Representations of degenerate poly-Bernoulli polynomials
aa r X i v : . [ m a t h . N T ] D ec REPRESENTATIONS OF DEGENERATE POLY-BERNOULLI POLYNOMIALS TAEKYUN KIM, DAE SAN KIM, JONGKYUM KWON, AND HYUNSEOK LEEA
BSTRACT . As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithmfunctions. Recently, as degenerate versions of such functions and polynomials, degenerate polyloga-rithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by meansof the degenerate polylogarithm functions, and some of their properties were investigated. The aim ofthis paper is to further study some properties of the degenerate poly-Bernoulli polynomials by usingthree formulas coming from the recently developed ‘ λ -umbral calculus.’ In more detail, among otherthings, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoullipolynomials and by higher-order degenerate derangement polynomials.
1. I
NTRODUCTION
Carlitz is the first one who initiated the study of degenerate versions of some special numbersand polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers (see [1]). Inrecent years, studying degenerate versions of some special numbers and polynomials regained inter-ests of some mathematicians with their interests not only in combinatorial and arithmetic propertiesbut also in applications to differential equations, identities of symmetry and probability theory (see[5,6,9,10,13,15,16] and the references therein). It is noteworthy that studying degenerate versionsis not only limited to polynomials but also can be extended to transcendental functions like gammafunctions (see [8]).The Rota’s theory of umbral calculus is based on linear functionals and differential operators(see [2-4,17-21]). The Sheffer sequences occupy the central position in the theory and are char-acterized by the generating functions where the usual exponential function enters. The motivationfor the paper [6] starts from the question that what if the usual exponential function is replacedby the degenerate exponential functions (see (2)). As it turns out, it corresponds to replacing thelinear functional by the family of λ -linear functionals (see (12)) and the differential operator by thefamily of λ -differential operators (see (14)). Indeed, these replacements lead us to define λ -Shefferpolynomials and degenerate Sheffer polynomials (see (16)).As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions.Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm func-tions were introduced and degenerate poly-Bernoulli polynomials were defined by means of thedegenerate polylogarithm functions, and some properties of the degenerate poly-Bernoulli polyno-mials were investigated (see [13]).The aim of this paper is to further study the degenerate poly-Bernoulli polynomials, which isa λ -Sheffer sequence and hence a degenerate Sheffer sequence, by using the above-mentioned λ -linear functionals and λ -differential operators. In more detail, these polynomials are investigatedby three different tools, namely a formula about representing a λ -Sheffer sequence by another (see(19)), a formula coming from the generating functions of λ -Sheffer sequences (see Theorem 1) anda formula arising from the definitions for λ -Sheffer sequences (see Theorems 6,7). Then, among Mathematics Subject Classification.
