A Note on Degenerate Multi-poly-Bernoulli numbers and polynomials
aa r X i v : . [ m a t h . N T ] M a y A NOTE ON DEGENERATE MULTI-POLY-BERNOULLI NUMBERS ANDPOLYNOMIALS
TAEKYUN KIM AND DAESAN KIMA
BSTRACT . In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polyno-mials which are defined by means of the multiple polylogarithms and degenerate versions of themulti-poly-Bernoulli numbers and polynomials. We investigate some properties for those numbersand polynomials. In addition, we give some identities and relations for the degenerate multi-poly-Bernoulli numbers and polynomials.
1. I
NTRODUCTION
For 0 = λ ∈ R , Carlitz considered the higher-order degenerate Bernoulli polynomials given by(1) (cid:18) t ( + λ t ) λ (cid:19) r ( + λ t ) x λ = ∞ ∑ n = β ( r ) n , λ ( x ) t n n ! , ( see [ ]) . When x = β ( r ) n , λ = β ( r ) n , λ ( ) are called the higher-order degenerate Bernoulli numbers.For k ∈ Z , the polylogarithm function is defined by(2) Li k ( x ) = ∞ ∑ n = x n n k , ( see [ , , ]) . Note that Li ( x ) = ∞ ∑ n = x n n ! = − log ( − x ) .As is known, the poly-Bernoulli polynomials are defined by(3) Li k ( − e − t ) e t − e xt = ∞ ∑ n = PB ( k ) n ( x ) t n n ! , ( see [ ]) . When x = PB ( k ) n = PB ( k ) n ( ) are called the poly-Bernoulli numbers. Note that PB ( ) n = B n ( x ) ,where B n ( x ) are ordinary Bernoulli polynomials given by ∞ ∑ n = B n ( x ) t n n ! = te t − e xt , ( see [ − ]) . The polyexponential function was introduced by Hardy. Kim-Kim recently introduced a modifiedversion of that, called the modified polyexponential function, which is given by(4) Ei k ( x ) = ∞ ∑ n = x n ( n − ) ! n k , ( see [ , , − ]) . Note that Ei ( x ) = e x − k (cid:0) log ( + t ) (cid:1) e t − e xt = ∞ ∑ n = B ( k ) n ( x ) t n n ! , ( see [ ]) . Mathematics Subject Classification.
Key words and phrases. degenerate multi-poly-Bernoulli polynomials; multiple poly-logarithm.
Note that B ( ) n ( x ) = B n ( x ) , ( n ≥ ) .The degenerate exponential functions are defined by(6) e x λ ( t ) = ( + λ t ) x λ , e λ ( t ) = e λ ( t ) = ( + λ t ) λ , ( see [ ]) . Here we observe that(7) e x λ ( t ) = ∞ ∑ n = ( x ) n , λ t n n ! , ( see [ ]) , where ( x ) , λ = ( x ) n , λ = x ( x − λ ) · · · ( x − ( n − ) λ , ( n ≥ ) .It is well known that the Stirling numbers of the second kind are defined by(8) 1 k ! ( e t − ) k = ∞ ∑ n = k S ( n , k ) t n n ! , ( n ≥ ) , ( see [ , , , , ]) . For k , k , . . . , k r ∈ Z , the multiple polylogarithm is defined by(9) Li k , k ,..., k r ( x ) = ∑ < n < n < ··· < n r x n r n k n k · · · n k r r , ( see [ ]) , where the sum is over all integers n , n , . . . , n r satisfying 0 < n < n < · · · < n r .About twenty years ago, the first author introduced the generalized Bernoulli numbers B ( k , k ,..., k r ) n of order r (see [10]) which are given by(10) r !Li k , k ,..., k r ( − e − t )( e t − ) r = ∞ ∑ n = B ( k , k ,..., k r ) n t n n ! . Actually, the r ! in (10) does not appear in [10]. However, the present definition is more conve-nient, since B ( , ,..., ) n = B ( r ) n are the Bernoulli numbers of order r (see (15)). These numbers wouldhave been called the