New degenerated polynomials arising from non-classical Umbral Calculus
aa r X i v : . [ m a t h . N T ] N ov NEW DEGENERATED POLYNOMIALS ARISING FROM NON-CLASSICAL UMBRALCALCULUS
ORLI HERSCOVICI AND TOUFIK MANSOURA
BSTRACT . We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbersbased on the Carlitz-Tsallis degenerate exponential function and concepts of the Umbral Calculus associatedwith it. Also, we present generalizations of some familiar identities and connection between these kinds ofBernoulli, Euler, and Genocchi polynomials. Moreover, we establish a new analogue of the Euler identity forthe degenerate Bernoulli numbers.
1. I
NTRODUCTION
The Bernoulli, Euler, and Genocchi numbers and polynomials are closely connected to each other [7].Their study attracts attention of many researchers (see [3, 8, 10, 13] and reference therein). Besides theclassical versions there exist their different q -analogues and parameterized versions [1, 2, 8, 13]. The de-generate versions of the Bernoulli and Euler numbers were defined and studied by Carlitz. They are basedon a degenerate exponential function e λ , µ ( t ) = ( + λ t ) µ . In this way the degenerate Bernoulli numbers ofCarlitz are defined by a generating function t ( + λ t ) µ − = ∞ ∑ n = β n ( λ ) t n n ! , (1.1)with condition λ µ =
1. Degenerate versions of the Bernoulli, Euler, and Genocchi polynomials were studiedby different researchers (see [9, 15] and references therein). The umbral calculus was one of the methodsused for study of these polynomials. However, those degenerate polynomials were defined in terms ofversions exponential functions with classical additive property and studied by techniques of the classicalumbral calculus. For example, the degenerate Bernoulli polynomials studied in [15] are defined as t ( + λ t ) λ − )( + λ t ) x λ = ∞ ∑ n = β n ( λ , x ) t n n ! , (1.2)where the degenerate Bernoulli numbers are evaluated, as usually, as β n ( λ ) = β n ( λ , ) . In this work we de-fine a new degenerate Bernoulli, Euler, and Genocchi polynomials and numbers and study them by applyingnon-classical umbral calculus. They are based on degenerated, or, in other words, deformed exponentialfunction that their additive property is deformed too. Our results generalize many of well known identitiesfor classical case. Moreover, we bring a new analogue of the Euler identity for the Bernoulli numbers andestablish connections between degenerate versions of the Bernoulli, Euler, and Genocchi polynomials.This paper is organized as following. We start from some definitions and useful theorems of umbral cal-culus. Each one from the following three sections considers, respectively, the degenerate Bernoulli, Euler,and Genocchi polynomials and numbers. The last section considers connections between these polynomialsand shows a way to find connections with other polynomials. Mathematics Subject Classification.
Key words and phrases. degenerate Bernoulli polynomials; degenerate Euler polynomials; degenerate Genocchi polynomials;Euler identity for Bernoulli numbers; Carlitz-Tsallis degenerated exponential function.
2. B
ACKGROUND AND DEFINITIONS
Let us consider the umbral calculus associated with the deformed exponential function e q ( x ) = ( + ( − q ) x ) − q , (2.1)defined by Carlitz in [6] with substitution λ = − q and by Tsallis in [14]. It is easy to see that this Carlitz-Tsallis exponential function (2.1) is an eigenfunction of the operator D q ; x ≡ ( + ( − q ) x ) ddx , where d / dx is the ordinary Newton’s derivative. This exponential function has the following extension as formal powerseries (see [5]): e q ( x ) = + ∞ ∑ n = Q n − ( q ) x n n ! , where Q n ( q ) = · q ( q − ) . . . ( nq − ( n − )) . Let us define a sequence c n ; q as following c n ; q = (cid:26) n ! Q n − ( q ) , n ≥ , , n = . Therefore, by using this notation, we can write e q ( x ) = ∞ ∑ n = x n c n ; q . (2.2)Define x ⊕ q y = x + y + ( − q ) xy (see, Borges [4]) and ( x + y ) nc = ∑ nk = c n ; q c k ; q c n − k ; q x k y n − k . Now we can statethe following Proposition. Proposition 2.1.
For any x , y ∈ C it holds that ∞ ∑ n = ( xt ⊕ q yt ) n c n ; q = ∞ ∑ n = ( x + y ) nc t n c n ; q . Proof.