Key words and phrases. degenerate poly-Bernoulli polynomials; degenerate derangement polynomials; λ -umbralcalculus. TAEKYUN KIM, DAE SAN KIM, JONGKYUM KWON, AND HYUNSEOK LEE other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerateBernoulli polynomials and by higher-order degenerate derangement polynomials. The rest of thissection is devoted to recalling the necessary facts that are needed throughout the paper, whichincludes the ‘ λ -umbral calculus’.For k ∈ Z , and 0 = λ ∈ R , the degenerate polylogarithm functions are defined by(1) Li k , λ ( x ) = ∞ ∑ n = ( − λ ) n − ( ) n , / λ ( n − ) ! n k x n , ( see [ ]) , where ( x ) , λ = , ( x ) n , λ = x ( x − λ ) · · · ( x − ( n − ) λ ) , ( n ≥ ) .For any λ ∈ R , the degenerate exponential functions are given by(2) e x λ ( t ) = n ∑ l = ( x ) n , λ n ! t n , e λ ( t ) = e λ ( t ) = ∞ ∑ n = ( ) n , λ n ! t n , ( see [ ]) . The compositional inverse log λ ( t ) of e λ ( t ) is given by(3) log λ ( t ) = ∞ ∑ n = λ n − n ! ( ) n , / λ ( t − ) n , ( see [ ]) . From (1) and (3), we have(4) Li , λ ( x ) = − log λ ( − x ) , and lim λ → Li k , λ ( x ) = Li k ( x ) , where Li k ( x ) are the polylogarithm functions defined byLi k ( x ) = ∞ ∑ n = x n n k , ( see [ , ]) . In [5], the degenerate poly-Bernoulli polynomials are defined by Kim-Kim as(5) Li k , λ ( − e λ ( − t )) e λ ( t ) − e x λ ( t ) = ∞ ∑ n = B ( k ) n , λ ( x ) t n n ! . When x = B ( k ) n , λ = B ( k ) n , λ ( ) are called the degenerate poly-Bernoulli numbers.It is well known that Carlitz’s degenerate Bernoulli polynomials of order r are defined by(6) (cid:18) te λ ( t ) − (cid:19) r e x λ ( t ) = ∞ ∑ n = β ( r ) n , λ ( x ) t n n ! , ( see [ ]) . For r = β ( r ) n , λ ( x ) = β n , λ ( x ) are called the degenerate Bernoulli polynomials.From (5), we note that(7) ∞ ∑ n = B ( ) n , λ ( x ) t n n ! = Li , λ ( − e λ ( − t )) e λ ( t ) − = te λ ( t ) − e x λ ( t ) = ∞ ∑ n = β n , λ ( x ) t n n ! . By (7), we get B ( ) n , λ ( x ) = β n , λ ( x ) , ( n ≥ ) , (see [7]).The degenerate Stirling numbers of the second kind appear as the coefficients in the expansion(8) ( x ) n , λ = n ∑ l = S , λ ( n , l )( x ) l , ( n ≥ ) , ( see [ ]) . As the inversion formula of (8), the degenerate Stirling numbers of first kind appear as the coeffi-cients in the expansion(9) ( x ) n = n ∑ l = S , λ ( n , l )( x ) l , λ , ( n ≥ ) , ( see [ ]) . EPRESENTATIONS OF DEGENERATE POLY-BERNOULLI POLYNOMIALS 3
Thus, by (8) and (9), we have(10) 1 k ! (cid:0) e λ ( t ) − (cid:1) k = ∞ ∑ n = k S , λ ( n , k ) t n n ! , and 1 k ! (cid:0) log λ ( + t ) (cid:1) k = ∞ ∑ n = k S , λ ( n , k ) t n n ! , ( k ≥ ) , ( see [ , , , ]) . In view of (6), the degenerate derangement polynomials of order r ( ∈ N ) are defined by(11) 1 ( − t ) r e − λ ( t ) e x λ ( t ) = ∞ ∑ n = d ( r ) n , λ ( x ) t n n ! , ( see [ ]) . When r = d n , λ ( x ) = d ( ) λ ( x ) are called the degenerate derangement polynomials.Note that lim λ → d n , λ ( x ) = d n ( x ) , where d n ( x ) are the derangement polynomials and d n = d n ( ) thederangement numbers (see [12,13,14,15]).We remark that the umbral calculus has long been studied by many people (see [2,3,4,6,18-20]). For the rest of this section, we will recall the necessary facts on the λ -linear functionals, λ -differential operators and λ -Sheffer sequences, and so on. The details on these can be found inthe recent paper [6].Let C be the field of complex numbers, F = (cid:26) f ( t ) = ∞ ∑ k = a k t k k ! (cid:12)(cid:12)(cid:12)(cid:12) a k ∈ C (cid:27) , and let P = C [ x ] = (cid:26) ∞ ∑ i = a i x i (cid:12)(cid:12)(cid:12)(cid:12) a i ∈ C with a i = i (cid:27) . For f ( t ) ∈ F with f ( t ) = ∞ ∑ k = a k t k k ! , and λ ∈ R , the λ -linear functional h f ( t ) |·i λ on P is defined by(12) h f ( t ) | ( x ) n , λ i λ = a n , ( n ≥ ) , ( see [ ]) . By (12), we get(13) h t k | ( x ) n , λ i = n ! δ n , k , ( n , k ≥ ) , ( see [ ]) , where δ n , k is the Kronecker’s symbol.The λ -differential operators on P are defined by(14) ( t k ) λ ( x ) n , λ = (cid:26) ( n ) k ( x ) n − k , λ , if 0 ≤ k ≤ n , , if k > n .For f ( t ) = ∞ ∑ k = a k t k k ! ∈ F , and by (14), we get (cid:0) f ( t ) (cid:1) λ ( x ) n , λ = n ∑ k = (cid:18) nk (cid:19) a k ( x ) n − k , λ , ( n ≥ ) , (15) (cid:0) e y λ ( t ) (cid:1) λ ( x ) n , λ = ( x + y ) n , λ , ( n ≥ ) , ( see [ ]) . Let f ( t ) be a delta series and let g ( t ) be an invertible series. Then there exists a unique sequence S n , λ ( x ) ( deg S n , λ ( x ) = n ) of polynomials satisfying the orthogonality conditions(16) D g ( t ) (cid:0) f ( t ) (cid:1) k (cid:12)(cid:12)(cid:12) S n , λ ( x ) E λ = n ! δ n , k , ( n , k ≥ ) , ( see [ ]) . TAEKYUN KIM, DAE SAN KIM, JONGKYUM KWON, AND HYUNSEOK LEE