multi-poly-Bernoulli numbers, since it is a multiple version of poly-Bernoullinumbers (see (3)). Furthermore, we may consider the multi-poly-Bernoulli polynomials, which arenatural extensions of the multi-poly-Bernoulli numbers, given by(11) r !Li k , k ,..., k r ( − e − t )( e t − ) r e xt = ∞ ∑ n = B ( k , k ,..., k r ) n ( x ) t n n ! . The multi-poly-Bernoulli polynomials are multiple versions of the poly-Bernoulli polynomials in(3). We let the interested reader refer to [1] for the detailed properties on those polynomials.In this paper, we consider the degenerate multi-poly-Bernoulli numbers and polynomials (see(16)) which are defined by means of the multiple polylogarithms and degenerate versions of themulti-poly-Bernoulli numbers and polynomials studied earlier in the literature (see [2]). We inves-tigate some properties for those numbers and polynomials. In addition, we give some identities andrelations for the degenerate multi-poly-Bernoulli numbers and polynomials.2. D
EGENERATE MULTI - POLY -B ERNOULLI NUMBERS AND POLYNOMIALS
From (9), we note that ddx Li k , k ,..., k r ( x ) = ddx ∑ < n < n < ··· < n r x n r n k n k · · · n k r r (12) = x ∑ < n < n < ··· < n r x n r n k n k · · · n k r − r = x Li k , k ,..., k r − ( x ) . NOTE ON DEGENERATE MULTI-POLY-BERNOULLI NUMBERS AND POLYNOMIALS 3
Let us take k r =
1. Then we have ddx Li k ,..., k r − , ( x ) = ∑ < n < n < ··· < n r − n k n k · · · n k r − r − ∞ ∑ n r = n r − + x n r − . (13) = − x ∑ < n < n < ··· < n r − x n r − n k n k · · · n k r − r − = − x Li k , k ,..., k r − ( x ) Thus, by (13), we get(14) Li k , ( x ) = Z − x Li k ( x ) dx . By integration by parts, from (14), we note thatLi , ( x ) = Z − x ( − log ( − x )) dx = (cid:0) − log ( − x ) (cid:1) . By induction, we get Li1 , , . . . , | {z } r − times ( x ) = ( − ) r r ! (cid:0) log ( − x ) (cid:1) r , ( r ∈ N ) , (15) = ∞ ∑ l = r S ( l , r )( − ) l − r t l l ! = ∞ ∑ l = r | S ( l , r ) | t l l !where S ( l , r ) (respectively, | S ( l , r ) | ) are the signed (respectively, unsigned) Stirling numbers ofthe first kind.Now, we consider the degenerate multi-poly-Bernoulli polynomials which are degenerate versionsof the multi-poly-Bernoulli polynomials in (11) and given by(16) r !Li k , k ,..., k r ( − e − t ) (cid:0) e λ ( t ) − (cid:1) r e x λ ( t ) = ∞ ∑ n = β ( k , k ,..., k r ) n , λ ( x ) t n n ! . When x = β ( k , k ,..., k r ) n , λ = β ( k , k ,..., k r ) n , λ ( ) are called the degenerate multi-poly-Bernoulli numbers.From (16), we note that ∞ ∑ n = β ( k , k ,..., k r ) n , λ ( x ) t n n ! = r !Li k , k ,..., k r ( − e − t ) (cid:0) e λ ( t ) − (cid:1) r e x λ ( t )= ∞ ∑ l = β ( k , k ,..., k r ) l , λ t l l ! ∞ ∑ m = ( x ) m , λ t m m !(17) = ∞ ∑ n = (cid:18) n ∑ l = (cid:18) nl (cid:19) ( x ) n − l , λ β ( k , k ,..., k r ) l , λ (cid:19) t n n ! . Thus, by (17), we get(18) β ( k , k ,..., k r ) n , λ ( x ) = n ∑ l = (cid:18) nl (cid:19) ( x ) n − l , λ β ( k , k ,..., k r ) l , λ , ( n ≥ ) . TAEKYUN KIM AND DAESAN KIM
From (1), (15) and (16), we note that(19) β r − times z }| { ( , , . . . , ) n , λ ( x ) = β ( r ) n , λ ( x ) , ( n ≥ ) . Proposition 1.
