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Some identities related to degenerate Stirling numbers of the second kind

  • Taekyun Kim EMAIL logo , Dae San Kim und Hye Kyung Kim
Veröffentlicht/Copyright: 16. November 2022
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 55 Heft 1

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.

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

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

  3. Ethics approval and consent to participate: The authors declare that there is no ethical problem in the production of this article.

  4. Consent for publication: The authors want to publish this article in this journal.

  5. 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

© 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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  14. On a first-order differential system with initial and nonlocal boundary conditions
  15. On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints
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  17. Some properties of graded generalized 2-absorbing submodules
  18. Eigenvalue inclusion sets for linear response eigenvalue problems
  19. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation
  20. More properties of generalized open sets in generalized topological spaces
  21. An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
  22. An accurate and efficient local one-dimensional method for the 3D acoustic wave equation
  23. On a weighted elliptic equation of N-Kirchhoff type with double exponential growth
  24. On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces
  25. Entire and meromorphic solutions for systems of the differential difference equations
  26. Multiplication operators on the Banach algebra of bounded Φ-variation functions on compact subsets of ℂ
  27. Mannheim curves and their partner curves in Minkowski 3-space E13
  28. Characterizations of the group invertibility of a matrix revisited
  29. Iterates of q-Bernstein operators on triangular domain with all curved sides
  30. Data analysis-based time series forecast for managing household electricity consumption
  31. A robust study of the transmission dynamics of zoonotic infection through non-integer derivative
  32. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing
  33. Review Article
  34. Remarks on some variants of minimal point theorem and Ekeland variational principle with applications
  35. Special Issue on Recent Methods in Approximation Theory - Part I
  36. Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces
  37. Approximation of integrable functions by general linear matrix operators of their Fourier series
  38. Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation
  39. Approximation of conic sections by weighted Lupaş post-quantum Bézier curves
  40. On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation
  41. Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings
  42. Linear barycentric rational collocation method for solving biharmonic equation
  43. A note on the convergence of Phillips operators by the sequence of functions via q-calculus
  44. Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
  45. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part I
  46. Positive solutions for fractional differential equation at resonance under integral boundary conditions
  47. Source term model for elasticity system with nonlinear dissipative term in a thin domain
  48. A numerical study of anomalous electro-diffusion cells in cable sense with a non-singular kernel
  49. On Opial-type inequality for a generalized fractional integral operator
  50. Special Issue on Advances in Integral Transforms and Analysis of Differential Equations with Applications
  51. Mathematical analysis of a MERS-Cov coronavirus model
  52. Rapid exponential stabilization of nonlinear continuous systems via event-triggered impulsive control
  53. Novel soliton solutions for the fractional three-wave resonant interaction equations
  54. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach
  55. Special Issue on Problems, Methods and Applications of Nonlinear Analysis
  56. Some recent results on singular p-Laplacian equations
  57. Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity
  58. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems
  59. Existence of solutions for a nonlinear problem at resonance
  60. Asymptotic stability of solutions for a diffusive epidemic model
  61. Special Issue on Computational and Numerical Methods for Special Functions - Part I
  62. Fully degenerate Bernoulli numbers and polynomials
  63. Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform
  64. Some identities related to degenerate Stirling numbers of the second kind
  65. Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind
  66. λ-q-Sheffer sequence and its applications
  67. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part I
  68. General decay for a nonlinear pseudo-parabolic equation with viscoelastic term
  69. Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces
  70. Computation of solution of integral equations via fixed point results
  71. Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points
  72. Notes on continuity result for conformable diffusion equation on the sphere: The linear case
https://doi.org/10.1515/dema-2022-0170
Schlagwörter für diesen Artikel
degenerate Stirling numbers of the second kind; degenerate Fubini polynomials; degenerate Bell polynomials
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Artikel in diesem Heft