Here S n , λ ( x ) is called the λ -Sheffer sequence for (cid:0) g ( t ) , f ( t ) (cid:1) , which is denoted by S n , λ ( x ) ∼ (cid:0) g ( t ) , f ( t ) (cid:1) λ . The sequence S n , λ ( x ) is the λ -Sheffer sequence for ( g ( t ) , f ( t )) if and only if(17) 1 g ( f ( t )) e y λ (cid:0) f ( t ) (cid:1) = ∞ ∑ n = S n , λ ( y ) t n n ! , ( see [ ]) , for all y ∈ C , where f ( t ) is the compositional inverse of f ( t ) such that f ( f ( t )) = f ( f ( t )) = t .Let S n , λ ( x ) ∼ (cid:0) g ( t ) , f ( t ) (cid:1) λ . Then, from Theorem 16 of [6], we recall that(18) ( f ( t )) λ S n , λ ( x ) = nS n − , λ ( x ) , ( n ≥ ) . For S n , λ ( x ) ∼ ( g ( t ) , f ( t )) λ , r n , λ ( x ) ∼ ( h ( t ) , l ( t )) , we have(19) S n , λ ( x ) = m ∑ k = C n , k r k , λ ( x ) , ( n ≥ ) , ( see [ ]) , where C n , k = k ! (cid:28) h ( f ( t )) g ( f ( t )) (cid:0) l ( f ( t )) (cid:1) k (cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ (cid:29) λ . In this paper, we study the properties of degenerate poly-Bernoulli polynomial arising from de-generate polylogarithmic function and give some identities of those polynomials associated withspecial polynomials which are derived from the properties of λ -Sheffer sequences.2. R EPRESENTATIONS OF DEGENERATE POLY -B ERNOULLI POLYNOMIALS
For S n , λ ( x ) ∼ ( g ( t ) , f ( t )) λ , ( n ≥ ) , we have * g ( f ( t )) e x λ (cid:0) f ( t ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ = ∞ ∑ k = S k , λ ( x ) k ! (cid:10) t k | ( x ) n , λ (cid:11) λ (20) = S n , λ ( x ) , ( n ≥ ) . From (20), we note that S n , λ ( x ) = * g ( f ( t )) e x λ (cid:0) f ( t ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ = ∞ ∑ j = j ! * g ( f ( t ) (cid:0) f ( t ) (cid:1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ ( x ) j , λ (21) = n ∑ j = j ! * g ( f ( t ) (cid:0) f ( t ) (cid:1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ ( x ) j , λ . Therefore, by (21), we obtain the following theorem.
Theorem 1.
For S n , λ ( x ) ∼ ( g ( t ) , f ( t )) λ , we haveS n , λ ( x ) = n ∑ j = j ! * g ( f ( t ) (cid:0) f ( t ) (cid:1) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ ( x ) j , λ . From (5) and (17), we have(22) B ( k ) n , λ ( x ) ∼ (cid:18) e λ ( t ) − k , λ ( − e λ ( − t )) , t (cid:19) λ . By Theorem 1, we get the following corollary.
Corollary 2.
For n ≥ , we have B ( k ) n , λ ( x ) = n ∑ j = (cid:18) nj (cid:19) B ( k ) n − j , λ ( x ) j , λ . EPRESENTATIONS OF DEGENERATE POLY-BERNOULLI POLYNOMIALS 5
From (22), we note that(23) (cid:18) e λ ( t ) − k , λ ( − e λ ( − t )) (cid:19) λ B ( k ) n , λ ( x ) = ( x ) n , λ ∼ ( , t ) λ . By (23), and noting that ( x ) n , λ = ∑ nl = ∑ lj = S , λ ( n , l ) S , λ ( l , j )( x ) j , λ , we get B ( k ) n , λ ( x ) = (cid:18) Li k , λ ( − e λ ( − t )) e λ ( t ) − (cid:19) λ ( x ) n , λ (24) = n ∑ l = l ∑ j = S , λ ( n , l ) S , λ ( l , j ) (cid:18) Li k , λ ( − e λ ( − t )) e λ ( t ) − (cid:19) λ ( x ) j , λ = n ∑ l = l ∑ j = S , λ ( n , l ) S , λ ( l , j ) j ∑ m = B ( k ) m , λ m ! ( t m ) λ ( x ) j , λ = n ∑ l = l ∑ j = j ∑ m = (cid:18) jm (cid:19) S , λ ( n , l ) S , λ ( l , j ) B ( k ) m , λ ( x ) j − m , λ = n ∑ l = l ∑ m = l ∑ j = m (cid:18) jm (cid:19) S , λ ( n , l ) S , λ ( l , j ) B ( k ) m , λ ( x ) j − m , λ = n ∑ l = l ∑ m = l − m ∑ j = (cid:18) j + mm (cid:19) S , λ ( n , l ) S , λ ( l , j + m ) B ( k ) m , λ ( x ) j , λ . Therefore, by (24), we obtain the following theorem.