For n ≥ , we have β ( k , k ,..., k r ) n , λ ( x ) = n ∑ l = (cid:18) nl (cid:19) ( x ) n − l , λ β ( k , k ,..., k r ) l , λ , β r − times z }| { ( , , . . . , ) n , λ ( x ) = β ( r ) n , λ ( x ) . From (16), we note that ∞ ∑ n = β ( k ,..., k r ) n , λ ( x ) t n n ! = r ! (cid:0) e λ ( t ) − (cid:1) r Li k ,..., k r (cid:0) − e − t (cid:1) e x λ ( t ) (20) = r ! (cid:0) e λ ( t ) − (cid:1) r ∑ < n < ··· < n r − n k · · · n k r − r − ∞ ∑ n r = n r − + (cid:0) − e − t (cid:1) n r n k r r e x λ ( t )= r ! (cid:0) e λ ( t ) − (cid:1) r ∑ < n < ··· < n r − (cid:0) − e − t (cid:1) n r − n k · · · n k r − r − ∞ ∑ n r = (cid:0) − e − t (cid:1) n r ( n r + n r − ) k r e x λ ( t )= r ! e x λ ( t ) (cid:0) e λ ( t ) − (cid:1) r ∑ < n < ··· < n r − (cid:0) − e − t (cid:1) n r − n k · · · n k r − r − ∞ ∑ n r = n r ! ( n r + n r − ) k r ∞ ∑ l = n r ( − ) l − n r S ( l , n r ) t l l !To proceed further, we observe that, for any integer k , we have(21) ( x + y ) − k = ∞ ∑ m = ( − ) m (cid:18) k + m − m (cid:19) x − k − m y m . Now, from (20) and (21), we have ∞ ∑ n = β ( k ,..., k r ) n , λ ( x ) t n n ! = rte λ ( t ) − (cid:18) ( r − ) ! e x λ ( t ) (cid:0) e λ ( t ) − (cid:1) r − ∑ < n < ··· < n r − (cid:0) − e − t (cid:1) n r − n k · · · n k r − − mr − (cid:19) × t ∞ ∑ l = (cid:18) l ∑ n r = n r ! ( − ) l − n r S ( l , n r ) ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m n − k r − mr (cid:19) t l l ! = rte λ ( t ) − ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m ∞ ∑ j = β ( k ,..., k r − − m ) j , λ ( x ) t j j ! × ∞ ∑ l = (cid:18) l + ∑ n r = n r ! ( − ) l − n r − S ( l + , n r ) l + n − k r − mr (cid:19) t l l ! = rte λ ( t ) − ∞ ∑ k = (cid:18) k ∑ l = l + ∑ n r = ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m (cid:18) kl (cid:19) × n r ! ( − ) l − n r − S ( l + , n r ) l + n − k r − mr β ( k , k ,..., k r − − m ) k − l , λ ( x ) (cid:19) t k k ! NOTE ON DEGENERATE MULTI-POLY-BERNOULLI NUMBERS AND POLYNOMIALS 5 = r ∞ ∑ p = β p , λ t p p ! ∞ ∑ k = (cid:18) k ∑ l = l + ∑ n r = ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m × (cid:18) kl (cid:19) n r ! ( − ) l − n r − S ( l + , n r ) l + n − k r − mr β ( k , k ,..., k r − − m ) k − l , λ (cid:19) t k k ! = ∞ ∑ n = (cid:18) n ∑ k = k ∑ l = l + ∑ n r = ∞ ∑ m = r (cid:18) k r + m − m (cid:19)(cid:18) kl (cid:19)(cid:18) nk (cid:19) ( − ) m × n r ! ( − ) l − n r − S ( l + , n r ) l + n − k r − mr β ( k , k ,..., k r − − m ) k − l , λ ( x ) β n − k , λ (cid:19) t n n ! , (22)where β n , λ are the Carlitz’s degenerate Bernoulli numbers with β ( ) n , λ = β n , λ . Therefore, by (22), weobtain the following theorem. Theorem 2.
For k , k , . . . , k r ∈ Z and n ≥ , we have β ( k , k ,..., k r ) n , λ ( x ) = r n ∑ k = k ∑ l = l + ∑ n r = ∞ ∑ m = (cid:18) k r + m − m (cid:19)(cid:18) kl (cid:19)(cid:18) nk (cid:19) ( − ) m × n r ! ( − ) l − n r − S ( l + , n r ) l + n − k r − mr β n − k , λ β ( k , k ,..., k r − − m ) k − l , λ ( x ) . (23)Replacing k r by − k r in (23), we obtain the following corollary. Corollary 3.