It follows from (2.2) that e q ( xt ) e q ( yt ) = ∞ ∑ n = x n t n c n ; q · ∞ ∑ k = y k t k c k ; q = ∞ ∑ n = n ∑ k = c n ; q c k ; q c n − k ; q x k y n − k t n c n ; q = ∞ ∑ n = ( x + y ) nc t n c n ; q = e q (( x + y ) c t ) , (2.3)From another side we have e q ( xt ) e q ( yt ) = ( + ( − q ) xt ) − q ( + ( − q ) yt ) − q = (cid:0) + ( − q )( xt + yt + ( − q ) xyt ) (cid:1) − q = e q ( xt ⊕ q yt ) = ∞ ∑ n = ( xt ⊕ q yt ) n c n ; q , (2.4)Therefore, by comparing (2.3) with (2.4), we obtain that e q ( xt ⊕ q yt ) = e q (( x + y ) c t ) , or, in more detailednotation, ∞ ∑ n = ( xt ⊕ q yt ) n c n ; q = ∞ ∑ n = ( x + y ) nc t n c n ; q , which completes the proof. (cid:3) From here and on we will define (cid:0) nk (cid:1) c ; q = c n ; q c k ; q c n − k ; q . Clearly, ( x + y ) nc = ∑ nk = (cid:0) nk (cid:1) c ; q x k y n − k .Let P be the algebra of polynomials in a single variable x over the field C and P ∗ be the vector spaceof all linear functionals on P . The notation h L | p ( x ) i denotes the action of a linear functional L on a poly-nomial p ( x ) , and the vector space operations on P ∗ are defined by h α L + α L | p ( x ) i = α h L | p ( x ) i + EW DEGENERATED POLYNOMIALS 3 α h L | p ( x ) i for any constants α , α ∈ C . Let F denote the algebra of formal power series in a singlevariable t over the field C : F = ( f ( t ) = ∑ k ≥ a k t k c k ; q (cid:12)(cid:12)(cid:12) a k ∈ C ) . The formal power series f ( t ) defines a linear functional on P by setting(2.5) h f ( t ) | x n i = c n ; q a n , ∀ n ≥ , and in particular (cid:10) t k | x n (cid:11) = c n ; q δ n , k , for all n , k ≥
0, where δ n , k is the Kronecker delta function.Let f L ( t ) = ∑ k ≥ h L | x k i c k ; q t k , then we get h f L ( t ) | x n i = h L | x n i . Thus the map L F L ( t ) is a vector spaceisomorphism from P ∗ onto F . Henceforth, F denotes both the algebra of formal power series in t andthe vector space of all linear functionals in P (see [11]), so that F is an umbral algebra, and the umbralcalculus is the study of the umbral calculus. Note that the umbral calculus considered here is non-classicalbecause it is associated with the sequence c n ; q instead of classical n !. From (2.2)-(2.5) one can easily seethat (cid:10) e q ( yt ) | x n (cid:11) = y n and, respectively, (cid:10) e q ( yt ) | ( p ( x ) (cid:11) = p ( y ) . For all f ( t ) ∈ F and for all polynomials p ( x ) ∈ P we have f ( t ) = ∑ k ≥ (cid:10) f ( t ) | x k (cid:11) c k ; q t k and p ( x ) = ∑ k ≥ (cid:10) t k | p ( x ) (cid:11) c k ; q x k . For f ( t ) , . . . , f m ( t ) ∈ F , we have (see [11, 12]) h f ( t ) · · · f m ( t ) | x n i = ∑ i + ··· + i m = ni j ≥ c n ; q c i ; q · · · c i m ; q (cid:10) f ( t ) | x i (cid:11) · · · (cid:10) f m ( t ) | x i m (cid:11) . Let us define a linear operator D c q ; t as following D c q ; t t n = ( c n ; q c n − q t n − , for integer n ≥ , , n = . Therefore for any polynomial p ( x ) = ∑ nj = a j x j we have (cid:10) t k | p ( x ) (cid:11) = c k ; q a k = D kc q ; x p ( ) , and, in particular, (cid:10) t | p ( x ) (cid:11) = p ( ) .For any f ( t ) ∈ F the linear operator f ( t ) on F is defined by (see [11]) f ( t ) x n = ∑ nk = c n ; q c k ; q c n − k ; q a k x n − k ,which leads to t k x n = ( c n ; q c n − k ; q x n − k , for integer n ≥ k , , n < k . (2.6)For f ( t ) , g ( t ) ∈ F , it holds that h f ( t ) g ( t ) | p ( x ) i = h g ( t ) | f ( t ) p ( x ) i = h f ( t ) | g ( t ) p ( x ) i . The degree of f ( t ) (denoted by o ( f ( t )) ) is the smallest k such that t k does not vanish. If o ( f ( t )) = f ( t ) is called invertible and has a multiplicative inverse denoted by f − ( t ) or 1 / f ( t ) . If o ( f ( t )) = f ( t ) is called delta series and has a compositional inverse ¯ f ( t ) satisfying f ( ¯ f ( t )) = ¯ f ( f ( t )) = t .For a delta series f ( t ) ∈ F and an invertible series g ( t ) ∈ F we say that a polynomial sequence s n ( x ) is a Sheffer sequence for the pair ( g ( t ) , f ( t )) and denote it by s n ( x ) ∼ ( g ( t ) , f ( t )) if for all n , k ≥ (cid:10) g ( t ) f ( t ) k | s n ( x ) (cid:11) = c n ; q δ n , k . Thus, s n ( x ) ∼ ( g ( t ) , f ( t )) if and only if1 g ( ¯ f ( t )) e q ( y ¯ f ( t )) = ∞ ∑ n = s n ( y ) c n ; q t n , (2.7) ORLI HERSCOVICI AND TOUFIK MANSOUR for all y ∈ C (see [12]). The following statements are equivalent s n ( x ) ∼ ( g ( t ) , f ( t )) , g ( t ) s n ( x ) ∼ ( , f ( t )) , f ( t ) s n ( x ) = c n ; q c n − q s n − ( x ) . (2.8)Moreover, the following theorem holds. Theorem 2.2.
