n + 1 ) , ( n ≥ 1 ) (see [1,2,3, 4,5,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20]).

As the inversion formula of (4), the degenerate Stirling numbers of the second kind are defined as

(5) ( x ) n , λ = ∑ k = 0 n S 2 , λ ( n , k ) ( x ) k , ( n ≥ 0 ) ( see [5, 8–14] ) .

From (5), we note that

(6) S 2 , λ ( n + 1 , k ) = S 2 , λ ( n , k − 1 ) + ( k − n λ ) S 2 , λ ( n , k ) ,

where n , k ∈ Z with n ≥ k ≥ 0 (see [5]).

Carlitz considered the degenerate Bernoulli polynomials given by

(7) t e λ ( t ) − 1 e λ x ( t ) = ∑ n = 0 ∞ β n , λ ( x ) t n n ! ( see [1,2] ) .

When x = 0 , β n , λ = β n , λ ( 0 ) are called the degenerate Bernoulli numbers.

By (7), we easily obtain

(8) β n , λ ( x ) = ∑ l = 0 n n l ( x ) n − l , λ β l , λ , ( n ≥ 0 ) ( see [1] ) .

Recently, Kim-Kim introduced the degenerate Fubini polynomials given by

(9) 1 1 − x ( e λ ( t ) − 1 ) = ∑ n = 0 ∞ F n , λ ( x ) t n n ! ( see [12] ) .

Thus, by (9), we obtain

(10) F n , λ ( x ) = ∑ k = 0 n S 2 , λ ( n , k ) k ! x k ( see [12] ) .

From (5) and (6), we note that

(11) 1 k ! ( e λ ( t ) − 1 ) k = ∑ n = k ∞ S 2 , λ ( n , k ) t n n ! , ( k ≥ 0 ) , and 1 k ! ( log λ ( 1 + t ) ) k = ∑ n = k ∞ S 1 , λ ( n , k ) t n n ! ( see [5] ) .

By (11), we obtain

(12) ∑ l = 0 k k l ( − 1 ) l ( l ) n , λ = ( − 1 ) k k ! S 2 , λ ( n , k ) , ( n ≥ k ≥ 0 ) .

Recently, the degenerate Bell polynomials are considered by Kim et al., which are given by

(13) e x ( e λ ( t ) − 1 ) = ∑ n = 0 ∞ ϕ n , λ ( x ) t n n ! ( see [9,10,13] ) .

From (13), we note that

(14) ϕ n , λ ( x ) = ∑ k = 0 n S 2 , λ ( n , k ) x k = e − x ∑ k = 0 ∞ ( k ) n , λ k ! x k , ( n ≥ 0 ) ( see [13] ) .

2 Some identities related to degenerate Stirling numbers of the second kind

From (6), we note that

(15) S 2 , λ ( n + 1 , n ) = S 2 , λ ( n , n − 1 ) + ( n − n λ ) S 2 , λ ( n , n ) = n ( 1 − λ ) + S 2 , λ ( n , n − 1 ) = n ( 1 − λ ) + ( n − 1 ) ( 1 − λ ) + S 2 , λ ( n − 1 , n − 2 ) ,

where n is a nonnegative integer.

Continuing this process, we have

(16) S 2 , λ ( n + 1 , n ) = n ( 1 − λ ) + ( n − 1 ) ( 1 − λ ) + ⋯ + 1 ( 1 − λ ) + S 2 , λ ( 1 , 0 ) = ( 1 − λ ) ( 1 + 2 + ⋯ + n ) = ( 1 − λ ) n + 1 2 .

Therefore, by (16), we obtain the following lemma.

Lemma 1

For n ≥ 0 , we have

S 2 , λ ( n + 1 , n ) = ( 1 − λ ) n + 1 2 .

From (12), we note that

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

Theorem 2

For n ≥ 0 , we have

∑ l = 0 n n l ( − 1 ) l ( l ) n + 1 , λ = ( n + 1 ) ! n 2 ( − 1 ) n ( 1 − λ ) .

Now, we observe that

(18) ∑ n = 0 ∞ ( x + y ) n , λ t n n ! = ( 1 + λ t ) x + y λ = ( 1 + λ ) x λ ( 1 + λ t ) y λ = ∑ n = 0 ∞ ∑ l = 0 n n l ( x ) l , λ ( y ) n − l , λ t n n ! .

By comparing the coefficients on both sides of (18), we obtain

(19) ( x + y ) n , λ = ∑ l = 0 n n l ( x ) l , λ ( y ) n − l , λ , ( n ≥ 0 ) .