Theorem 3.
For n ≥ , we haveB ( k ) n , λ ( x ) = n ∑ l = l ∑ m = l − m ∑ j = (cid:18) j + mm (cid:19) S , λ ( n , l ) S , λ ( l , j + m ) B ( k ) m , λ ( x ) j , λ . Now, we observe that B ( k ) n , λ ( y ) = * Li k , λ ( − e λ ( − t )) e λ ( t ) − e y λ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ (25) = * Li k , λ ( − e λ ( − t )) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) te λ ( t ) − e y λ ( t ) (cid:19) λ ( x ) n , λ + λ = n ∑ l = (cid:18) nl (cid:19) β l , λ ( y ) (cid:28) t Li k , λ (cid:0) − e λ ( − t ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − l , λ (cid:29) λ TAEKYUN KIM, DAE SAN KIM, JONGKYUM KWON, AND HYUNSEOK LEE = n ∑ l = (cid:18) nl (cid:19) β l , λ ( y ) * t ∞ ∑ m = ( − λ ) m − ( ) m , / λ m k − ( − ) m m ! (cid:0) e λ ( − t ) − (cid:1) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − l , λ + λ = n ∑ l = (cid:18) nl (cid:19) β ( y ) l , λ * t ∞ ∑ j = (cid:18) j ∑ m = ( − λ ) m − ( ) m , / λ m k − ( − ) j − m S , λ ( j , m ) t j j ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − l , λ + λ = n ∑ l = (cid:18) nl (cid:19) β l , λ ( y ) ∞ ∑ j = (cid:18) j + ∑ m = ( − λ ) m − ( ) m , / λ m k − ( − ) j + − m S , λ ( j + , m )( j + ) ! (cid:19) h t j | ( x ) n − l , λ i λ = n ∑ l = (cid:18) nl (cid:19) β l , λ ( y ) n − l + ∑ m = ( − λ ) m − ( ) m , / λ m k − ( − ) n − l + − m ( n − l + ) ! S , λ ( n − l + , m )( n − l ) ! = n ∑ l = n − l + ∑ m = ( − ) n − l (cid:0) nl (cid:1) λ m − ( n − l + ) m k − ( ) m , / λ S , λ ( n − l + , m ) β l , λ ( y ) . Therefore, by (25), we obtain the following theorem.
Theorem 4.
For n ≥ , we haveB ( k ) n , λ ( x ) = n ∑ l = n − l + ∑ m = ( − ) n − l (cid:0) nl (cid:1) λ m − ( n − l + ) m k − ( ) m , / λ S , λ ( n − l + , m ) β l , λ ( x ) . By (6) and (17), we note that(26) β ( s ) n , λ ( x ) ∼ (cid:18) ( e λ ( t ) − ) s t s , t (cid:19) λ From (19), (22) and (26), we have(27) B ( k ) n , λ ( x ) = n ∑ m = C n , m β ( s ) m , λ ( x ) , where C n , m = m ! * Li k , λ ( − e λ ( − t )) e λ ( t ) − ( e λ ( t ) − ) s t s t m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n , λ + λ (28) = (cid:18) nm (cid:19)* Li k , λ ( − e λ ( − t )) e λ ( t ) − ( e λ ( t ) − ) s t s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − m , λ + λ = (cid:18) nm (cid:19) n − m ∑ l = S , λ ( l + s , s ) (cid:0) l + ss (cid:1) l ! * Li k , λ ( − e λ ( − t )) e λ ( t ) − t l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − m , λ + λ = (cid:18) nm (cid:19) n − m ∑ l = S , λ ( l + s , s ) (cid:0) l + ss (cid:1) (cid:18) n − ml (cid:19)* Li k , λ ( − e λ ( − t )) e λ ( t ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x ) n − m − l , λ + λ = (cid:18) nm (cid:19) n − m ∑ l = S , λ ( l + s , s ) (cid:0) l + ss (cid:1) (cid:18) n − ml (cid:19) B ( k ) n − m − l , λ . Therefore, by (27) and (28), we obtain the following theorem.