For k , k , . . . , k r ∈ Z and n ≥ , we have β ( k , k ,..., − k r ) n , λ ( x ) = r n ∑ k = k ∑ l = l + ∑ n r = ∞ ∑ m = (cid:18) k r m (cid:19)(cid:18) kl (cid:19)(cid:18) nk (cid:19) × n r ! ( − ) l − n r − S ( l + , n r ) l + n k r − mr β n − k , λ β ( k , k ,..., k r − − m ) k − l , λ ( x ) . From (16), we have ∞ ∑ n = (cid:18) β ( k ,..., k r ) n , λ ( x + ) − β ( k ,..., k r ) n , λ ( x ) (cid:19) t n n ! = r ! e x λ ( t ) (cid:0) e λ ( t ) − (cid:1) r − Li k ,..., k r (cid:0) − e − t (cid:1) . = r ! e x λ ( t ) (cid:0) e λ ( t ) − (cid:1) r − ∑ < n < ··· < n r − n k · · · n k r − r − ∞ ∑ n r = n r − + (cid:0) − e − t (cid:1) n r n k r r = r ! (cid:0) e λ ( t ) − (cid:1) r − e x λ ( t ) ∑ < n < ··· < n r − (cid:0) − e − t (cid:1) n r − n k · · · n k r − r − ∞ ∑ n r = (cid:0) − e − t (cid:1) n r ( n r + n r − ) k r = r ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m ( r − ) ! e x λ ( t ) (cid:0) e λ ( t ) − (cid:1) r − Li k ,..., k r − − m (cid:0) − e − t (cid:1) × ∞ ∑ n r = n − k r − mr n r ! ∞ ∑ l = n r ( − ) l − n r S ( l , n r ) t l l ! = r ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m ∞ ∑ j = β ( k ,..., k r − , k r − − m ) j , λ ( x ) t j j ! × ∞ ∑ l = (cid:18) l ∑ n r = n − k r − mr n r ! ( − ) l − n r S ( l , n r ) (cid:19) t l l ! TAEKYUN KIM AND DAESAN KIM = ∞ ∑ n = (cid:18) r ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m n ∑ l = l ∑ n r = (cid:18) nl (cid:19) n − k r − mr n r ! × ( − ) l − n r S ( l , n r ) β ( k , k ,..., k r − , k r − − m ) n − l , λ ( x ) (cid:19) t n n ! . Therefore, we obtain the following theorem.
Theorem 4.
For n , r ≥ and k , k , . . . , k r ∈ Z , we have r (cid:18) β ( k ,..., k r ) n , λ ( x + ) − β ( k ,..., k r ) n , λ ( x ) (cid:19) = ∞ ∑ m = (cid:18) k r + m − m (cid:19) ( − ) m n ∑ l = l ∑ n r = (cid:18) nl (cid:19) n − k r − mr × n r ! ( − ) l − n r S ( l , n r ) β ( k ,..., k r − , k r − − m ) n − l , λ ( x ) . By the definition of degenerate multi-poly-Bernoulli polynomials, we get ∞ ∑ n = β ( k , k ,..., k r ) n , λ ( x + y ) t n n ! = r ! (cid:0) e λ ( t ) − (cid:1) r Li k , k ,..., k r ( − e − t ) e x + y λ ( t ) . = r ! (cid:0) e λ ( t ) − (cid:1) r Li k , k ,..., k r ( − e − t ) e x λ ( t ) e y λ ( t )= ∞ ∑ l = β ( k , k ,..., k r ) l , λ ( x ) t l l ! ∞ ∑ m = ( y ) m , λ t m m ! = ∞ ∑ n = (cid:18) n ∑ l = (cid:18) nl (cid:19) β ( k ,..., k r ) l , λ ( x )( y ) n − l , λ (cid:19) t n n ! . Thus, we note that β ( k ,..., k r ) n , λ ( x + y ) = n ∑ l = (cid:18) nl (cid:19) β ( k ,..., k r ) l , λ ( x )( y ) n − l , λ , ( n ≥ ) . Finally, we note that β ( k ,..., k r ) n , λ ( x ) is not a Sheffer sequence.3. C ONCLUSION
In [3], Carlitz initiated study of degenerate versions of Bernoulli and Euler polynomials, namelythe degenerate Bernoulli and Euler polynomials. In recent years, some mathematicians intensivelystudied various versions of many special numbers and polynomials and quite a few interesting re-sults were found about them (see [6,7,9,11-16,18,20]). Here we would like to mention that study ofdegenerate versions is not limited only to polynomials but can be extended also to transcendentalfunctions like gamma functions (see [12]).In this paper, we considered the degenerate multi-poly-Bernoulli numbers and polynomials whichare defined by means of the multiple polylogarithms. They are degenerate versions of the multi-poly-Bernoulli numbers and polynomials, and multiple versions of the degenerate poly-Bernoullinumbers and polynomials ( r = NOTE ON DEGENERATE MULTI-POLY-BERNOULLI NUMBERS AND POLYNOMIALS 7 details. It is one of our future projects to continue this line of research and find applications notonly in mathematics but also in science and engineering.R
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WANGWOON U NIVERSITY , S
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