From (12) and (19), we note that

(20) ∑ k = 0 n n k ( − 1 ) k ( x k + y ) m , λ = ∑ k = 0 n n k ( − 1 ) k ∑ j = 0 m m j ( x k ) j , λ ( y ) m − j , λ = ∑ k = 0 n n k ( − 1 ) k ∑ j = 0 m m j x j ( k ) j , λ ∕ x ( y ) m − j , λ = ∑ j = 0 m m j x j ( y ) m − j , λ ∑ k = 0 n n k ( − 1 ) k ( k ) j , λ ∕ x = ∑ j = n m m j x j ( y ) m − j , λ n ! ( − 1 ) n S 2 , λ x ( j , n ) .

By (20), we obtain the following theorem.

Theorem 3

For m , n with m ≥ n ≥ 0 , we have

∑ k = 0 n n k ( − 1 ) k ( x k + y ) m , λ = ∑ j = n m m j x j ( y ) m − j , λ n ! ( − 1 ) n S 2 , λ ∕ x ( j , n ) .

The degenerate formal power series is given by

(21) f λ ( t ) = a 0 + a 1 ( t ) 1 , λ + a 2 ( t ) 2 , λ + a 3 ( t ) 3 , λ + ⋯ = ∑ k = 0 ∞ a k ( t ) k , λ ∈ C [ [ t ] ] .

Thus, by (12) and (21), we obtain

(22) ∑ k = 0 n n k ( − 1 ) k f λ ( k ) = ∑ k = 0 n n k ( − 1 ) k ∑ j = 0 ∞ a j ( k ) j , λ = ∑ j = 0 ∞ a j ∑ k = 0 ∞ n k ( − 1 ) k ( k ) j , λ = ∑ j = 0 ∞ a j n ! ( − 1 ) n S 2 , λ ( j , n ) = ( − 1 ) n n ! ∑ j = n ∞ a j S 2 , λ ( j , n ) .

Let Δ be the difference operator with Δ f ( x ) = f ( x + 1 ) − f ( x ) . Then, we have

(23) Δ n f ( x ) = ∑ k = 0 n n k ( − 1 ) n − k f ( x + k ) , ( n ≥ 0 ) .

Also, we observe that

(24) ( 1 + Δ ) n f ( 0 ) = f ( x ) , ( n ≥ 0 ) .

Assume that f ( x ) is a continuous function. Then, we have

(25) f ( x ) = lim n → x f ( n ) = ( 1 + Δ ) x f ( 0 ) = ∑ k = 0 ∞ x k Δ k f ( 0 ) .

This is called the Newton’s interpolation of f ( x ) . Let us take f ( x ) = ( x ) m , λ , ( m ≥ 0 ) .

Then, by (23) and (25), we obtain

(26) ∑ k = 0 m S 2 , λ ( m , k ) ( x ) k = ( x ) m , λ = ∑ k = 0 ∞ 1 k ! ∑ j = 0 k k j ( − 1 ) k − j ( j ) m , λ ( x ) k .

By comparing the coefficients on both sides of (26), we have

1 k ! ∑ j = 0 k k j ( − 1 ) k − j ( j ) m , λ = S 2 , λ ( m , k ) , if k ≤ m , 0 , if k > m .

In particular, for k = m , we have

1 k ! ∑ j = 0 k k j ( − 1 ) k − j ( j ) k , λ = 1 , ( k ≥ 0 ) .

We note that

(27) Δ x n = x + 1 n − x n = x n − 1 , ( n ≥ 1 ) .

From (27), we note that

(28) n + 1 k + 1 = ∑ j = k n j k , ( n , k ≥ 0 ) .

Recently, Kim-Kim introduced the degenerate differential operator given by

(29) x d d x k , λ = x d d x x d d x − λ ⋯ x d d x − ( k − 1 ) λ ( see [8] ) .

From (14) and (29), we note that

(30) x d d x m , λ e x = x d d x m , λ ∑ k = 0 ∞ x k k ! = ∑ k = 0 ∞ ( k ) m , λ k ! x k = e x e − x ∑ k = 0 ∞ ( k ) m , λ k ! x k = e x ϕ m , λ ( x ) ,

where m is a positive integer.

Therefore, by (30), we obtain the following theorem.

Theorem 4

For m ∈ N ∪ { 0 } , we have

e − x x d d x m , λ e x = ϕ m , λ ( x ) .

Now, we observe that

(31) x d d x m , λ 1 1 − x = x d d x x d d x − λ ⋯ x d d x − ( m − 1 ) λ ∑ n = 0 ∞ x n = ∑ n = 0 ∞ ( n ) m , λ x n .

