EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6085 ISSN 1307-5543 – ejpam.com Published by New York Business Global Probabilistic Degenerate Poly r-Stirling Numbers of the Second Kind and r-Bell Polynomials Si Hyeon Lee Kwangwoon University, Seoul 139-701, Republic of Korea Abstract. We introduce degenerate poly r-Stirling numbers of the second kind and poly r-Bell polynomials by using degenerate polyexponential function and investigate some properties of these number and polynomials. 2020 Mathematics Subject Classifications: 11B73 Key Words and Phrases: Stirling numbers of the second kind, Bell polynomials 1. Introduction Recently, degenerate Sitrling numbers of the second kind and degenerate Bell poly- nomials have been studied by many researchers (see [1],[2],[3],[4],[5],[6],[7],[8]). These numbers and polynomials were explored from a view of probabilistic perspective (see [9],[10],[11],[12],[13],[14],[15],[16],[17],[18],[19]). The outline of the paper is as follows. In section 1, we recall some definitions. In section 2, we consider a probabilistic polyexponen- tial function using a degenerate polyexponential function, and then define the probabilistic degenerate poly r-Stirling numbers of the second kind and probabilistic degenerate poly r-Bell polynomials. In Theorem 2.1, we derive an expression for S (r,k,Y ) 2,λ (n + r, l + r). In Theorem 2.2, we get an expression for S (r,k,Y ) 2,λ (n + r, l + r) as sum of the products. In Theorem 2.3, we get expression for S (r,k,Y ) 2,λ (n + r, l + 2r). In Theorem 2.4, we find some relation for S (r,k,Y ) 2,λ (n+ r, l+ r). In Theorem 2.5 we get an expression for Bel (r,k,Y ) n,λ (x). In Theorem 2.6 we derive an expression for Bel (r,k,Y ) n,λ (x). In ([20],[4],[13],[21],[14],[22]) and ([15],[23],[24],[25],[19]) researchers studied degenerate exponential function. For any nonzero λ ∈ R, the degenerate exponentials exλ(t), which are defined by exλ(t) = ∞∑ n=0 (x)n,λ tn n! . (1) DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6085 Email address: ugug11@naver.com (S. H. Lee) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 2 of 10 where (x)0,λ = 1, (x)n,λ = x(x− λ) · · · (x− (n− 1)λ), (n ≥ 1). The degenerate Stirling numbers of the second kind are defined 1 k! (eλ(t)− 1)k = ∞∑ n=k { n k } λ tn n! , (see[26], [27], [5], [6], [7], [28]). (2) The degenerate r-Stirling numbers of the second kind are given by 1 k! (eλ(t)− 1)kerλ(t) = ∞∑ n=k S (r) 2,λ(n+ r, k + r) tn n! , (see[20], [29]). (3) The degenerate r-Bell