Theorem 5.
For n ≥ , we haveB ( k ) n , λ ( x ) = n ∑ m = (cid:18) nm (cid:19)(cid:18) n − m ∑ l = (cid:0) n − ml (cid:1)(cid:0) l + ss (cid:1) S , λ ( l + s , s ) B ( k ) n − m − l , λ (cid:19) β ( s ) m , λ ( x ) . EPRESENTATIONS OF DEGENERATE POLY-BERNOULLI POLYNOMIALS 7
For n ≥
0, we let P n = (cid:8) p ( x ) ∈ C [ x ] (cid:12)(cid:12) deg p ( x ) ≤ n (cid:9) . Then P n is an ( n + ) -dimensional vector space over C .From (11), we note that d n , λ ( x ) ∼ (( − t ) e λ ( t ) , t ) λ , ( n ≥ ) . For p ( x ) ∈ P n , we let(29) p ( x ) = n ∑ l = C l d l , λ ( x ) . By (16), we have h ( − t ) e λ ( t ) t m | p ( x ) i λ = n ∑ l = C l (cid:10) ( − t ) e λ ( t ) t m (cid:12)(cid:12) d l , λ ( x ) i λ (30) = n ∑ l = C l m ! δ m , l = C m m ! , where 0 ≤ m ≤ n .Therefore, by (29) and (30), we obtain the following theorem. Theorem 6.
For p ( x ) ∈ P n , we have p ( x ) = n ∑ l = C l d l , λ ( x ) , where C l = l ! (cid:10) ( − t ) e λ t l (cid:12)(cid:12) p ( x ) (cid:11) λ . Let p ( x ) = B ( k ) n , λ ( x ) ∈ P n . Then we have(31) B ( k ) n , λ ( x ) = n ∑ l = C l d l , λ ( x ) , where C l = l ! (cid:10) ( − t ) e λ ( t ) t l | B ( k ) n , λ ( x ) (cid:11) λ (32) = (cid:18) nl (cid:19)(cid:10) ( − t ) e λ ( t ) (cid:12)(cid:12) B ( k ) n − l , λ ( x ) (cid:11) λ = (cid:18) nl (cid:19)(cid:10) ( − t ) | B ( k ) n − l , λ ( x + ) (cid:11) λ = (cid:18) nl (cid:19)(cid:10) (cid:12)(cid:12) B ( k ) n − l , λ ( x + ) (cid:11) λ − (cid:18) nl (cid:19) ( n − l ) (cid:10) | B ( k ) n − l − , λ ( x + ) (cid:11) λ = (cid:18) nl (cid:19) B ( k ) n − l , λ − n (cid:18) n − l (cid:19) B ( k ) n − l − , λ ( ) . Thus, by (31) and (32), we get(33) B ( k ) n , λ ( x ) = n ∑ l = (cid:18)(cid:18) nl (cid:19) B ( k ) n − l , λ − n (cid:18) n − l (cid:19) B ( k ) n − l − , λ (cid:19) d l , λ ( x ) . From (11), we note that d ( r ) n , r ( x ) ∼ (cid:0) ( − t ) r e λ ( t ) , t (cid:1) λ .Let us assume that(34) p ( x ) = n ∑ m = C ( r ) m d ( r ) m , λ ( x ) ∈ P n . TAEKYUN KIM, DAE SAN KIM, JONGKYUM KWON, AND HYUNSEOK LEE
Then, by (16), we get (cid:10) ( − t ) r e λ ( t ) t m (cid:12)(cid:12) p ( x ) (cid:11) λ = n ∑ l = C ( r ) l (cid:10) ( − t ) r e λ ( t ) t m (cid:12)(cid:12) d ( r ) l , λ ( x ) (cid:11) λ (35) = m ! C ( r ) m , ( ≤ m ≤ n ) . Therefore, by (34) and (35), we obtain the following theorem.
Theorem 7.