Let s n ( x ) ∼ ( g ( t ) , f ( t )) . Then (i) for any polynomial p ( x ) , p ( x ) = ∑ ∞ n = h g ( t ) f ( t ) n | p ( x ) i c n ; q s n ( x ) (Polynomial Expansion, Theorem 6.2.3in [12]); (ii) s n ( x ) = ∑ nk = h g ( ¯ f ( t )) − ¯ f ( t ) k | x n i c k ; q x k (Conjugate Representation, Theorem 6.2.5 in [12]);Moreover, a sequence s n ( x ) ∼ ( g ( t ) , f ( t )) for some invertible g ( t ) , if and only ife q ( yt ) s n ( x ) = n ∑ k = c n ; q c k ; q c n − k ; q p k ( y ) s n − k ( x ) for all constants y, where p ( x ) ∼ ( , f ( t )) (The Sheffer Identity, Theorem 6.2.8 in [12]). Moreover, if s n ( x ) ∼ ( g ( t ) , t ) then, from (2.7) we obtain1 g ( t ) x n = s n ( x ) , or x n = g ( t ) s n ( x ) . (2.9)Let us define now an inverse operator for operator D c q ; x as following I c q ; x x n = Z x n d c q x = c n ; q c n + q x n + . (2.10)Obviously, I c q ; x ( D c q ; x x n ) = D c q ; x ( I c q ; x x n ) = x n .3. B ERNOULLI POLYNOMIALS
Let us define degenerate Bernoulli polynomials B n ; q ( x ) and numbers B n ; q as te q ( t ) − e q ( xt ) = ∞ ∑ n = B n ; q ( x ) t n c n ; q . (3.1) te q ( t ) − = ∞ ∑ n = B n ; q t n c n ; q . (3.2)The first few values of degenerate Bernoulli polynomials and numbers are listed in Table 1. Let us denote n B n ; q ( x ) B n ; q x − q − q x − x + − q − q − ( q − ) x + qx + ( q − ) x + q − q + ( q − ) q − q + ( q − ) ( q − q + ) x − ( q − q ) x − ( q − ) x − ( q − q + ) x − ( q − q + q − )( q − )( q − ) − q − q + q − ( q − )( q − ) T ABLE
1. Degenerate Bernoulli polynomials and numbers.by B q the umbra of the Bernoulli numbers sequence, that is, te q ( t ) − = e q ( B q t ) = ⇒ t = e q ( B q t ) e q ( t ) − e q ( B q t ) . EW DEGENERATED POLYNOMIALS 5
By applying (2.3), we have t = ∑ ∞ n = ( B q + ) nc t n c n ; q − ∑ ∞ n = B n ; q t n c n ; q . Thus, ( B q + ) nc − B n ; q = δ , n , which is ageneralization of a well-known identity for the classical Bernoulli numbers. From (3.1) we obtain, ∞ ∑ n = B n ; q ( x ) t n c n ; q = te q ( t ) − e q ( xt ) = ∞ ∑ n = B n ; q t n c n ; q · ∞ ∑ k = t k x k c k ; q = ∞ ∑ n = n ∑ k = (cid:18) nk (cid:19) c ; q B k ; q x n − k t n c n ; q , and, by extracting the coefficients of t n c n ; q , we get an analogue of the well known identity for the Bernoullipolynomials. Proposition 3.1.
For all n ∈ N , the degenerated Bernoulli polynomials B n ; q ( x ) defined by (3.1) satisfy B n ; q ( x ) = n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q x k . Moreover, for all n ≥ and x ∈ C , it holds that B n ; q ( x ) = ( B q + x ) nc and B n ; q ( ) − B n ; q = δ , n . Note that the identity of the Proposition 3.1 can be obtained immediately from noticing that B n ; q ( x ) ∼ ( e q ( t ) − t , t ) and applying Theorem 2.2(ii). From (2.9), we obtain te q ( t ) − x k = B k ; q ( x ) , or x k = e q ( t ) − t B k ; q ( x ) . (3.3)By applying (2.8), we get t B n ; q ( x ) = c n ; q c n − q B n − q ( x ) . Lemma 3.2.
For any polynomial p ( x ) = n ∑ k = a k x k ∈ P, it holds that (cid:28) e q ( yt ) − t | p ( x ) (cid:29) = Z y p ( u ) d c q u . Proof.
