aa r X i v : . [ m a t h . N T ] J u l LAH-BELL NUMBERS AND POLYNOMIALS
DAE SAN KIM AND TAEKYUN KIMA
BSTRACT . In this paper, we introduce the Lah-Bell numbers and their natural extensions, namelythe Lah-Bell polynomials, and derive some basic properties of such numbers and polynomials by us-ing elementary methods. In addition, we consider the degenerate Lah-Bell numbers and polynomialsas degenerate versions of the Lah-Bell numbers and polynomials.
1. I
NTRODUCTION
As is well known, the Stirling number of the second kind S ( n , k ) , ( n ≥ k ≥ ) , is the number ofways to partition a set with n elements into k non-empty subsets. The n -th Bell number B n , ( n ≥ ) is the number of ways to partition a set with n elements into non-empty subsets. Thus, we have(1) B n = n ∑ k = S ( n , k ) , ( n ≥ ) . The Bell polynomials B n ( x ) are natural extensions of the Bell numbers (see (6), (10)).The unsigned Lah number L ( n , k ) counts the number of ways a set of n elements can be parti-tioned into k nonempty linearly ordered subsets. In view of the relationship between the Stirlingnumbers of the second kind and the Bell numbers, it is very natural and meaningful to define then-th Lah-Bell number B Ln , ( n ≥ ) , as the number of ways a set of n elements can be partitioned intonon-empty linearly ordered subsets . Thus, we have(2) B Ln = n ∑ k = L ( n , k ) , ( n ≥ ) . The Lah-Bell polynomials B Ln ( x ) are also defined as natural extensions of the Lah-Bell numbers(see (23), Lemma 4) .The aim of this paper is to study the Lah-Bell numbers and polynomials and to derive someof their basic properties by using elementary methods. We also consider the degenerate Lah-Bellnumbers and polynomials (see (42), (44)) as degenerate versions of the Lah-Bell numbers andpolynomials.In more detail, for the Lah-Bell numbers and polynomials we derive the generating functions,some relations with Bell numbers and polynomials, and Dobinski-like formulas. We show the con-nections between Lah numbers and Stirling numbers, recurrence relations and derivatives for theLah-Bell polynomials. We introduce the bivariate Lah-Bell polynomials, and find the generatingfunction of them and their connection with the bivariate Bell polynomials studied in [15]. In addi-tion, as degenerate versions of the Lah-Bell polynomials and numbers, we introduce the degenerateLah-Bell polynomials and numbers. Then, for the degenerate Lah-Bell polynomials we deduce thegenerating function, an explicit expression and their relation with the degenerate Bell polynomials. Mathematics Subject Classification.
Key words and phrases.
Lah-Bell numbers; Lah-Bell polynomials; bivariate Lah-Bell polynomials; degenerate Lah-Bell numbers; degenerate Lah-Bell polynomials.
We recall that the falling factorial sequence is given by ( x ) = , ( x ) n = x ( x − ) · · · ( x − n + ) , ( n ≥ ) . Then, we have(3) x n = n ∑ k = S ( n , k )( x ) k , ( n ≥ ) , ( see [ − ]) . From (3), we can derive the following equation.(4) 1 k ! (cid:0) e t − (cid:1) k = ∞ ∑ n = k S ( n , k ) t n n ! , ( k ≥ ) . By (1) and (4), we get(5) e ( e t − ) = ∞ ∑ n = B n t n n ! , ( see [ ]) . The Bell polynomials are given by(6) e x ( e t − ) = ∞ ∑ n = B n ( x ) t n n ! , ( see [ , , ]) . When x = B n = B n ( ) are the n -th Bell numbers. From (6), we have(7) B n ( x ) = e − x ∞ ∑ k = k n k ! x k , ( n ≥ ) , ( see [ , , ]) . It is well known that the Stirling number of the first kind S ( n , k ) counts the number of permuta-tions of n elements consisting of k disjoint cycles. As an inversion formula of (3), we have(8) ( x ) n = n ∑ k = S ( n , k ) x k . From (8), we note that(9) 1 k ! (cid:0) log ( + t ) (cid:1) k = ∞ ∑ n = k S ( n , k ) t n n ! , ( k ≥ ) , ( see [ − , − ]) . By (4) and (6), we get(10) B n ( x ) = n ∑ K = S ( n , k ) x k , ( n ≥ ) , ( see [ ]) . For n , k ≥
