EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 3, 2022, 1054-1066 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some identities on λ -analogues of r-Stirling numbers of the second kind Dae San Kim1, Hye Kyung Kim2*, Taekyun Kim3 1 Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea 2 Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea 3 Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Abstract. Recently, the λ -analogues of r-Stirling numbers of the first kind were studied by Kim-Kim. The aim of this paper is to introduce the λ -analogues of r-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the λ -analogues of Whitney-type r-Stirling numbers of the second and derive similar results to the case of the λ -analogues of r-Stirling numbers of the second kind. In addition, we consider the λ -analogues of Dowling polynomials and deduce a Dobinski-like formula. 2020 Mathematics Subject Classifications: 11B73, 11B83 Key Words and Phrases: λ -anlogues of r-Stirling numbers of the second, λ -analogues of Whitney-type r-Stirling numbers of the second, λ -analogues of Dowling polynomials 1. Introduction Carlitz [3] initiated a study of the degenerate Bernoulli and Euler polynomials and numbers, which are degenerate versions of the Bernoulli and Euler polynomials and numbers. In recent years, studying degenerate versions of special numbers and polynomials regained interests of some mathematicians. They have been explored with various tools and many fascinating results have been revealed. It is remakable that this quest for degenerate versions is not just restricted to polynomials but also extended to transcendental functions, like gamma functions. In addition, it also led to the introduction of λ -umbral calculus and λ -Sheffer sequences. The degenerate Stirling numbers of the first kind and of the second kind, which are degenerate versions of the Stirling numbers of the first kind and of the second kind, appear frequently when we study degenerate versions of some special numbers and polynomials. They arise naturally when we replace the powers of x by the generalized falling factorial polynomials (x)k,λ in the ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4441 Email addresses: dskim@sogang.ac.kr (D. S. Kim), hkkim@cu.ac.kr (H. K. Kim), tkkim@kw.ac.kr (T. Kim) https://www.ejpam.com 1054 © 2022 EJPAM All rights reserved. D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1055 defining equations of the Stirling numbers of both kinds (see (1)), while the λ -analogues of Stirling numbers of the first kind and of the second kind appear when we replace the falling factorials by the generalized falling factorials. The aim of this paper is to introduce the λ -analogues of r-Stirling numbers of the second kind and to investigate some properties, recurrence relations and certain identities on those numbers. We also introduce the λ -analogues of Whitney-type r-Stirling numbers of the second and derive similar results to the case of the λ -analogues of r-Stirling numbers of the second kind. In ad- dition, we consider the λ -analogues of Dowling polynomials, which are a natural extension of the analogues of Whitney-type Stirling numbers of the second kind, and deduce a Dobinski-like formula. The