Let us consider the action of this linear functional on a monomial x j . From (2.2), we obtain (cid:28) e q ( yt ) − t (cid:12)(cid:12)(cid:12) x j (cid:29) = (cid:28) e q ( yt ) − (cid:12)(cid:12)(cid:12) t x j (cid:29) . (3.4)The operator t is the inverse of the operator t defined as tx j = c j ; q c j − q x j − . Therefore, by applying t to bothsides of this equation, we obtain x j = t c j ; q c j − q x j − and, thus, t x j − = c j − q c j ; q x j . So by (3.4), we have (cid:28) e q ( yt ) − t (cid:12)(cid:12)(cid:12) x j (cid:29) = (cid:28) e q ( yt ) − (cid:12)(cid:12)(cid:12) c j ; q c j + q x j + (cid:29) = c j ; q c j + q y j + = Z y x j d c q x . By linearity, we complete the proof. (cid:3)
By applying this Lemma to the polynomials B n ; q ( x ) and using (3.3), we obtain Z B n ; q ( u ) d c q u = (cid:28) e q ( t ) − t (cid:12)(cid:12)(cid:12) B n ; q ( x ) (cid:29) = (cid:28) (cid:12)(cid:12)(cid:12) e q ( t ) − t B n ; q ( x ) (cid:29) = h | x n i = (cid:10) t | x n (cid:11) = c n ; q δ n , . (3.5)From another side, by Proposition 3.1 we have Z B n ; q ( u ) d c q u = Z n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q u k d c q u = n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q Z u k d c q u , (3.6) ORLI HERSCOVICI AND TOUFIK MANSOUR and, by using the definition (2.10), we obtain Z B n ; q ( u ) d c q u = n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q c k ; q c k + q u k + (cid:12)(cid:12)(cid:12) = n ∑ k = c n ; q c k + q c n − k ; q B n − k ; q = c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q B k ; q . (3.7)Hence, by comparing (3.5) with (3.7) and bringing into consideration that c q =
1, we can state the follow-ing result.
Proposition 3.3.
For all integer n ≥ , it holds that ∑ nk = (cid:0) n + k (cid:1) c ; q B k ; q = δ n , . Remark 3.4.
This Proposition brings another formulation and proof of the Corollary of the Proposition 3.1at x = . One of the very important aspects in the theory of orthogonal polynomials is a connection betweendifferent kinds of polynomials. Let us consider the equation (3.3). It can be rewritten as tx k = ( e q ( t ) − ) B k ; q ( x ) . (3.8)Therefore, by (2.6), we obtain c k ; q c k − q x k − = e q ( t ) B k ; q ( x ) − B k ; q ( x ) = k ∑ j = (cid:18) kj (cid:19) c ; q B k ; q ( x ) − B k ; q ( x ) = k − ∑ j = (cid:18) kj (cid:19) c ; q B k ; q ( x ) . Thus we can state the following result.
Proposition 3.5.
For all integer n ≥ it holds that x n = c n ; q c n + q ∑ nk = (cid:0) n + k (cid:1) c ; q B k ; q ( x ) . Moreover, for all integern ≥ , [ x n ] B n ; q ( x ) = . Now, we are ready to present an analogue of the Euler identity for Bernoulli polynomials and numbers.
Theorem 3.6.
For all integer n ≥ , the degenerate Bernoulli polynomials defined by (3.1) satisfy n ∑ k = (cid:18) nk (cid:19) c ; q B k ; q ( x ) B n − k ; q ( y ) = − ( n − ) B n ; q (( x + y ) c ) − n B n − q (( x + y ) c ) ( n − ) − q ( n − )( n − ) q − ( n − )+ d B n ; q (( x + y ) c ) + ( − q ) c n ; q c n − q \ B n − q (( x + y ) c ) , (3.9) where d B n ; q ( u ) = ∑ nk = (cid:0) nk (cid:1) c ; q k B n − k ; q u k .Proof. Let b ( t ) = te q ( t ) − . Therefore b ( t ) = ( − qt ) b ( t ) − ( + ( − q ) t ) tb ′ ( t ) , (3.10)where b ′ ( t ) = ddt b ( t ) . By multiplying both sides by e q ( xt ) e q ( yt ) and replacing b ′ ( t ) e q ( ut ) = ( b ( t ) e q ( ut )) ′ − b ( t ) e ′ q ( ut ) in accordance with Leibniz rule, we obtain b ( t ) e q ( xt ) e q ( yt ) = ( − qt ) b ( t ) e q (( x + y ) c t ) − ( t + ( − q ) t ) (cid:2) ( b ( t ) e q (( x + y ) c t )) ′ − b ( t ) e ′ q (( x + y ) c t ) (cid:3) . (3.11) EW DEGENERATED POLYNOMIALS 7