0, the unsigned Lah numbers are given by(11) L ( n , k ) = (cid:18) n − k − (cid:19) n ! k ! , ( see [ , , , ]) . The rising factorial sequence is given by(12) h x i = , h x i n = x ( x + ) · · · ( x + n − ) , ( n ≥ ) . AH-BELL NUMBERS AND POLYNOMIALS 3
From (11), we note that h x i n = n ∑ k = L ( n , k )( x ) k , (13) ( x ) n = n ∑ k = ( − ) n − k L ( n , k ) h x i k , (14) L ( n , k ) = n ∑ j = n ( − ) n − j S ( n , j ) S ( j , k ) , (15) L ( n , k ) = (cid:18) nk (cid:19)(cid:18) n − k − (cid:19) ( n − k ) ! = (cid:18) n ! k ! (cid:19) kn ( n − k ) ! , (16) L ( n , k + ) = n − kk ( k + ) L ( n , k ) , ( n ≥ , k ≥ ) . (17) 2. L AH -B ELL NUMBERS AND POLYNOMIALS
In this section, we study the Lah-Bell numbers B Ln in (2) and their extension, namely the Lah-Bellpolynomials B Ln ( x ) (see (23)). In addition, we investigate the degenerate Lah-Bell numbers B Ln , λ By using (11), we easily get(18) 1 k ! (cid:18) t − t (cid:19) k = ∞ ∑ n = k L ( n , k ) t n n ! , ( k ≥ ) . From (2) and (18), we note that e t − t = ∞ ∑ k = k ! (cid:18) t − t (cid:19) k = ∞ ∑ k = ∞ ∑ n = k L ( n , k ) t n n !(19) = ∞ ∑ n = (cid:18) n ∑ k = L ( n , k ) (cid:19) t n n ! = ∞ ∑ n = B Ln ( n , k ) t n n ! . Therefore, by (19), we obtain the following lemma.
Lemma 1.
The generating function of Lah-Bell numbers is given bye (cid:0) − t − (cid:1) = ∞ ∑ n = B Ln t n n ! . Replacing t by 1 − e − t in Lemma 1, we get e e t − = ∞ ∑ k = B Lk k ! (cid:0) − e − t (cid:1) k = ∞ ∑ k = ( − ) k B Lk ∞ ∑ n = k S ( n , k )( − ) n t n n !(20) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k B Lk S ( n , k ) (cid:19) t n n ! . On the other hand,(21) e e t − = ∞ ∑ n = B n t n n ! . Therefore, by (20) and (21), we obtain the following theorem.
DAE SAN KIM AND TAEKYUN KIM
Theorem 2.
For n ≥ , we have B n = n ∑ k = ( − ) n − k B Lk S ( n , k ) . We observe that e − t − = e ∞ ∑ k = k ! ( − t ) − k (22) = e ∞ ∑ k = k ! ∞ ∑ n = h k i n t n n ! = ∞ ∑ n = (cid:26) e ∞ ∑ k = h k i n k ! (cid:27) t n n ! . Therefore, by Lemma 1 and (22), we obtain the following Dobinski-like formula for Lah-Bellnumbers.
Theorem 3.
For n ≥ , we have B Ln = e ∞ ∑ k = h k i n k ! . In view of (10), we define Lah-Bell polynomials by(23) B Ln ( x ) = n ∑ l = L ( n , l ) x l , ( n ≥ ) . From (23), we note that ∞ ∑ n = B Ln ( x ) t n n ! = ∞ ∑ n = (cid:18) n ∑ k = x k L ( n , k ) (cid:19) t n n !(24) = ∞ ∑ k = x k (cid:18) ∞ ∑ n = k L ( n , k ) t n n ! (cid:19) = ∞ ∑ k = x k k ! (cid:18) − t − (cid:19) k = e x (cid:0) − t − (cid:1) . This shows the following result.
Lemma 4.
The generating function of Lah-Bell polynomials is given bye x (cid:0) − t − (cid:1) = ∞ ∑ n = B Ln ( x ) t n n ! . Replacing t by 1 − e − t in Lemma 4, we get e x ( e t − ) = ∞ ∑ k = B Lk ( x ) k ! (cid:0) − e − t (cid:1) k = ∞ ∑ k = ( − ) k B Lk ( x ) ∞ ∑ n = k S ( n , k )( − ) n t n n !(25) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B Lk ( x ) (cid:19) t n n ! . Therefore, by (6) and (25), we obtain the following theorem.