outline of this paper is as follows. In Section 1, we recall the generalized falling fac- torial sequence, the degenerate exponential functions, the Stirling numbers of both kinds, the λ -analogues of r-Stirling numbers of the first kind and the λ -analogues of unsigned r-Stirling numbers of the first kind. In Section 2, we introduce the λ -analogues of r-Stirling numbers of the second as the coefficients appearing when powers of x+ r are expressed in terms of the degen- erate falling factorial sequence. In the special case of r = 0, we get the λ -analogues of Stirling numbers of the second kind. In Theorem 1, we obtain the generating function of the λ -analogues of r-Stirling numbers of the second. We express those numbers in terms of the forward difference operator in Theorem 2. In Theorems 3 and 6, we find an expression of the analogues of r-Stirling numbers of the second kind in terms of the analogues of Stirling numbers of the second kind and vice versa. In Theorems 4 and 5, we get recurrence relations for the analogues of r-Stirling numbers of the second kind. In Theorem 7, we derive an identity connecting the analogues of r-Stirling numbers of the second kind and some values of the higher order Bernoulli polynomials. In Section 3, introduced are the λ -analogues of Whitney-type r-Stirling numbers of the second. In case of r = 1, we get the λ -analogues of Whitney-type Stirling numbers of the second. The gener- ating function of those numbers are derived in Theorem 12. In Theorem 13, we obtain an identity relating the λ -analogues of Whitney-type r-Stirling numbers of the second, the λ -analogues of Whitney-type Stirling numbers of the second and some values of higher order Bernoulli numbers. We introduce the λ -analogues of Dowling polynomials, and deduce the generating function and Dobinski-like formula in Theorems 10 and 11, respectively. In the rest of this section, we recall the facts that are needed in this paper. Throughout this paper, let λ be any nonzero real number. The generalized falling factorial sequence is defined by (x)0,λ = 1, (x)n,λ = x(x−λ ) · · ·(x− (n−1)λ ), (n ≥ 1), (see [2], [5], [7], [8], [11], [10]). (1) It is known that the degenerate exponential functions are defined by ex λ (t) = (1+λ t) x λ = ∞ ∑ n=0 (x)n,λ tn n! , (see [3], [6], [9]). (2) When x = 1, we use the notation as eλ (t) = e1 λ (t). For n ≥ 0, the Stirling numbers of the first kind are defined by (x)n = n ∑ k=0 S1(n,k)xk, (see [1], [2], [4], [6], [12], [14]), (3) D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1056 where the falling factorial sequence is given by (x)0 = 1, (x)n = x(x−1) · · ·(x−n+1), (n ≥ 1). As the inversion formula of (3), the Stirling numbers of the second kind are defined as xn = n ∑ k=0 { n k } (x)k, (see [4], [8], [11], [12], [14], [13]). (4) The λ -analogues of Stirling numbers of the first kind are defined by (x)n,λ = n ∑ k=0 S1,λ (n,k)x k, (n ≥ 0), (see [8]). (5) For r ∈ N∪{0}, the λ -analogues of r-Stirling numbers of the first kind are defined by (x+ r)n,λ = n ∑ k=0 S(r)1,λ (n,k)x k, (see [8]). (6) Also, the λ -analogues of unsigned r-Stirling numbers of the first kind are given by ⟨x+ r⟩n,λ = n ∑ k=0 [ n+ r k+ r ] r,λ xk, (n ≥ 0), (see [8]). (7) Note that lim λ→1 [ n+ r k+ r ] r,λ = [ n+ r k+ r ] r are the ordinary unsigned r-Stirling numbers of the first kind. 2. λ -analogues of r-Stirling numbers of the second kind First, we consider the λ -analogues of Stirling numbers of the second kind as the inversion formula of (5), which are defined by xn = n ∑ k=0 { n k } λ (x)k,λ , (n ≥ 0). (8) From (8), we note that ext = ∞ ∑ n=0 tn n! xn = ∞ ∑ n=0 tn n! n ∑ k=0 { n k } λ (x)k,λ = ∞ ∑ k=0 ( ∞ ∑ n=k { n k } λ tn n! ) (x)k,λ . (9) On the other hand, by (1), we get ext = ( eλ t −1+1 ) x λ = ∞ ∑ k=0 ( x λ k ) (eλ t −1)k (10) D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1057 = ∞ ∑ k=0 λ −k 1 k! ( eλ t −1 )k (x)k,λ . From (9) and (10), we note that 1 λ k 1 k! ( eλ t −1 )k = ∞ ∑ n=k { n k } λ tn n! . (11) Note that lim λ→0 { n k } λ = { n k } are the Stirling numbers of the second kind which are defined by xn = n ∑ k=0 { n k } (x)k, (n ≥ 0). For r ∈ N∪ {0}, we consider the λ -analogues of r-Stirling numbers of the second kind as the inversion formula of (6) which are defined by (x+ r)n = n ∑ k=0 { n+ r k+ r } r,λ (x)k,λ , (n ≥ 0). (12) From (12), we note that e(x+r)t = ∞ ∑ n=0 (x+ r)n tn n! = ∞ ∑ n=0 ( n ∑ k=0 { n+ r k+ r } r,λ (x)k,λ ) tn n! (13) = ∞ ∑ k=0 ( ∞ ∑ n=k { n+ r k+ r } r,λ tn n! ) (x)k,λ . On the other hand, by (1), we get e(x+r)t = ertext = ert(eλ t −1+1) x λ (14) = ert ∞ ∑ k=0 ( x λ k ) (eλ t −1)k = ∞ ∑ k=0 1 k (eλ t −1)kert 1 λ k (x)k,λ . From (13) and (14), we note that 1 λ k 1 k! ( eλ t −1 )kert = ∞ ∑ n=k { n+ r k+ r } r,λ tn n! . (15) Theorem 1. The generating function of the λ -analogues of r-Stirling numbers of the second kind is given by 1 λ k 1 k! ( eλ t −1 )kert = ∞ ∑ n=k { n+ r k+ r } r,λ tn n! , (k ≥ 0). Note that lim λ→1 { n+ r k+ r } r,λ = { n+ r k+ r } r are the ordinary r-Stirling numbers of the second kind which are defined by (x+ r)n = n ∑ k=0 { n+ r k+ r } r (x)k, (n ≥ 0), (see [11]]). D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1058 Let △ be the difference operator with △ f (x) = f (x+1)− f (x). Then we have △k f (x) = k ∑ l=0 ( k l ) (−1)k−l f (x+ l). (16) From Theorem 1, we note that ∞ ∑ n=k { n+ r k+ r } r,λ tn n! = 1 λ k 1 k! ( eλ t −1 )kert (17) = 1 λ k 1 k! k ∑ l=0 ( k l ) (−1)k−le(lλ+r)t = ∞ ∑ n=0 ( 1 λ k 1 k! k ∑ l=0 ( k l ) (−1)k−l(lλ + r)n ) tn n! . Let us take f (x) = ( x λ )n in (8). Then we have △k ( r λ )n = k ∑ l=0 ( k l ) (−1)k−l ( l + r λ )n (18) = k ∑ l=0 ( k l ) (−1)k−l(λ l + r)n λ −n. By (17) and (18), we get ∞ ∑ n=k { n+ r k+ r } r,λ tn n! = ∞ ∑ n=0 λ n−k 1 k! △k ( r λ )n tn n! . (19) Theorem 2. For k ≥ 0, we have λ n−k 1 k! △k ( r λ )n = { {n+r k+r } r,λ , if n ≥ k, 0, if 0 ≤ n < k. By Theorem 1, we get ∞ ∑ n=k { n+ r k+ r } r,λ tn n! = 1 λ k 1 k! ( eλ t −1 )kert (20) = ∞ ∑ l=k { l k } λ t l l! ∞ ∑ m=0 rm tm m! = ∞ ∑ n=k ( n ∑ l=k { l k } λ rn−l ( n l )) tn n! . Therefore, by comparing the coefficients on both sides of (20), we obtain the following theorem. D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1059 Theorem 3. For n,k ∈ Z with n ≥ k ≥ 0, we have{ n+ r k+ r } r,λ = n ∑ l=k ( n l ){ l k } λ rn−l. For n ≥ 1, we have (x+ r)n+1 = (x+ r)n(x+ r) = n ∑ k=0 { n+ r k+ r } r,λ (x+ r)(x)k,λ = n ∑ k=0 { n+ r k+ r } r,λ (x− kλ + kλ + r)(x)k,λ (21) = n ∑ k=0 { n+ r k+ r } r,λ (x)k+1,λ + n ∑ k=0 { n+ r k+ r } r,λ (x)k,λ (λk+ r) = n+1 ∑ k=1 { n+ r k−1+ r } r,λ (x)k,λ + n ∑ k=0 (λk+ r) { n+ r k+ r } r,λ (x)k,λ = n+1 ∑ k=0 ({ n+ r k−1+ r } r,λ +(λk+ r) { n+ r k+ r } r,λ ) (x)k,λ . On the other hand, by (12), we get (x+ r)n+1 = n+1 ∑ k=0 { n+1+ r k+ r } r,λ (x)k,λ . (22) Therefore, by (21) and (22), we obtain the following theorem. Theorem 4. For n,k ∈ Z with n ≥ k ≥ 1, we have{ n+1+ r k+ r } r,λ = { n+ r k−1+ r } r,λ +(λk+ r) { n+ r k+ r } r,λ . Now, we observe that 1 λ k 1 k! ( eλ t −1 )k 1 λ m 1 m! ( eλ t −1 )mert (23) = 1 λ k+m 1 (k+m)! (eλ t −1)k+mert (k+m)! k!m! = ( k+m k ) ∞ ∑ n=m+k { n+ r k+m+ r } r,λ tn n! . On the other hand, by (15), we get 1 λ k 1 k! ( eλ t −1 )k 1 λ m 1 m! (eλ t −1)mert (24) D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1060 = ∞ ∑ l=k { l k } λ t l l! ∞ ∑ j=m { j+ r m+ r } r,λ t j j! = ∞ ∑ n=m+k n−m ∑ l=k { l k } λ { n− l + r m+ r } r,λ ( n l ) tn n! . Therefore, by (23) and (24), we obtain the following theorem. Theorem 5. For m,n,k ≥ 0 with n ≥ m+ k, we have( m+ k k ){ n+ r k+m+ r } r,λ = n−m ∑ l=k ( n l ){ l k } λ { n− l + r m+ r } r,λ . From the definition of the λ -analogues of the Stirling numbers of the second kind, we have ∞ ∑ n=k { n k } λ tn n! = 1 λ k 1 k! ( et −1 )k = 1 λ k 1 k! ( eλ t −1 )kerte−rt (25) = ∞ ∑ l=k { l + r k+ r } r,λ t l l! ∞ ∑ m=0 (−r)m tm m! = ∞ ∑ n=k ( n ∑ l=k ( n l ){ l + r k+ r } r,λ (−1)n−lrn−l ) tn n! . Therefore, by comparing the coefficients on both sides of (25) we obtain the following theorem. Theorem 6. For n,k ≥ 0 with n ≥ k, we have{ n k } λ = n ∑ l=k ( n l ){ l + r k+ r } r,λ (−1)n−lrn−l. For m ∈ N, the higher-order Bernoulli polynomials are defined by( t et −1 )m ext = ∞ ∑ n=0 B(m) n (x) tn n! , (see [1,3,7]). (26) From (26), we note that ∞ ∑ n=k { n+ r k+ r } r,λ tn n! = 1 λ k 1 k! ( eλ t −1 )kert (27) = 1 tm 1 λ k+m (k+m)! k! 1 (k+m)! (eλ t −1)k+m ( λ t eλ t −1 )m ert = ( k+m k ) m! tm ∞ ∑ l=k+m { l k+m } λ t l l! ( ∞ ∑ j=0 B(m) j r λ ) λ jt j j! = ( k+m k ) ∞ ∑ l=k { l +m k+m } λ m!l! (l +m)! t l l! ∞ ∑ j=0 B(m) j ( r λ ) λ jt j j! D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1061 = ( k+m k ) ∞ ∑ n=k ( n ∑ l=k (n l )(l+m l ){l +m k+m } λ λ n−lB(m) n−l ( r λ )) tn n! . Therefore, by comparing the coefficients on both sides of (27), we obtain the following theorem. Theorem 7. For n,k ≥ 0 with n ≥ k, we have{n+r k+r } r,λ(k+m k ) = n ∑ l=k (n l )(l+m l ){l +m k+m } λ B(m) n−l ( r λ ) λ n−l. 3. Further Remarks For m,n ≥ 0, we define λ -analogues of the Whitney-type Stirling numbers of the second kind as (mx+1)n = n ∑ k=0 Wm,λ (n,k)m k(x)k,λ , (n ≥ 0). (28) From (28), we note that e(mx+1)t = ∞ ∑ m=0 (mx+1)n tn n! = ∞ ∑ n=0 n ∑ k=0 Wm,λ (n,k)m k(x)k,λ tn n! = ∞ ∑ k=0 ∞ ∑ n=k Wm,λ (n,k) tn n! mk(x)k,λ . (29) On the other hand, by (11), we get e(mx+1)t = et(eλmt −1+1 ) x λ = et ∞ ∑ k=0 ( x λ k ) (eλmt −1)k = ∞ ∑ k=0 ( 1 λ k 1 k! ( eλmt −1 m )k et ) mk(x)k,λ . (30) Therefore, by (29) and (30), we obtain the generating function Wm,λ (n,k), (n,k ≥ 0). Theorem 8. For k ≥ 0, we have 1 k! 1 λ k ( eλmt −1 m )k et = ∞ ∑ n=k Wm,λ (n,k) tn n! . From Theorem 8, we note that ∞ ∑ n=k { n+1 k+1 } λ tn n! = d dt { ∞ ∑ n=k { n+1 k+1 } λ tn+1 (n+1)! } (31) D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1062 = d dt ( 1 (k+1)! 