It follows that ∞ ∑ n = n ∑ k = (cid:18) nk (cid:19) c ; q B k ; q ( x ) B n − k ; q ( y ) t n c n ; q = ( − qt ) ∞ ∑ n = B n ; q (( x + y ) c ) t n c n ; q − ( t + ( − q ) t ) ∞ ∑ n = B n ; q (( x + y ) c ) t n c n ; q ! ′ + ( t + ( − q ) t ) ∞ ∑ n = B n ; q t n c n ; q · ∞ ∑ k = ( x + y ) kc t k c k ; q ! ′ . (3.12)After differentiating and applying the Cauchy product, one can extract the coefficients of t n c n ; q for n ≥ n ∑ k = (cid:18) nk (cid:19) c ; q B k ; q ( x ) B n − k ; q ( y ) = B n ; q (( x + y ) c ) − c n ; q c n − q q B n − q (( x + y ) c ) − n B n ; q (( x + y ) c ) − c n ; q c n − q ( − q )( n − ) B n − q (( x + y ) c )+ n − ∑ k = (cid:18) nk (cid:19) c ; q ( n − k ) B k ; q · ( x + y ) n − kc + n − ∑ k = (cid:18) n − k (cid:19) c ; q ( n − k − )( − q ) B k ; q · ( x + y ) n − k − c c n ; q c n − q . (3.13)Rearranging the summation indexes, denoting d B n ; q ( u ) = ∑ nk = (cid:0) nk (cid:1) c ; q k B n − k ; q u k , and gathering the similarterms complete the proof. (cid:3) An analogue of the Euler identity for the degenerate Bernoulli numbers follows immediately from theprevious Theorem by assuming x = y = d B n ; q ( ) = n ≥ Theorem 3.7.
For all integer n ≥ , the degenerate Bernoulli numbers defined by (3.2) satisfy n ∑ k = (cid:18) nk (cid:19) c ; q B k ; q B n − k ; q = − n B n ; q − n B n − q ( n − ) − q ( n − )( n − ) q − ( n − ) .
4. E
ULER POLYNOMIALS
Let us define degenerate Euler polynomials E n ; q ( x ) and values E n ; q = E n ; q ( ) as2 e q ( t ) + e q ( xt ) = ∞ ∑ n = E n ; q ( x ) t n c n ; q , (4.1) 2 e q ( t ) + = ∞ ∑ n = E n ; q t n c n ; q . (4.2)The first five degenerate Euler polynomials and their special values are listed in Table 2. It is easy to seethat E n ; q ( x ) ∼ (cid:16) e q ( t )+ , t (cid:17) . Therefore, by (2.8), we obtain t E n ; q ( x ) = c n ; q c n − q E n − q ( x ) and D kc q ; x E n ; q ( x ) = c n ; q c n − k ; q E n − k ; q ( x ) . From (2.9), we have 2 e q ( t ) + x k = E k ; q ( x ) , or x k = e q ( t ) + E k ; q ( x ) . (4.3) ORLI HERSCOVICI AND TOUFIK MANSOUR n E n ; q ( x ) E n ; q x − − x q − x − q + q − q q ( q − q ) x − qx +( − q ) x − q + q − q ( q − ) − q + q − q ( q − ) ( q − q + q ) x − ( q − q ) x − ( q − q ) x − ( q − q + ) x − q + q − q + q ( q − )( q − ) − q + q − q + q ( q − )( q − ) T ABLE
2. Degenerate Euler polynomials and values.Let us denote by E q the umbra of the Euler values sequence, that is,2 e q ( t ) + = e q ( E q t ) = ⇒ = e q ( E q t ) e q ( t ) + e q ( E q t ) . By applying (2.3), we obtain 2 = ∑ ∞ n = ( E q + ) nc t n c n ; q + ∑ ∞ n = E n ; q t n c n ; q . Thus, ( E q + ) nc + E n ; q = δ , n , which is ageneralization of a well-known identity for the classical case. From (4.1)-(4.2), we get ∞ ∑ n = E n ; q ( x ) t n c n ; q = e q ( t ) + · e q ( xt ) = ∞ ∑ n = E n ; q t n c n ; q · ∞ ∑ k = x k t k c k ; q = ∞ ∑ n = n ∑ k = c n ; q c k ; q c n − k ; q E n − k ; q x k t n c n ; q , which leads to the following proposition. Proposition 4.1.
For all n ∈ N , the degenerated Euler polynomials E n ; q ( x ) defined by (4.1) satisfy E n ; q ( x ) = n ∑ k = (cid:18) nk (cid:19) c ; q E n − k ; q x k . Moreover, for all n ≥ and x ∈ C , E n ; q ( x ) = ( E q + x ) nc and E n ; q ( ) + E n ; q = δ , n . From the Theorem 2.2, by assuming y =
1, we obtain e q ( t ) E n ; q ( x ) = ∑ nk = c n ; q c k ; q c n − k ; q E k ; q ( x ) . Applyingthis identity to (4.3) leads to the next result. Proposition 4.2.
For all integer n ≥ , it holds thatx n = n ∑ k = (cid:18) nk (cid:19) c ; q E k ; q ( x ) + E n ; q ( x ) . Moreover, for all n ≥ , [ x n ] E n ; q ( x ) = . By substituting x = Corollary 4.3.