It is easy to show that

(32) 1 1 − x = ∫ 0 ∞ e − t ( 1 − x ) d t = ∫ 0 ∞ e − t e x t d t ,

and

(33) x d d x m , λ e x t = ∑ k = 0 ∞ t k k ! x d d x m , λ x k = ∑ k = 0 ∞ ( k ) m , λ k ! ( t x ) k = e x t e − x t ∑ k = 0 ∞ ( k ) m , λ k ! ( t x ) k = e x t ϕ m , λ ( x t ) = e x t ∑ k = 0 m S 2 , λ ( m , k ) x k t k .

Thus, by (32) and (33), we obtain

(34) x d d x m , λ 1 1 − x = ∫ 0 ∞ e − t x d d x m , λ e x t d t = ∑ k = 0 m S 2 , λ ( m , k ) x k ∫ 0 ∞ e − t ( 1 − x ) t k d t = 1 1 − x ∑ k = 0 m S 2 , λ ( m , k ) x 1 − x k ∫ 0 ∞ e − y y k d y = 1 1 − x ∑ k = 0 m S 2 , λ ( m , k ) x 1 − x k k ! .

From (10), (31), and (34), we obtain the following theorem.

Theorem 5

For m ≥ 0 , we have

x d d x m , λ 1 1 − x = 1 1 − x F m , λ x 1 − x .

In particular,

1 1 − x F m , λ x 1 − x = ∑ n = 0 ∞ ( n ) m , λ x n .

By (7), we obtain

(35) ∑ n = 0 ∞ β n , λ t n n ! = t e λ ( t ) − 1 = log λ ( 1 + e λ ( t ) − 1 ) e λ ( t ) − 1 = 1 e λ ( t ) − 1 ∑ k = 1 ∞ λ k − 1 ( 1 ) k , 1 ∕ λ k ! ( e λ ( t ) − 1 ) k = ∑ k = 0 ∞ λ k ( 1 ) k + 1 , 1 ∕ λ ( k + 1 ) 1 k ! ( e λ ( t ) − 1 ) k = ∑ n = 0 ∞ ∑ k = 0 n λ k ( 1 ) k + 1 , 1 ∕ λ k + 1 S 2 , λ ( n , k ) t n n ! .

By comparing the coefficients on both sides of (35), we obtain the following theorem.

Theorem 6

For n ≥ 0 , we have

β n , λ = ∑ k = 0 n λ k ( 1 ) k + 1 , 1 ∕ λ k + 1 S 2 , λ ( n , k ) .

We observe that

(36) ∑ k = 0 n − 1 e λ k ( t ) = 1 e λ ( t ) − 1 ( e λ n ( t ) − 1 ) = 1 t t e λ ( t ) − 1 ( e λ n ( t ) − 1 ) = 1 t ∑ m = 0 ∞ ( β m , λ ( n ) − β m , λ ) t m m ! = ∑ m = 0 ∞ β m + 1 , λ ( n ) − β m + 1 , λ m + 1 t m m ! .

On the other hand, by (1), we obtain

(37) ∑ k = 0 n − 1 e λ k ( t ) = ∑ m = 0 ∞ ∑ k = 0 n − 1 ( k ) m , λ t m m ! .

From (8), (36), and (37), we have

(38) ∑ k = 0 n − 1 ( k ) m , λ = ( 0 ) m , λ + ( 1 ) m , λ + ( 2 ) m , λ + ⋯ + ( n − 1 ) m , λ = 1 m + 1 ( β m + 1 , λ ( n ) − β m + 1 , λ ) = 1 m + 1 ∑ l = 0 m + 1 m + 1 l ( n ) m + 1 − l , λ β l , λ − β m + 1 , λ = 1 m + 1 ∑ l = 0 m m + 1 l β l , λ ( n ) m + 1 − l , λ ,

where m is a nonnegative integer.

Therefore, by (38), we obtain the following lemma.

Lemma 7

For n ∈ N and m ≥ 0 , we have

∑ k = 0 n − 1 ( k ) m , λ = 1 m + 1 ∑ l = 0 m m + 1 l ( n ) m + 1 − l , λ β l , λ .

Carlitz introduced the degenerate Euler polynomials given by

(39) 2 e λ ( t ) + 1 e λ x ( t ) = ∑ n = 0 ∞ E n , λ ( x ) t n n ! .

When x = 0 , E n , λ = E n , λ ( 0 ) are called the degenerate Euler numbers.