AH-BELL NUMBERS AND POLYNOMIALS 5
Theorem 5.
For n ≥ , we haveB n ( x ) = n ∑ k = ( − ) n − k S ( n , k ) B Lk ( x ) . By Theorem 3 and Lemma 4, we get e x (cid:0) − t − (cid:1) = e − x ∞ ∑ k = x k k ! ( − t ) − k = ∞ ∑ k = (cid:18) e − x ∞ ∑ k = h k i n k ! x k (cid:19) t n n ! . (26)Therefore, by Lemma 4 and (26), we obtain the following Doinski-like formula for Lah-Bell poly-nomials. Theorem 6.
For n ≥ , the following Dobinski-like formula holds:B Ln ( x ) = e − x ∞ ∑ k = h k i n k ! x k . Replacing t by − log ( − t ) in (6), we get e x (cid:0) − t − (cid:1) = ∞ ∑ k = B k ( x ) k ! (cid:0) − log ( − t ) (cid:1) k = ∞ ∑ k = ( − ) k B k ( x ) ∞ ∑ n = k ( − ) n S ( n , k ) t n n !(27) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B k ( x ) (cid:19) t n n ! . Therefore, by Lemma 4 and (27), we obtain the following theorem.
Theorem 7.
For n ≥ , we haveB Ln ( x ) = n ∑ k = ( − ) n − k S ( n , k ) B k ( x ) . Replacing t by − log ( − t ) in (4), we have1 k ! (cid:18) − t − (cid:19) k = ∞ ∑ l = k S ( l , k ) l ! (cid:0) − log ( − t ) (cid:1) l (28) = ∞ ∑ l = k ( − ) l S ( l , k ) ∞ ∑ n = l S ( n , l )( − ) n t n n ! = ∞ ∑ n = k (cid:18) n ∑ l = k ( − ) n − l S ( n , l ) S ( l , k ) (cid:19) t n n ! . On the other hand,(29) 1 k ! (cid:18) − t − (cid:19) k = ∞ ∑ n = k L ( n , k ) t n n ! . By (28) and (29), we get(30) L ( n , k ) = n ∑ l = k ( − ) n − l S ( n , l ) S ( l , k ) , ( n ≥ ) . DAE SAN KIM AND TAEKYUN KIM
By replacing t by 1 − e − t in (29), we get1 k ! (cid:0) e t − (cid:1) k = ∞ ∑ l = k L ( l , k ) l ! (cid:0) − e − t (cid:1) l (31) = ∞ ∑ l = k ( − ) l L ( l , k ) ∞ ∑ n = l S ( n , l )( − ) n t n n ! = ∞ ∑ n = k (cid:18) n ∑ l = k ( − ) n − l S ( n , l ) L ( l , k ) (cid:19) t n n ! . Therefore, by (4) and (31), we get(32) S ( n , k ) = n ∑ l = k ( − ) n − l S ( n , l ) L ( l , k ) , where n , k ≥
0, with n ≥ k . Theorem 8.
For n , k ≥ , with n ≥ k, we haveS ( n , k ) = n ∑ l = k ( − ) n − l S ( n , l ) L ( l , k ) , and L ( n , k ) = n ∑ l = k ( − ) n − l S ( n , l ) S ( l , k ) . From Lemma 4, we note that(33) x ddt (cid:18) − t (cid:19) e x (cid:0) − t − (cid:1) = ∞ ∑ n = B Ln + ( x ) t n n ! . On the other hand, x ddt (cid:18) − t (cid:19) e x (cid:0) − t − (cid:1) = x ∞ ∑ l = ( l + ) ! t l l ! ∞ ∑ m = B Lm ( x ) t m m !(34) = ∞ ∑ n = (cid:18) x n ∑ m = (cid:18) nm (cid:19) ( n − m + ) ! B Lm ( x ) (cid:19) t n n ! . Therefore, by (33) and (34), we obtain the following theorem.
Theorem 9.
For n ≥ , we haveB Ln + ( x ) = x n ∑ m = (cid:18) nm (cid:19) ( n − m + ) ! B Lm ( x ) . Again, from Lemma 4, we observe that ∞ ∑ n = ddx B Ln ( x ) t n n ! = ddx e x (cid:0) − t − (cid:1) = (cid:18) − t − (cid:19) e x (cid:0) − t − (cid:1) = ∞ ∑ l = l ! t l l ! ∞ ∑ m = B Lm ( x ) t m m ! = ∞ ∑ n = (cid:18) n − ∑ m = (cid:18) nm (cid:19) ( n − m ) ! B Lm ( x ) (cid:19) t n n ! . AH-BELL NUMBERS AND POLYNOMIALS 7
Thus, we have(35) ddx B Ln ( x ) = n − ∑ m = (cid:18) nm (cid:19) ( n − m ) ! B Lm ( x ) , ( n ≥ ) . Therefore, by (35), we obtain the following theorem.
Theorem 10.
For n ≥ , we haveddx B Ln ( x ) = n − ∑ m = (cid:18) nm (cid:19) ( n − m ) ! B Lm ( x ) . For n ≥
0, the bivariate Bell polynomials are defined by(36) B n ( x , y ) = n ∑ k = S ( n , k )( x ) k y k , ( see [ ]) . Letting y → y / x and then x → ∞ , we see that the bivariate Bell polynomial B n ( x , y ) reduces to theunivariate Bell polynomial B n ( y ) .From (36), we note that(37) ∞ ∑ n = B n ( x , y ) t n n ! = (cid:0) + y ( e t − ) (cid:1) x , ( see [ ]) . In view of (36), we define bivariate Lah-Bell polynomials by(38) B Ln ( x , y ) = n ∑ k = L ( n , k )( x ) k y k , ( n ≥ ) . From, (38), we note that ∞ ∑ n = B Ln ( x , y ) t n n ! = ∞ ∑ n = (cid:18) n ∑ k = L ( n , k )( x ) k y k (cid:19) t n n !(39) = ∞ ∑ k = ( x ) k y k ∞ ∑ n = k L ( n , k ) t n n ! = ∞ ∑ k = (cid:18) xk (cid:19) y k (cid:18) − t − (cid:19) k = (cid:18) + y (cid:18) − t − (cid:19)(cid:19) x . Therefore, by (39), we obtain the following lemma.
Lemma 11.
The generating function of bivariate Lah-Bell polynomais is given by (cid:18) + y (cid:18) − t − (cid:19)(cid:19) x = ∞ ∑ n = B Ln ( x , y ) t n n ! . Remark 12.
Letting y → y / x and x → ∞ , we note the bivariate Lah-Bell polynomial B Ln ( x , y ) re-duces to the univariate Lah-Bell polynomial B Ln ( y ) , ( n ≥ ) . DAE SAN KIM AND TAEKYUN KIM
Replacing t by 1 − e − t in Lemma 11, we get (cid:0) + y ( e t − ) (cid:1) x = ∞ ∑ k = B Lk ( x , y ) k ! (cid:0) − e − t (cid:1) k = ∞ ∑ k = ( − ) k B Lk ( x , y ) ∞ ∑ n = k S ( n , k ) ( − t ) n n !(40) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B Lk ( x , y ) (cid:19) t n n ! . Replacing t by − log ( − t ) in (37), we get (cid:18) + y (cid:18) − t − (cid:19)(cid:19) x = ∞ ∑ k = B k ( x , y )( − ) k k ! (cid:0) log ( − t ) (cid:1) k (41) = ∞ ∑ k = ( − ) k B k ( x , y ) ∞ ∑ n = k ( − ) n S ( n , k ) t n n ! = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B k ( x , y ) (cid:19) t n n ! . Therefore, by (37), Lemma 11, (40) and (41), we obtain the following theorem.
Theorem 13.