For all integer n ≥ , it holds that − E n ; q = ∑ n − k = (cid:0) nk (cid:1) c ; q E k ; q .
5. G
ENOCCHI POLYNOMIALS
Let us define degenerate Genocchi polynomials G n ; q ( x ) and numbers G n ; q as2 te q ( t ) + e q ( xt ) = ∞ ∑ n = G n ; q ( x ) t n c n ; q , (5.1) 2 te q ( t ) + = ∞ ∑ n = G n ; q t n c n ; q . (5.2) EW DEGENERATED POLYNOMIALS 9
The first few values of degenerate Genocchi polynomials and numbers are listed in Table 3. It is easy to see n G n ; q ( x ) G n ; q x − q − q qx − x − q + q ( q − ) − q q ( q − ) ( q − q ) x − qx +( − q ) x − ( q − q + ) q ( q − )( q − ) − q − q + q ( q − )( q − ) T ABLE
3. Degenerate Genocchi polynomials and numbers.that G n ; q ( x ) ∼ (cid:16) e q ( t )+ t , t (cid:17) . Moreover, by comparing (4.1) with (5.1), one can immediately conclude that deg ( G n ; q ( x )) = n −
1. Therefore, by (2.8), we obtain t G n ; q ( x ) = c n ; q c n − q G n − q ( x ) and D kc q ; x G n ; q ( x ) = c n ; q c n − k ; q G n − k ; q ( x ) . From (2.9), we have 2 te q ( t ) + x k = G k ; q ( x ) or x k = e q ( t ) + t G k ; q ( x ) . (5.3)Let us denote by G q the umbra of the Genocchi numbers sequence, that is,2 te q ( t ) + = e q ( G q t ) = ⇒ t = e q ( G q t ) e q ( t ) + e q ( G q t ) . By applying (2.3), we get that 2 t = ∑ ∞ n = ( G q + ) nc t n c n ; q + ∑ ∞ n = G n ; q t n c n ; q . So, ( G q + ) nc + G n ; q = δ , n , which is ageneralization of a well-known identity for the classical case.From the definitions of degenerate Genocchi numbers and polynomials (5.1)-(5.2), we obtain ∞ ∑ n = G n ; q ( x ) t n c n ; q = te q ( t ) + · e q ( xt ) = ∞ ∑ n = G n ; q t n c n ; q · ∞ ∑ k = x k t k c k ; q = ∞ ∑ n = n ∑ k = c n ; q c k ; q c n − k ; q G n − k ; q x k t n c n ; q , and we can state the following proposition. Proposition 5.1.
For all n ∈ N , the degenerated Genocchi polynomials G n ; q ( x ) defined by (5.1) satisfy G n ; q ( x ) = ∑ nk = (cid:0) nk (cid:1) c ; q G n − k ; q x k . Moreover, for all n ∈ N and x ∈ C , G n ; q ( x ) = ( G q + x ) nc and G n ; q ( ) + G n ; q = δ , n . From the Theorem 2.2 with y =
1, we obtain e q ( t ) G n ; q ( x ) = ∑ nk = c n ; q c k ; q c n − k ; q G k ; q ( x ) . Applying this identityto the equation (5.3) leads to the next result. Proposition 5.2.
For all integer n ≥ , it holds thatx n = c n ; q c n + q n + ∑ k = (cid:18) n + k (cid:19) c ; q G k ; q ( x ) + c n ; q c n + q G n + q ( x ) . Moreover, for all n ≥ , [ x n ] G n + q ( x ) = c n + q c n ; q . By substituting x = Corollary 5.3.
For all integer n ≥ , it holds that − G n + q = ∑ nk = (cid:0) n + k (cid:1) c ; q G k ; q .
6. C
ONNECTIONS BETWEEN POLYNOMIALS
We already have shown a connection between monomials p ( x ) = x n and degenerated Bernoulli B n ; q ( x ) ,degenerated Euler E n ; q ( x ) , and degenerated Genocchi G n ; q ( x ) polynomials. Let us assume now that apolynomial p ( x ) ∈ P of degree n can be expressed as a linear combination of the deformed Bernoullipolynomials p ( x ) = ∑ nk = b k B k ; q ( x ) . Therefore, by Theorem 2.2(i), we obtain n ∑ k = b k B k ; q ( x ) = n ∑ k = D e q ( t ) − t · t k (cid:12)(cid:12)(cid:12) p ( x ) E c k ; q B k ; q ( x ) , where b k = c k ; q (cid:28) e q ( t ) − t · t k (cid:12)(cid:12)(cid:12) p ( x ) (cid:29) = c k ; q (cid:28) e q ( t ) − t · (cid:12)(cid:12)(cid:12) t k p ( x ) (cid:29) = c k ; q (cid:28) e q ( t ) − t · (cid:12)(cid:12)(cid:12) D kc q ; x p ( x ) (cid:29) = c k ; q Z D kc q ; x p ( x ) d c q x . Thus, we can state the following statement.