For n ∈ N with n ≡ 1 (mod 2), we have

(40) 2 ∑ k = 0 n − 1 ( − 1 ) k e λ k ( t ) = 2 e λ ( t ) + 1 ( 1 + e λ n ( t ) ) = ∑ l = 0 ∞ ( E l , λ + E l , λ ( n ) ) t l l ! .

On the other hand, by (1), we obtain

(41) 2 ∑ k = 0 n − 1 ( − 1 ) k e λ k ( t ) = ∑ l = 0 ∞ 2 ∑ k = 0 n − 1 ( − 1 ) k ( k ) l , λ t l l ! .

From (40) and (41), we have

(42) 2 ∑ k = 0 n − 1 ( − 1 ) k ( k ) l , λ = E l , λ + E l , λ ( n ) ,

where n ∈ N with n ≡ 1 (mod 2) and l ≥ 0 .

For m ∈ N , we have

(43) ( 1 ) m , λ + ( 2 ) m , λ + ⋯ + ( n ) m , λ = ∑ k = 1 n ( k ) m , λ = ∑ k = 0 n ∑ l = 0 m S 2 , λ ( m , l ) ( k ) l = ∑ k = 0 n ∑ l = 0 m S 2 , λ ( m , l ) k l l ! = ∑ l = 0 m S 2 , λ ( m , l ) l ! ∑ k = 0 n k l = ∑ l = 0 m S 2 , λ ( m , l ) l ! ∑ k = l n k l .

From (30) and (43), we have

(44) ( 1 ) m , λ + ( 2 ) m , λ + ⋯ + ( n ) m , λ = ∑ l = 0 m S 2 , λ ( m , l ) l ! ∑ k = l n k l = ∑ l = 0 m S 2 , λ ( m , l ) l ! n + 1 l + 1 .

By (44), we obtain the following theorem.

Theorem 8

For m , n ∈ N , we have

( 1 ) m , λ + ( 2 ) m , λ + ⋯ + ( n ) m , λ = ∑ l = 0 m S 2 , λ ( m , l ) l ! n + 1 l + 1 .

In particular, from Lemma 7, we obtain the following corollary.

Corollary 9

For m , n ∈ N , we have

1 m + 1 ∑ l = 0 m m + 1 l ( n ) m + 1 − l , λ β l , λ = ∑ l = 0 m n l + 1 S 2 , λ ( m , l ) l !

Equivalently, we have

1 m + 1 ( β m + 1 , λ ( n ) − β m + 1 , λ ) = ∑ l = 1 m + 1 n l S 2 , λ ( m , l − 1 ) ( l − 1 ) !

Remark

Recently, several authors have studied the special numbers and polynomials related to the Stirling numbers of the first and second kind [19, 20,21, 22,23, 24,25].

3 Conclusion

Studies on degenerate versions of some special polynomials and numbers began with Carlitz’s article [1], where he studied the degenerate Bernoulli and Euler polynomials. With regained interest, intensive studies have been conducted for degenerate versions of quite a few special polynomials and numbers by employing tools such as combinatorial methods, generating functions, umbral calculus techniques, p -adic analysis, differential equations, special functions, probability theory, and analytic number theory.

In this article, we further studied some identities and properties related to the degenerate Stirling numbers of the second kind in connection with the degenerate Bell polynomials, the degenerate Fubini polynomials, the degenerate Bernoulli polynomials, and the degenerate Euler polynomials.

It is one of our future projects to continue to explore various degenerate versions of many special polynomials and numbers by using aforementioned tools.

Acknowledgments

The author would like to thank the referees for the detailed and valuable comments that helped improve the original manuscript in its present form. The authors also thank the Jangjeon Institute for Mathematical Science for the support of this research.

Funding information: This work was supported by the Basic Science Research Program through the the National Research Foundation of Korea (NRF-2021R1F1A1050151).
Conflict of interest: The authors declare no conflict of interest.
Ethics approval and consent to participate: The authors declare that there is no ethical problem in the production of this article.
Consent for publication: The authors want to publish this article in this journal.
Data availability statement: Not applicable.

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Received: 2022-06-17

Revised: 2022-09-17

Accepted: 2022-10-04

Published Online: 2022-11-16

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

Some identities related to degenerate Stirling numbers of the second kind

Abstract

1 Introduction and preliminaries

2 Some identities related to degenerate Stirling numbers of the second kind

Lemma 1

Theorem 2

Theorem 3

Theorem 4

Theorem 5

Theorem 6

Lemma 7

Theorem 8

Corollary 9

Remark

3 Conclusion

Acknowledgments

References

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