polynomials are defined by ex(eλ(t)−1)erλ(t) = ∞∑ n=0 Bel (r) n,λ(x) tn n! , (see[3], [30], [29]). (4) The degenerate polyexponential function is defined by Eik,λ(x) = ∞∑ n=1 (1)n,λ (n− 1)!nk xn, (k ∈ Z, |x| < 1), (see[[31], [4], [5], [10], [16], [17], [18], [32], [33]). (5) In this paper, let Y be a random variable such that the moment generating function of Y and satisfy E[eY t] = ∞∑ n=0 E[Y n] tn n! , (|t| < r), (see[1], [9], [2], [34], [35], [8], [36]). (6) for some r > 0. Let (Yk)k≥1 be a sequence of mutually copies of random variable Y and let Sk = Y1 + Y2 + · · ·+ Yk, (k ≥ 1) with S0 = 0. The probabilistic degenerate Stirling numbers of the second kind associated with Y are defined by 1 k ( E[eYλ (t)]− 1 )k = ∞∑ n=k { n k } Y,λ tn n! , (see[1], [11], [15], [37], [8], [38], [19]). (7) 2. Probabilistic degenerate poly r-Stirling numbers of the second kind and r-Bell polynomials In this section, we consider degenerate probabilistic polyexponential function which is given by EiYk,λ(t) = ∞∑ n=1 E[(Y )n,λ] (n− 1)!nk tn. (8) S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 3 of 10 From (8), we note that EiY1,λ(t) = ∞∑ n=1 E[(Y )n,λ] n! tn = E[eYλ (t)]− 1 (9) and Ei11,λ(t) = ∞∑ n=1 (1)n,λ n! tn = Ei1,λ(t). (10) Now, we consider probabilistic degenerate poly r-Stirling numbers of the second kind which are given by 1 l! ( EiYk,λ(t) )l ( E[eYλ (t)] )r = ∞∑ n=l S (r,k,Y ) 2,λ (n+ r, l + r) tn n! . (11) When k = 1, Y = 1, S (r,1,1) 2,λ (n+ r, l + r) = S (r) 2,λ(n+ r, l + r). From (11), we have ∞∑ n=l S (r,k,Y ) 2,λ (n+ r, l + r) tn n! = 1 l! ( EiYk,λ(t) )l ( E[eYλ ](t) )r (12) = 1 l! ( ∞∑ i1=1 E[(Y )i1,λ] (i1 − 1)!ik1 ti1 ) · · ·  ∞∑ il=1 E[(Y )il,λ] (il − 1)!ikl til (E[eYλ ](t) )r = 1 l! ∞∑ m=l ∑ i1+···+il=m ( m i1, · · · , il ) E[(Y )i1,λ] · · ·E[(Y )il,λ] ik−1 1 ik−1 2 · · · ik−1 l tm m! ∞∑ j=0 E[(Sr)j,λ] tj j! = 1 l! ∞∑ n=l n∑ m=l ∑ i1+···+il=m ( m i1, · · · , il )( n m ) E[(Y )i1,λ] · · ·E[(Y )il,λ] ik−1 1 ik−1 2 · · · ik−1 l E[(Sr)j,λ] tn n! . Thus, we have the following theorem. Theorem 1. For n ≥ l, we have S (r,k,Y ) 2,λ (n+r, l+r) = ∞∑ n=l n∑ m=l ∑ i1+···+il=m ( m i1, · · · , il )( n m ) E[(Y )i1,λ] · · ·E[(Y )il,λ] ik−1 1 ik−1 2 · · · ik−1 l E[(Sr)j,λ]. From (11), we get ∞∑ n=l S (r,k,Y ) 2,λ (n+ r, l + r) tn n! = 1 l! ( EiYk,λ(t) )l ( E[eYλ (t)]− 1 + 1 )r (13) = 1 l! ( EiYk,λ(t) )l ∞∑ i=0 ( r i )( E[eYλ (t)]− 1 )i S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 4 of 10 = 1 l! ( EiYk,λ(t) )l ∞∑ i=0 (r)i ∞∑ m=i { m i } λ,Y tm m! = ∞∑ j=l S (k,Y ) 2,λ (j, l) tj j! ∞∑ m=i m∑ i=0 (r)i { m i } λ,Y tm m! = ∞∑ n=l+i n∑ m=i m∑ i=0 ( n m ) (r)i { m i } λ,Y S (k,Y ) 2,λ (n−m, l) tn n! . Thus, we get the following theorem. Theorem 2. For l ≥ i, we have S (r,k,Y ) 2,λ (n+ r, k + r) = n∑ m=i m∑ i=0 ( n m ) (r)i { m i } λ,Y S (k,Y ) 2,λ (n−m, l). From (11), we have ∞∑ n=k S (r,k,Y ) 2,λ (n+ r, l + r) tn n! ( EiYk,λ(t) ) = 1 l! ( EiYl,λ(t) )l+1 ( E[eYλ (t)] )r (14) = (l + 1)! l! ∞∑ n=k S (r,k,Y ) 2,λ (n+ r, l + r + 1) tn n! = (l + 1) ∞∑ n=k S (r,k,Y ) 2,λ (n+ r, l + r + 1) tn n! . The left hand side of (14), we have ∞∑ n=k S (r,k,Y ) 2,λ (n+ r, l + r) tn n! ( EiYk,λ(t) ) = ∞∑ m=k S (r,k,Y ) 2,λ (m+ r, l + r) tm m! ∞∑ i=1 E[(Y )i,λ] (i− 1)!ik ti (15) = ∞∑ n=k+1 n∑ m=k ( n m ) S (r,k,Y ) 2,λ (m+ r, l + r) E[(Y )n−m,λ] (n−m− 1)!(n−m)k tn n! . By comparing the coefficients of (14) and (15), we get the following theorem. Theorem 3. For n ≥ k + 1, we have S (r,k,Y ) 2,λ (n+ r, l + r + 1) = 1 l + 1 n∑ m=k ( n m ) S (r,k,Y ) 2,λ (m+ r, l + r) E[(Y )n−m,λ] (n−m− 1)!(n−m)k . S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 5 of 10 From (11), we have ∞∑ n=l S (r,k,Y ) 2,λ (n+ r, l + r) tn n! = 1 l! ( EiYl,λ(t) )l ( E[eYλ (t)] )r (16) = 1 r! ( EiYk,λ(t) )r ( E[eYλ (t)] )r ( EiYk,λ(t) )l−r = ∞∑ m=r S (r,k,Y ) 2,λ (m+ r, 2r) tm m! (l − r)! ∞∑ i=l−r S (k,Y ) 2,λ (i, l − r) ti i! = ∞∑ n=l n∑ m=r (l − r)! ( n m ) S (r,k,Y ) 2,λ (m+ r, 2r)S (k,Y ) 2,λ (n−m, l − r) tn n! . Thus, we get the following theorem. Theorem 4. For n ≥ l, we have S (r,k,Y ) 2,λ (n+ r, l + r) = n∑ m=r ( n m ) (l − r)!S (r,k,Y ) 2,λ (m+ r, 2r)S (k,Y ) 2,λ (n−m, l − r). Now, we consider probabilistic degenerate poly r-Bell polynomials associated with Y which are given by ex(EiYk,λ(t)) ( E[eYλ (t)] )r = ∞∑ n=0 Bel (r,k,Y ) n,λ (x) tn n! . (17) From (17), we have ∞∑ n=0 Bel (r,k,Y ) n,λ (x) tn n! = ex(EiYk,λ(t)) ( E[eYλ (t)] )r (18) = ∞∑ l=0 xl ( EiYk,λ(t) )l l! ( E[eYλ (t)] )r = ∞∑ l=0 xl ∞∑ n=l S (r,k,Y ) 2,λ (n+ r, l + r) tn n! = ∞∑ n=l n∑ l=0 xlS (r,k,Y ) 2,λ (n+ r, l + r) tn n! . Thus, we have the following theorem. S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 6 of 10 Theorem 5. For n ≥ l, we have Bel (r,k,Y ) n,λ (x) = n∑ l=0 xlS (r,k,Y ) 2,λ (n+ r, l + r). From (17), we ovserve that ∞∑ n=0 Bel (r,k,Y ) n,λ (x) tn n! = e x 2 (EiYk,λ(t)) ∞∑ k=0 Bel (r,k,Y ) k,λ (x 2 ) tn n! (19) = ∞∑ l=0 (x 2 )l 1 l! ( EiYk,λ(t) )l ∞∑ m=0 Bel (r,k,Y ) m,λ (x 2 ) tn n! = ∞∑ l=0 xl 2ll! ( ∞∑ i1=1 E[E(Y )i1,λ] (i1 − 1)!ik1 ti1 )( ∞∑ i2=1 E[E(Y )i2,λ] (i2 − 1)!ik2 ti2 ) · · ·  ∞∑ il=1 E[E(Y )il,λ] (i1 − 1)!ik1 til  × ∞∑ m=0 Bel (r,k,Y ) m,λ (x 2 ) tm m! = ∞∑ l=0 xl 2ll! ∞∑ j=l ∑ i1+i2+···+il=j ( j i1i2 · · · il ) E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] ik−1 1 ik−2 2 · · · ik−1 l tj j! ∞∑ m=0 ×Bel (r,k,Y ) m,λ (x 2 ) tm m! = ∞∑ n=0 n∑ j=0 j∑ l=0 ∑ i1+i2+···+il=j ( n j )( j i1i2 · · · il ) xl 2ll! E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] (i1i2 · · · il)k−1 ×Bel (r,k,Y ) n−j,λ (x 2 ) tn n! and ∞∑ n=0 Bel (r,k,Y ) n,λ (x) tn n! = e 2 3 x(EiYk,λ(t)) ∞∑ k=0 Bel (r,k,Y ) k,λ (x 3 ) tn n! (20) = ∞∑ l=0 ( 2 3 )l xl l! ( EiYk,λ(t) )l ∞∑ m=0 Bel (r,k,Y ) m,λ (x 3 ) tn n! = ∞∑ l=0 ( 2 3 )l xl l! ( ∞∑ i1=1 E[E(Y )i1,λ] (i1 − 1)!ik1 ti1 )( ∞∑ i2=1 E[E(Y )i2,λ] (i2 − 1)!ik2 ti2 ) · · ·  ∞∑ il=1 E[E(Y )il,λ] (i1 − 1)!ik1 til  × ∞∑ m=0 Bel (r,k,Y ) m,λ (x 3 ) tm m! S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 7 of 10 = ∞∑ l=0 ( 2 3 )l xl l! ∞∑ j=l ∑ i1+i2+···+il=j ( j i1i2 · · · il ) E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] ik−1 1 ik−2 2 · · · ik−1 l tj j! ∞∑ m=0 ×Bel (r,k,Y ) m,λ (x 3 ) tm m! = ∞∑ n=0 n∑ j=0 j∑ l=0 ∑ i1+i2+···+il=j ( n j )( j i1i2 · · · il )( 2 3 )l xl l! E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] (i1i2 · · · il)k−1 ×Bel (r,k,Y ) n−j,λ (x 3 ) tn n! . Repeating this process α times, we have ∞∑ n=0 Bel (r,k,Y ) n,λ (x) tn n! = e α−1 α x(EiYk,λ(t)) ∞∑ k=0 Bel (r,k,Y ) k,λ (x α ) tn n! (21) = ∞∑ l=0 ( α− 1 α )l xl l! ( EiYk,λ(t) )l ∞∑ m=0 Bel (r,k,Y ) m,λ (x α ) tn n! = ∞∑ l=0 ( α− 1 α )l xl l! ( ∞∑ i1=1 E[E(Y )i1,λ] (i1 − 1)!ik1 ti1 )( ∞∑ i2=1 E[E(Y )i2,λ] (i2 − 1)!ik2 ti2 ) · · ·  ∞∑ il=1 E[E(Y )il,λ] (i1 − 1)!ik1 til  × ∞∑ m=0 Bel (r,k,Y ) m,λ (x α ) tm m! = ∞∑ l=0 ( α− 1 α )l xl l! ∞∑ j=l ∑ i1+i2+···+il=j ( j i1i2 · · · il ) E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] ik−1 1 ik−2 2 · · · ik−1 l tj j! ∞∑ m=0 ×Bel (r,k,Y ) m,λ (x α ) tm m! = ∞∑ n=0 n∑ j=0 j∑ l=0 ∑ i1+i2+···+il=j ( n j )( j i1i2 · · · il )( α− 1 α )l xl l! E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] (i1i2 · · · il)k−1 ×Bel (r,k,Y ) n−j,λ (x α ) tn n! . By comparing the coefficients on both sides in (21), we have the following theorem. Theorem 6. For n, k ≥ 0, α ∈ N, we have Bel (r,k,Y ) n,λ (x) = n∑ j=0 j∑ l=0 ∑ i1+i2+···+il=j ( n j )( j i1i2 · · · il )( α− 1 α )l xl l! (22) S. H. Lee / Eur. J. Pure Appl. Math, 18 (2) (2025), 6085 8 of 10 × E[(Y )i1,λ]E[(Y )i2,λ] · · ·E[(Y )il,λ] (i1i2 · · · il)k−1 Bel (r,k,Y ) n−j,λ (x α ) . 3. 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