Proposition 6.1.
For any polynomial p ( x ) ∈ P of degree n, there exist constants b , b , . . . , b n such thatp ( x ) = ∑ nk = b k B k ; q ( x ) , where b k = c k ; q R D kc q ; x p ( x ) d c q x. Theorem 6.2.
Let us define (cid:0) n + k , m , n − k − m + (cid:1) c ; q = c n + q c k ; q c m ; q c n − k + − m ; q . Then, for all integer n ≥ , E n ; q ( x ) = c n ; q c n + q n ∑ k = n − k ∑ m = (cid:18) n + k , m , n − k − m + (cid:19) c ; q E m ; q B k ; q ( x ) , or E n ; q ( x ) = − c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q E n − k + q B k ; q ( x ) . Proof.
Let us assume that E n ; q ( x ) = ∑ nk = b k B k ; q ( x ) . Therefore, by Proposition (6.1), we have b k = c k ; q Z D kc q ; x E n ; q ( x ) d c q x = c k ; q Z c n ; q c n − k ; q E n − k ; q ( x ) d c q x = c n ; q c k ; q c n − k ; q · c n − k ; q c n − k + q E n − k + q ( x ) (cid:12)(cid:12)(cid:12) = c n ; q c k ; q c n − k + q (cid:0) E n − k + q ( ) − E n − k + q (cid:1) . In accordance with Proposition (6.1) for x =
1, we obtain b k = c n ; q c k ; q c n − k + q n − k + ∑ m = c n − k + q c m ; q c n − k + − m ; q E m ; q − E n − k + q ! = c n ; q c k ; q c n − k + q n − k ∑ m = c n − k + q c m ; q c n − k + − m ; q E m ; q = n − k ∑ m = c n ; q c k ; q c m ; q c n − k + − m ; q E m ; q = c n ; q c n + q n − k ∑ m = c n + q c k ; q c m ; q c n − k + − m ; q E m ; q . On the other side, by Proposition 6.1, we have b k = c n ; q c k ; q c n − k + q (cid:0) δ , n − k + − E n − k + q (cid:1) . EW DEGENERATED POLYNOMIALS 11
Therefore, E n ; q ( x ) = c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q (cid:0) δ , n − k + − E n − k + q (cid:1) B k ; q ( x )= − c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q E n − k + q B k ; q ( x ) , which completes the proof. (cid:3) Theorem 6.3.
For all integer n ≥ , G n ; q ( x ) = − c n ; q c n + q n − ∑ k = (cid:18) n + k (cid:19) c ; q G n − k + q B k ; q ( x ) , or G n ; q ( x ) = c n ; q c n + q n ∑ k = n − k ∑ m = (cid:18) n + k , m , n + − k − m (cid:19) c ; q G m ; q B k ; q ( x ) . Proof.
Let us assume that G n ; q ( x ) = ∑ nk = b k B k ; q ( x ) . Therefore, by Proposition (6.1), we have b k = c k ; q Z D kc q ; x G n ; q ( x ) d c q x = c k ; q Z c n ; q c n − k ; q G n − k ; q ( x ) d c q x = c n ; q c k ; q c n − k ; q · c n − k ; q c n − k + q G n − k + q ( x ) (cid:12)(cid:12)(cid:12) = c n ; q c k ; q c n − k + q (cid:0) G n − k + q ( ) − G n − k + q (cid:1) . By Proposition 6.1, we obtain b k = c n ; q c k ; q c n − k + q (cid:0) δ , n − k + − G n − k + q (cid:1) . Therefore G n ; q ( x ) = c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q (cid:0) δ , n − k + − G n − k + q (cid:1) B k ; q ( x )= B n ; q ( x ) − c n ; q c n + q n ∑ k = (cid:18) n + k (cid:19) c ; q G n − k + q B k ; q ( x ) , and, by the fact that G q =
1, we obtain the first statement of the theorem. From another side, by Proposi-tion (5.1) with x =
1, we obtain b k = c n ; q c k ; q c n − k + q (cid:0) G ; q ( n − k + ) − G n − k + q (cid:1) = c n ; q c k ; q c n − k + q n − k + ∑ m = (cid:18) n − k + n + − k − m (cid:19) c ; q G m ; q − G n − k + q ! = c n ; q c k ; q c n − k + q n − k ∑ m = c n − k + q c m ; q c n − k + − m ; q G m ; q = c n ; q c n + q n − k ∑ m = c n + q c k ; q c m ; q c n − k + − m ; q G m ; q = c n ; q c n + q n − k ∑ m = (cid:18) n + k , m , n + − k − m (cid:19) c ; q G m ; q , which completes the proof of the second statement of the theorem. (cid:3) Proposition 6.4.
For any polynomial p ( x ) ∈ P of degree n, there exist constants b , b , . . . , b n such thatp ( x ) = ∑ nk = b k E k ; q ( x ) , where b k = c k ; q ( D kcq ; x p )( ) − ( D kcq ; x p )( ) . Proof.
