S S symmetry Article Degenerate Stirling Polynomials of the Second Kind and Some Applications Taekyun Kim 1,*, Dae San Kim 2,*, Han Young Kim 1 and Jongkyum Kwon 3,* 1 Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea 2 Department of Mathematics, Sogang University, Seoul 121-742, Korea 3 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju 52828, Korea * Correspondence: [email protected] (T.K.); [email protected] (D.S.K.); [email protected] (J.K.) Received: 20 July 2019; Accepted: 13 August 2019; Published: 14 August 2019 Abstract: Recently, the degenerate l-Stirling polynomials of the second kind were introduced and investigated for their properties and relations. In this paper, we continue to study the degenerate l-Stirling polynomials as well as the r-truncated degenerate l-Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. Regarding applications, we show that both the degenerate l-Stirling polynomials of the second and the r-truncated degenerate l-Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. Keywords: degenerate l-Stirling polynomials of the second kind; r-truncated degenerate l-Stirling polynomials of the second kind; probability distribution 1. Introduction For n ≥ 0, the Stirling numbers of the second kind are defined as (see [1–26]) n n x = ∑ S2(n, k)(x)k, (1) k=0 where (x)0 = 1, (x)n = x(x − 1) ··· (x − n + 1), (n ≥ 1). From (1), we note that the generating function for S2(n, k) is given by (see [8,9,23,26]) ¥ n 1 t k t (e − 1) = ∑ S2(n, k) , (k ≥ 0). (2) k! n=k n! For l 2 R, the degenerate exponential function is defined by (see [10,11,27,28]) x 1 x l 1 l el(t) = (1 + lt) , el(t) = el(t) = (1 + lt) . (3) In view of (2), the degenerate l-Stirling polynomials of the second kind are defined by the generating function 1 ¥ tn ( ( ) − )k x ( ) = (x)( ) el t 1 el t ∑ S2,l n, k , (4) k! n=k n! where x 2 R and k is a nonnegative integer, (see [10,11]). (0) When x = 0, S2,l(n, k) = S2,l(n, k) are called the degenerate l-Stirling numbers of the second kind. Note that liml!0 S2,l(n, k) = S2(n, k), (n, k ≥ 0). Symmetry 2019, 11, 1046; doi:10.3390/sym11081046 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 1046 2 of 11 By letting l ! 0 in (4), we have the generating function for the Stirling polynomials of the second (x) kind S2 (n, k) (see [11,13,14,22]): ¥ n 1 t k xt (x) t (e − 1) e = ∑ S2 (n, k) . (5) k! n=k n! Let X be a discrete random variable with probability mass function P[X = x] = p(x). Then, the probability generating function of X is given by (see [4,17,18,20,22]) ¥ G(t) = E[tX] = ∑ tx p(x). (6) x=0 Suppose that X = (X1, X2, ··· , Xk) is a discrete random variable taking values in the k-dimensional nonnegative integer lattice. Then, the probability generating function of X is defined as (see [4]) X1 X2 Xk G(t) = G(t1, t2, ··· , tk) = E[t1 t2 ··· tk ] ¥ (7) = x1 x2 ··· xk ( ··· ) ∑ t1 t2 tk p x1, x2, xk , x1,x2,··· ,xk=0 where p(x1, x2, ··· xk) is the probability mass function of X = (X1, X2, ··· , Xk). X is the random variable with the zero-truncated Poisson distribution with parameter l if the probability mass function of X is (see [4,17,18,20,22]) 1 lx P[X = x] = p(x) = e−l , (8) 1 − e−l x! where x is a positive integer. For r 2 N, X is the random variable with r-truncated Poisson distribution with parameter l if the probability mass function of X is lx P[X = x] = p(x) = C(l, r)e−l , (9) x! −l r−1 lx −1 where C(l, r) = (1 − e ∑x=0 x! ) . In this paper, we study the degenerate l-Stirling polynomials, as a continuation of the previous work in [10], and also the r-truncated degenerate l-Stirling polynomials of the second kind which are derived from generating functions and Newton’s formula. We derive recurrence relations and various expressions for them. As applications, we show that both the degenerate l-Stirling polynomials of the second kind and the r-truncated degenerate l-Stirling polynomials of the second kind appear in the expressions of the probability distributions of appropriate random variables. 