For n ≥ , we have p ( x ) = n ∑ m = C ( r ) m d ( r ) m , λ ( x ) , where C ( r ) m = m ! (cid:10) ( − t ) r e λ ( t ) t m (cid:12)(cid:12) p ( x ) (cid:11) . We let(36) d n , λ ( x ) = n ∑ m = C ( r ) m d ( r ) m , λ ( x ) , where C ( r ) m = m ! (cid:10) ( − t ) r e λ ( t ) t m (cid:12)(cid:12) d n , λ ( x ) (cid:11) λ (37) = (cid:18) nm (cid:19)(cid:10) ( − t ) r (cid:12)(cid:12) d n − m , λ ( x + ) (cid:11) λ = (cid:18) nm (cid:19) r ∑ j = (cid:18) rj (cid:19) ( − ) j ( n − m ) j (cid:10) (cid:12)(cid:12) d n − m − j , λ ( x + ) i λ = (cid:18) nm (cid:19) r ∑ j = (cid:18) rj (cid:19)(cid:18) n − mj (cid:19) ( − ) j d n − m − j , λ ( ) j ! . By (36) and (37), we get(38) d n , λ ( x ) = n ∑ m = (cid:18) nm (cid:19)(cid:18) r ∑ j = (cid:18) rj (cid:19)(cid:18) n − mj (cid:19) j ! ( − ) j d n − m − j , λ ( ) (cid:19) d ( r ) m , λ ( x ) , ( n ≥ ) . Let us take p ( x ) = B ( k ) n , λ ( x ) ∈ P n , ( n ≥ ) . Then, by Theorem 7, we get(39) B ( k ) n , λ ( x ) = n ∑ m = C ( r ) m d ( r ) m , λ ( x ) , ( n ≥ ) , where C ( r ) m = m ! (cid:10) ( − t ) r e λ ( t ) t m (cid:12)(cid:12) B ( k ) n , λ ( x ) (cid:11) λ (40) = (cid:18) nm (cid:19)(cid:10) ( − t ) r (cid:12)(cid:12) B ( k ) n − m , λ ( x + ) (cid:11) λ = (cid:18) nm (cid:19) r ∑ j = (cid:18) rj (cid:19) ( − ) j (cid:18) n − mj (cid:19) j ! (cid:10) (cid:12)(cid:12) B ( k ) n − m − j ( x + ) (cid:11) λ = (cid:18) nm (cid:19) r ∑ j = (cid:18) rj (cid:19) ( − ) j (cid:18) n − mj (cid:19) j ! B ( k ) n − m − j , λ ( ) . Therefore, by (39) and (40), we obtain the following theorem.
EPRESENTATIONS OF DEGENERATE POLY-BERNOULLI POLYNOMIALS 9
Theorem 8.
For n ≥ , r ∈ N , we haveB kn , λ ( x ) = n ∑ m = (cid:18) nm (cid:19)(cid:18) r ∑ j = (cid:18) rj (cid:19) ( − ) j (cid:18) n − mj (cid:19) j ! B ( k ) n − m − j , λ ( ) (cid:19) d ( r ) m , λ ( x ) .
3. C
ONCLUSION
The study of degenerate versions of some special polynomials and numbers, which was initiatedby Carlitz, regained interests of some mathematicians and many interesting results were found notonly in their arithmetical and combinatorial aspects but also in applications to differential equations,identities of symmetry and probability theory.Recently, the ‘ λ -umbral calculus’ was developed by the motivation that what if the usual expo-nential function is replaced by the degenerate exponential functions in the generating function of aSheffer sequence. This question led us to the introduction of the concepts like λ -linear functionals, λ -differential operators and λ -Sheffer sequences.In this paper, we studied the degenerate poly-Bernoulli polynomials, which is a λ -Sheffer se-quence and hence a degenerate Sheffer sequence, by using three different formulas, namely aformula about representing a λ -Sheffer sequence by another, a formula coming from the gener-ating functions of λ -Sheffer sequences and a formula arising from the definitions for λ -Sheffersequences. Then, among other things, we represented the degenerate poly-Bernoulli polynomi-als by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangementpolynomials.It is one of our future projects to continue to investigate the degenerate special numbers andpolynomials by using the recently developed λ -umbral calculus. Acknowledgments
The authors would like to thank Jangjeon Research Institute for Mathematical Sciences for thesupport of this research.
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TK and DSK conceived of the framework and structured the whole paper; TK and DSK wrote thepaper; JK and HL checked the results of the paper and typed the paper; DSK and TK completed therevision of the article. All authors have read and agreed to the published version of the manuscript.
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