For n ≥ , we haveB Ln ( x , y ) = n ∑ k = ( − ) n − k S ( n , k ) B k ( x , y ) , and B n ( x , y ) = n ∑ k = ( − ) n − k S ( n , k ) B Lk ( x , y ) . For any 0 = λ ∈ R , the degenerate exponential functions are defined by e x λ ( t ) = ( + λ t ) x λ = ∞ ∑ n = ( x ) n , λ t n n ! , ( see [ , ]) , where ( x ) , λ = , ( x ) n , λ = x ( x − λ ) · · · ( x − ( n − ) λ ) , ( n ≥ ) .When x = e λ ( t ) = e λ ( t ) = ∑ ∞ n = ( ) n , λ n ! t n . Note that lim λ → e λ ( t ) = e t .Now, we define the degenerate Lah-Bell polynomials by(42) e x λ (cid:18) − t − (cid:19) = ∞ ∑ n = B Ln , λ ( x ) t n n ! . When x = B Ln , λ = B Ln , λ ( ) are called the degenerate Lah-Bell numbers.From (42), we note that ∞ ∑ n = B Ln , λ ( x ) t n n ! = ∞ ∑ k = ( x ) k , λ k ! (cid:18) − t − (cid:19) k = ∞ ∑ k = ( x ) k , λ ∞ ∑ n = k L ( n , k ) t n n !(43) = ∞ ∑ n = (cid:18) n ∑ k = L ( n , k )( x ) k , λ (cid:19) t n n ! . AH-BELL NUMBERS AND POLYNOMIALS 9
By comparing the coefficients on both sides of (43), we get(44) B Ln , λ ( x ) = n ∑ k = L ( n , k )( x ) k , λ , ( n ≥ ) . It is known that the degenerate Bell polynomials are given by(45) e x λ (cid:0) e t − (cid:1) = ∞ ∑ n = B n , λ ( x ) t n n ! , ( see [ ]) . Replacing t by 1 − e − t in (42), we get e x λ (cid:0) e t − (cid:1) = ∞ ∑ k = B Lk , λ ( x ) k ! (cid:0) − e − t (cid:1) k (46) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B Lk , λ ( x ) (cid:19) t n n ! . From (45) and (46), we note that(47) B n , λ ( x ) = n ∑ k = ( − ) n − k S ( n , k ) B Lk , λ ( x ) , ( n ≥ ) . Replacing t by − log ( − t ) in (45), we have e x λ (cid:18) t − t − (cid:19) = ∞ ∑ k = ( − ) k B k , λ ( x ) k ! (cid:0) log ( − t ) (cid:1) k (48) = ∞ ∑ n = (cid:18) n ∑ k = ( − ) n − k S ( n , k ) B k , λ ( x ) (cid:19) t n n ! . Thus, we have B Ln , λ ( x ) = n ∑ k = ( − ) n − k S ( n , k ) B k , λ ( x ) , ( n ≥ ) . Recall that the Laguerre polynomials L ( α ) n ( x ) of order α , ( α > − ) , are given by (see [12])(49) ( − t ) − α − e x tt − = ∞ ∑ n = L ( α ) n ( x ) t n n ! . By Lemma 4 and (49), ( − t ) − α − = ∞ ∑ m = B Lm ( x ) t m m ! ∞ ∑ l = L ( α ) l ( x ) t l l ! = ∞ ∑ n = (cid:18) n ∑ m = (cid:18) nm (cid:19) B Lm ( x ) L ( α ) n − m ( x ) (cid:19) t n n ! , which shows that h α + i n = n ∑ m = (cid:18) nm (cid:19) B Lm ( x ) L ( α ) n − m ( x ) . CONCLUSION
Taking into account the relationship between the Stirling numbers of the second kind and the Bellnumbers and in light of the combinatorial meaning of the unsigned Lah numbers, we introducedthe Lah-Bell numbers and their natural extensions, namely the Lah-Bell polynomials. We derivedsome basic properties of such numbers and polynomials by using elementary methods. We alsoconsidered the degenerate Lah-Bell numbers and polynomials as degenerate versions of the Lah-Bell numbers and polynomials.In more detail, for the Lah-Bell numbers and polynomials we derived the generating functions,some relations with Bell numbers and polynomials, and Dobinski-like formulas. We showed theconnections between Lah numbers and Stirling numbers, recurrence relations and derivatives forthe Lah-Bell polynomials. We introduced the bivariate Lah-Bell polynomials, and found the gen-erating function of them and their connection with the bivariate Bell polynomials. In addition, asdegenerate versions of the Lah-Bell polynomials and numbers, we introduced the degenerate Lah-Bell polynomials and numbers. Then, for the degenerate Lah-Bell polynomials we deduced thegenerating function, an explicit expression and their relation with the degenerate Bell polynomials.In recent years, one of our research areas of study has been to explore some special numbersand polynomials and their degenerate versions, and to discover their arithmetical and combinatorialproperties and some of their applications. We would like to continue to work on these by exploitingvarious means like generating functions, combinatorial methods, p -adic analysis, umbral calculus,differential equations and probability theory.R EFERENCES [1] Carlitz, L.
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OGANG U NIVERSITY , S
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WANGWOON U NIVERSITY , S
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EPUBLIC OF K OREA
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