2. The Degenerate l-Stirling Polynomials of the Second Kind Let x, t be real numbers and let n be a nonnegative integer. The difference operator 4 is defined by 4 f (x) = f (x + 1) − f (x). It is easy to show that n n (I + 4)n f (0) = f (n) = ∑ 4k f (0). (10) k=0 k From (10), we can derive Newton’s formula which is given by ¥ t f (t) = ∑ 4k f (0). (11) k=0 k Symmetry 2019, 11, 1046 3 of 11 Let us take f (t) = (t + x)n,l, (n ≥ 0). Then, by (11), we get ¥ t k (t + x)n,l = ∑ 4 (t + x)n,l k=0 k t=0 n t k = ∑ 4 (t + x)n,l (12) k=0 k t=0 n 1 k = ∑ 4 (t + x)n,l (t)k. k=0 k! t=0 Here, the generalized factorial sequence (x)n,l is given by (x)0,l = 1, (x)n,l = x(x − l) ··· (x − (n − 1)l), (n ≥ 1). (13) Let (x) 1 k 1 k S2,l (n, k) = 4 (t + x)n,l = 4 (x)n,l, (n, k ≥ 0). (14) k! t=0 k! Then, by (12) and (14), we get n ( + ) = (x)( )( ) ( ≥ ) t x n,l ∑ S2,l n, k t k, n 0 . (15) k=0 (x) In (18) below, we show that the S2,l (n, k) in (14) or (15) is really the degenerate l-Stirling polynomials of the second kind defined in (4). Using (13), (15), and interchanging the order of summations, we note that ¥ tn ¥ n tn x+y( ) = ( + ) = (x)( )( ) el t ∑ x y n,l ∑ ∑ S2,l n, k y k n=0 n! n=0 k=0 n! (16) ¥ ¥ tn = (x)( ) ( ) ∑ ∑ S2,l n, k y k. k=0 n=k n! On the other hand, ¥ + 1 x y( ) = x ( )( ( ) − + )y = ( ( ) − )k x ( )( ) el t el t el t 1 1 ∑ el t 1 el t y k. (17) k=0 k! By (16) and (17), we get 1 ¥ tn ( ( ) − )k x ( ) = (x)( ) ( ≥ ) el t 1 el t ∑ S2,l n, k , k 0 . (18) k! n=k n! Thus, we have shown that the definition of the degenerate l-Stirling polynomials of the second kind can be given as (14) or equivalently (15) or equivalently (18). Note that (x) (x) (x) S2,l (0, 0) = 1, S2,l (n, 0) = (x)n,l, S2,l (0, k) = 0, (k 6= 0). (19) Symmetry 2019, 11, 1046 4 of 11 We would like to derive a recurrence relation for the degenerate l-Stirling polynomials of the second kind. Now, we observe that n+1 (x)( + )( ) = ( + ) = ( + ) ( + − ) ∑ S2,l n 1, k t k t x n+1,l t x n,l t x nl k=0 n n = (x)( )( ) + ( − ) (x)( )( ) t ∑ S2,l n, k t k x nl ∑ S2,l n, k t k k=0 k=0 n+1 n = (x)( − )( ) + ( − ) (x)( )( ) t ∑ S2,l n, k 1 t k−1 x nl ∑ S2,l n, k t k k=1 k=0 (20) n+1 n = ( − + + − ) (x)( − )( ) + ( − ) (x)( )( ) ∑ t k 1 k 1 S2,l n, k 1 t k−1 x nl ∑ S2,l n, k t k k=1 k=0 n+1 n n = (x)( )( ) + (x)( )( ) + ( − ) (x)( )( ) ∑ S2,l n, k t k ∑ kS2,l n, k t k x nl ∑ S2,l n, k t k k=1 k=0 k=0 n+1 = (x)( − ) + ( + − ) (x)( ) ( ) ∑ S2,l n, k 1 k x nl S2,l n, k t k. k=0 Therefore, by comparing the coefficients on both sides of (20), we obtain the following theorem. Theorem 1. Let n, k be nonnegative integers. Then, we have (x) (x) (x) S2,l (n + 1, k) = S2,l (n, k − 1) + (k + x − nl)S2,l (n, k), (n ≥ k − 1). Note that (x) (x) S2,l (n, k) = 0, if k > n, S2,l (n, n) = 1. From (14), we have ( ) 1 S 0 (n, k) = 4k(0) = S (n, k), (n, k ≥ 0). (21) 2,l k! n,l 2,l Here, we want to derive an explicit expression for the degenerate l-Stirling polynomials of the second kind. By (18), we get ¥ tn 1 (x)( ) = x ( )( ( ) − )k ∑ S2,l n, k el t el t 1 n=k n! k! ¥ ¥ ( ) ( ) ··· ( ) 1 (x)m,l m 1 l1,l 1 l2,l 1 lk,l l = ∑ t ∑ ∑ t (22) k! m! l1!l2! ··· lk! m=0 l=k l1+···+lk=l ¥ n n n! (x) − (1)l ,l(1)l ,l ··· (1)l ,l t = ∑ ∑ n l,l ∑ 1 2 k , k! (n − l)! l1!l2! ··· lk! n! n=k l=k l1+···+lk=l where all lj’s are positive integers.