1 λ k+1 (e λ t −1)k+1 ) = 1 k! 1 λ k (e λ t −1)kete(λ−1)t = ∞ ∑ l=k W1,λ (l,k) t l l! ∞ ∑ j=0 (λ −1) j t j j! = ∞ ∑ n=k ( n ∑ l=k W1,λ (l,k)(λ −1)n−l ( n l )) tn n! . Therefore, by comparing the coefficients on both sides of (31), we obtain the following theorem. Theorem 9. For n,k ≥ 0 with n ≥ k, we have n ∑ l=k ( n l ) W1,λ (l,k)(λ −1)n−l = { n+1 k+1 } λ . Now, we consider the λ -analogues of Dowling polynomials which are defined by dm,λ (n,x) = n ∑ k=0 Wm,λ (n,k)x k, (n ≥ 0). (32) Thus, by (32), we get ∞ ∑ n=0 dm,λ (n,x) tn n! = ∞ ∑ n=0 n ∑ k=0 Wm,λ (n,k)x k tn n! (33) = ∞ ∑ k=0 xk ∞ ∑ n=k Wm,λ (n,k) tn n! (34) = et ∞ ∑ k=0 xk 1 k! 1 λ k ( eλmt −1 m )k = etex( eλmt−1 λm ). Theorem 10. For m ∈ N, we have etex( eλmt−1 λm ) = ∞ ∑ n=0 dm,λ (n,x) tn n! . When x = 1, dm,λ (n,1) = dm,λ (n) are called the λ -analogues of Dowling numbers. We define the λ -analogues of Bell polynomials by e x λ (eλ t−1) = ∞ ∑ n=0 φn,λ (x) tn n! . (35) Thus, we easily get φn,λ (x) = n ∑ k=0 { n k } λ xk, (n ≥ 0). When x = 1, φn,λ (1) = φn,λ are called the λ -analogues of Bell numbers. D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1063 From Theorem 10, we note that ∞ ∑ n=0 dm,λ (n,x) tn n! = etex( eλmt−1 λm ) = e− x λm etex eλmt λm (36) = e− x λm ∞ ∑ k=0 eλmkt+t λ kmk xk k! = e− x λm ∞ ∑ k=0 xk k!mkλ k ∞ ∑ n=0 (λmk+1)n tn n! = ∞ ∑ n=0 ( e− x λm ∞ ∑ k=0 xk k!mkλ k (λmk+1)n ) tn n! . Therefore, by comparing the coefficients on both sides of (36), we obtain the following Dobinski- like formula. Theorem 11. For n ≥ 0, we have dm,λ (n,x) = e− x λm ∞ ∑ k=0 xk k!mkλ k (λmk+1)n. For r ∈ N, m,n ≥ 0, we consider the λ -analogues of the Whitney- type r-Stirling numbers of the second kind defined by (mx+ r)n = n ∑ k=0 W (r) m,λ (n,k)m k(x)k,λ , (n ≥ 0). (37) From (37), we note that e(mx+r)t = ∞ ∑ n=0 (mx+ r)n tn n! = ∞ ∑ n=0 ( n ∑ k=0 W (r) m,λ (n,k)m k(x)k ) tn n! (38) = ∞ ∑ k=0 ( ∞ ∑ n=k W (r) m,λ (n,k) tn n! ) mk(x)k. On the other hand, by (1), we get e(mx+r)t = ert(emλ t −1+1) x λ = ∞ ∑ k=0 ( x λ k ) (emλ t −1)kert (39) = ∞ ∑ k=0 1 k! 1 λ k (e mλ t −1)kert(x)k,λ = ∞ ∑ k=0 ( 1 k! 1 λ k ( emλ t −1 m )k ert ) mk(x)k,λ . By (38) and (39), we get 1 k! 1 λ k ( eλmt −1 m )k ert = ∞ ∑ n=k W (r) m,λ (n,k) tn n! , (40) D. S. Kim, H. K. Kim, T. Kim / Eur. J. Pure Appl. Math, 15 (3) (2022), 1054-1066 1064 where k is a nonnegative integer. Therefore, by (40), we obtain the following theorem. Theorem 12. For k ≥ 0, we have 1 k! 1 λ k ( eλmt −1 m )k ert = ∞ ∑ n=k W (r) m,λ (n,k) tn n! . From (15) and Theorem 11, we have W (r) 1,λ (n,k) = { n+ r k+ r } r,λ , W (0) 1,λ (n,k) = { n k } λ and W (1) m,λ (n,k) =Wm,λ (n,k), (n,k ≥ 0). Now, we observe that ∞ ∑ n=k W (r) m,λ (n,k) tn n! = 1 k! 1 λ k ( eλmt −1 m )k ert (41) = 1 k! 1 λ k+α ( eλmt −1 m )k+α et 1 tα ( λmt eλmt −1 )α e(r−1)t = (k+α)! k! 1 tα 1 (k+α)! 1 λ k+α ( eλmt −1 m )k+α et ∞ ∑ j=0 B(α) j ( r−1 mλ ) λ jm j t j j! = α! tα ( k+α k ) ∞ ∑ l=k+α Wm,λ (l,k+α) t l l! ∞ ∑ j=0 B(α) j ( r− j mλ ) λ jm j t j j! = ( k+α k )( ∞ ∑ l=k Wm,λ (l +α,k+α)(l+α l ) t l l! ) ∞ ∑ j=0 B(α) j ( r− j mλ ) λ jm j t j j! = ( k+α k ) ∞ ∑ n=k ( n ∑ l=k (n l )(l+α l )Wm,λ (l +α,k+α)B(α) n−l ( r−n+ l mλ ) λ n−lmn−l ) tn n! , where α is a positive integer. Comparing the coefficients on both sides of (41), we have the following theorem. Theorem 13. For n,k ∈ N∪{0} and α ∈ N, we