Theorem (2.2)(i) gives p ( x ) = n ∑ k = b k E k ; q ( x ) = n ∑ k = D e q ( t )+ t k (cid:12)(cid:12)(cid:12) p ( x ) E c k ; q E k ; q ( x ) . Therefore b k = c k ; q (cid:28) e q ( t ) + t k (cid:12)(cid:12)(cid:12) p ( x ) (cid:29) = c k ; q (cid:28) e q ( t ) + (cid:12)(cid:12)(cid:12) t k p ( x ) (cid:29) = c k ; q (cid:28) e q ( t ) + (cid:12)(cid:12)(cid:12) D kc q ; x p ( x ) (cid:29) = c k ; q ( D kc q ; x p )( ) + ( D kc q ; x p )( ) , a required. (cid:3) Theorem 6.5.
For all integer n ≥ , B n ; q ( x ) = n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q E k ; q ( x ) + c n ; q c n − q E n − q ( x ) . Proof.
Let us assume that B n ; q ( x ) = ∑ nk = b k E k ; q ( x ) . Therefore, by Proposition (6.1), we have b k = ( D kc q ; x B n ; q ( x ))( ) + ( D kc q ; x B n ; q ( x ))( ) c k ; q = c k ; q (cid:18) c n ; q c n − k ; q B n − k ; q ( ) + c n ; q c n − k ; q B n − k ; q (cid:19) . So by Proposition 3.1, we obtain b k = (cid:18) nk (cid:19) c ; q ( B n − k ; q + δ , n − k + B n − k ; q ) = (cid:18) nk (cid:19) c ; q (cid:0) B n − k ; q + δ , n − k (cid:1) . Therefore, we get B n ; q ( x ) = n ∑ k = (cid:18) nk (cid:19) c ; q (cid:0) B n − k ; q + δ , n − k (cid:1) E k ; q ( x )= n ∑ k = (cid:18) nk (cid:19) c ; q B n − k ; q E k ; q ( x ) + c n ; q c n − q E n − q ( x ) , which completes the proof. (cid:3)
7. C
ONCLUSION
We defined and studied new analogs of the Bernoulli, Euler, and Genocchi polynomials and numbers.Classical identities for them including the Euler identity for Bernoulli numbers were extended. Moreover,we established connections between these polynomials and proved the formulae which enable to expandother Sheffer-type polynomials in terms of degenerate Bernoulli, degenerate Euler, or degenerate Genocchipolynomials defined in this work.
Acknowledgement . The research of the first author was supported by the Ministry of Science andTechnology, Israel. R
EFERENCES[1] M. Acikgoz, S. Araci, and U. Duran, New extensions of some known special polynomials under the theory of multiple q -calculus, Turkish J. Anal. Num. Theory (2015), no. 5, 128–139.[2] M. Acikgoz and Y. Simsek, On multiple interpolation functions of the N¨orlund type q -Euler polynomials, Abst. Appl. Anal. (2009), Article ID 382574.[3] S. Araci, M. Acikgoz, and E. S¸en, Some new formulae for Genocchi numbers and polynomials involving Bernoulli and Eulerpolynomials,
Int. J. Math. Math. Sci. , (2014), Article ID 760613, 7 pp.[4] E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Phys. A (2004), 95–101.
EW DEGENERATED POLYNOMIALS 13 [5] E.P. Borges, On a q -generalization of circular and hyperbolic functions, J. Phys. A (1998), 5281–5288.[6] L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) (1956), 28–33.[7] K. Dilcher and C. Vignat, General convolution identities for Bernoulli and Euler polynomials, J. Math. Anal. Appl. (2016),1478–1498.[8] Y. He, S. Araci, H.M. Srivastava, and M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials,
Appl. Math. Comput. , (2015), 31–41.[9] D. Lim, Some identities of degenerate Genocchi polynomials, Bull. Korean Math. Soc. (2016), no. 2, 569–579.[10] H. Ozden, Y. Simsek, and H. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Eulerand Genocchi polynomials, Comput. Math. Appl. , (2010), 2779–2787.[11] S. Roman, The theory of the umbral calculus. I, J. Math. Anal. Appl. (1982), no. 1, 58–115.[12] S. Roman, The umbral calculus. Pure and Applied Mathematics, 111. Academic Press, Inc., New York, 1984.[13] Y. Simsek, Complete sum of products of ( h , q ) -extension of Euler polynomials and numbers, J. Difference Equ. Appl. , (2010),no. 11, 1331–1348.[14] C. Tsallis, Possible generalizations of Boltzmann-Gibbs statistics, J. Stat. Phys. (1988), no. 1/2, 479–487.[15] P.T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, J. Number Theory , (2008),738–758.D EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF H AIFA , 3498838 H
AIFA , I
SRAEL
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NIVERSITY OF H AIFA , 3498838 H
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SRAEL
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