  1. Regular Articles
  2. On some summation formulas
  3. A study of a meromorphic perturbation of the sine family
  4. Asymptotic behavior of even-order noncanonical neutral differential equations
  5. Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation
  6. Starlike and convexity properties of q-Bessel-Struve functions
  7. Mathematical modeling and optimal control of the impact of rumors on the banking crisis
  8. On linear chaos in function spaces
  9. Convergence of generalized sampling series in weighted spaces
  10. Persistence landscapes of affine fractals
  11. Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces
  12. Various notions of module amenability on weighted semigroup algebras
  13. Regularity and normality in hereditary bi m-spaces
  14. On a first-order differential system with initial and nonlocal boundary conditions
  15. On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints
  16. Local linear approach: Conditional density estimate for functional and censored data
  17. Some properties of graded generalized 2-absorbing submodules
  18. Eigenvalue inclusion sets for linear response eigenvalue problems
  19. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation
  20. More properties of generalized open sets in generalized topological spaces
  21. An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
  22. An accurate and efficient local one-dimensional method for the 3D acoustic wave equation
  23. On a weighted elliptic equation of N-Kirchhoff type with double exponential growth
  24. On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces
  25. Entire and meromorphic solutions for systems of the differential difference equations
  26. Multiplication operators on the Banach algebra of bounded Φ-variation functions on compact subsets of ℂ
  27. Mannheim curves and their partner curves in Minkowski 3-space E13
  28. Characterizations of the group invertibility of a matrix revisited
  29. Iterates of q-Bernstein operators on triangular domain with all curved sides
  30. Data analysis-based time series forecast for managing household electricity consumption
  31. A robust study of the transmission dynamics of zoonotic infection through non-integer derivative
  32. A Dai-Liao-type projection method for monotone nonlinear equations and signal processing
  33. Review Article
  34. Remarks on some variants of minimal point theorem and Ekeland variational principle with applications
  35. Special Issue on Recent Methods in Approximation Theory - Part I
  36. Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces
  37. Approximation of integrable functions by general linear matrix operators of their Fourier series
  38. Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation
  39. Approximation of conic sections by weighted Lupaş post-quantum Bézier curves
  40. On the generalized growth and approximation of entire solutions of certain elliptic partial differential equation
  41. Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings
  42. Linear barycentric rational collocation method for solving biharmonic equation
  43. A note on the convergence of Phillips operators by the sequence of functions via q-calculus
  44. Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
  45. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part I
  46. Positive solutions for fractional differential equation at resonance under integral boundary conditions
  47. Source term model for elasticity system with nonlinear dissipative term in a thin domain
  48. A numerical study of anomalous electro-diffusion cells in cable sense with a non-singular kernel
  49. On Opial-type inequality for a generalized fractional integral operator
  50. Special Issue on Advances in Integral Transforms and Analysis of Differential Equations with Applications
  51. Mathematical analysis of a MERS-Cov coronavirus model
  52. Rapid exponential stabilization of nonlinear continuous systems via event-triggered impulsive control
  53. Novel soliton solutions for the fractional three-wave resonant interaction equations
  54. The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach
  55. Special Issue on Problems, Methods and Applications of Nonlinear Analysis
  56. Some recent results on singular p-Laplacian equations
  57. Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity
  58. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems
  59. Existence of solutions for a nonlinear problem at resonance
  60. Asymptotic stability of solutions for a diffusive epidemic model
  61. Special Issue on Computational and Numerical Methods for Special Functions - Part I
  62. Fully degenerate Bernoulli numbers and polynomials
  63. Wigner-Ville distribution and ambiguity function associated with the quaternion offset linear canonical transform
  64. Some identities related to degenerate Stirling numbers of the second kind
  65. Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind
  66. λ-q-Sheffer sequence and its applications
  67. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part I
  68. General decay for a nonlinear pseudo-parabolic equation with viscoelastic term
  69. Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces
  70. Computation of solution of integral equations via fixed point results
  71. Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points
  72. Notes on continuity result for conformable diffusion equation on the sphere: The linear case
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