have( k+α k )−1 W (r) m,λ (n,k) = n ∑ l=k (n l )(l+α l )Wm,λ (l +α,k+α)B(α) n−l ( r−n+ l mλ ) λ n−lmn−l. 4. Conclusion In this paper, we introduced the λ -analogues of r-Stirling numbers of the second which appear as the coefficients when powers of x+ r are expressed in terms of the degenerate falling factorial REFERENCES 1065 sequence. We obtained some properties, recurrence relations and certain identities on such num- bers. We also introduced the λ -analogues of Whitney-type r-Stirling numbers of the second and derived similar results to the case of the λ -analogues of r-Stirling numbers of the second. Fur- thermore, the λ -analogues of Dowling polynomials were introduced as a natural extension of the analogues of Whitney-type Stirling numbers of the second kind and a Dobinski-like formula for them was deduced. It is one of our future projects to continue to study analogues of some special numbers and polynomials and to find their applications in physics, science and engineering. Funding This work was supported by the Basic Science Research Program, the National Research Foun- dation of Korea, (NRF-2021R1F1A1050151). References [1] S. Araci. A new class of bernoulli polynomials attached to polyexponential functions and related identities. Advanced Studies in Contemporary Mathematics., 31(2):195–204, 2021. [2] M. S. Aydin, M. Acikgoz, and S. Araci. A new construction on the degenerate hurwitz-zeta function associated with certain applications. Proceedings of the Jangjeon Mathematical Society, 25(2):195–203, 2022. [3] L. Carlitz. Degenerate stirling, bernoulli and eulerian numbers. Utilitas Mathematica., 15:51–88, 1979. [4] L. Comtet. Advanced combinatorics. The art of finite and infinite expansions., volume ISBN: 90-277-0441-4. 1974. [5] W. A. Khan, M. Ghayasuddin, and D. Srivastava. A new class of partially degenerate laguerre-based hermite-genocchi polynomials. Advanced Studies in Contemporary Math- ematics (Kyungshang)., 32(1):71–83, 2022. [6] D. S. Kim and T. Kim.A. note on a new type of degenerate bernoulli numbers. Russian Journal of Mathematical Physics., 27(2):227–235, 2020. [7] H. K. Kim. Central lah numbers and central lah-bell numbers. Advanced Studies in Contem- porary Mathematics., 32(1):103–111, 2022. [8] T. Kim and D. S. Kim. Some identities on λ -analogues of r-stirling numbers of the first kind. Filomat., 34(2):451–460, 2020. [9] T. Kim and D. S. Kim. On some degenerate differential and degenerate difference operators. Russian Journal of Mathematical Physics., 29(1):37–46, 2022. REFERENCES 1066 [10] T. Kim, D. S. Kim, L.-C. Jang, H .Lee, and H. Kim. Representations of degenerate hermite polynomials. Advances in Applied Mathematics., 139(102359), 2022. [11] T. Kim, D. S. Kim, H. Lee, and J.-W. Park. A note on degenerate r-stirling numbers. Journal of Inequalities and Applications., 2020:225:12 pp, 2020. [12] S. Roman. The umbral calculus., volume ISBN: 0-12-594380-6. 1984. [13] Y. Simsek. Identities and relations related to combinatorial numbers and polynomials. Pro- ceedings of the Jangjeon Mathematical Society., 20(1):127–135, 2017. [14] Y. Simsek. Construction of generalized leibnitz type numbers and their properties. Advanced Studies in Contemporary Mathematics (Kyungshang)., 31(3):311–323, 2021.