A new class of generalized Bernoulli polynomials and Euler polynomials

0.13 MB·English

by  Nazim I. Mahmudov

A new class of generalized Bernoulli polynomials and Euler
polynomials
N. I. Mahmudov
Eastern Mediterranean University
Gazimagusa, TRNC, Mersiin 10, Turkey
2
1 Email: [email protected]
0
2 February 1, 2012

n
a
J
Abstract
1
3 The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli
polynomialsandEulerpolynomialsbasedontheq-integers. Theq-analoguesofwell-knownformulasare

] derived. The q-analogue of the Srivastava–Pint´er addition theorem is obtained. We give new identities
A
involvingq-Bernstein polynomials.
C
.
h
1 Introduction
t
a
m Throughoutthispaper,wealwaysmakeuseofthe followingnotation: Ndenotesthesetofnaturalnumbers,
[ N denotes the set of nonnegative integers, R denotes the set of real numbers, C denotes the set of complex
0
numbers.
1
v The q-shifted factorial is defined by
3
3 n−1 ∞
6 (a;q) =1, (a;q) = 1−qja , n∈N, (a;q) = 1−qja , |q|<1, a∈C.
0 n ∞
6
j=0 j=0
. Y(cid:0) (cid:1) Y(cid:0) (cid:1)
1
0 The q-numbers and q-numbers factorial is defined by
2
1 [a] = 1−qa (q 6=1); [0] !=1; [n] !=[1] [2] ...[n] n∈N, a∈C
: q 1−q q q q q q
v
i
X respectively. The q-polynomial coefficient is defined by
r
a n (q;q)
= n .
k (q;q) (q;q)
(cid:20) (cid:21)q n−k k
The q-analogue of the function (x+y)n is defined by
n
(x+y)nq := nk q12k(k−1)xn−kyk, n∈N0.
k=0(cid:20) (cid:21)q
X
The q-binomial formula is known as
n−1 n
(1−a)n =(a;q) = 1−qja = n q21k(k−1)(−1)kak.
q n k
j=0 k=0(cid:20) (cid:21)q
Y(cid:0) (cid:1) X
1
In the standard approach to the q-calculus two exponential functions are used:
∞ zn ∞ 1 1
e (z)= = , 0<|q|<1, |z|< ,
q [n] ! (1−(1−q)qkz) |1−q|
n=0 q k=0
X Y
∞ q21n(n−1)zn ∞
E (z)= = 1+(1−q)qkz , 0<|q|<1, z ∈C.
q [n] !
n=0 q k=0
X Y(cid:0) (cid:1)
From this form we easily see that e (z)E (−z)=1. Moreover,
q q
D e (z)=e (z), D E (z)=E (qz),
q q q q q q
where D is defined by
q
f(qz)−f(z)
D f(z):= , 0<|q|<1, 06=z ∈C.
q qz−z
The above q-standard notation can be found in [1].
Over 70 years ago, Carlitz extended the classical Bernoulli and Euler numbes and polynomials and
introduced the q-Bernoulli and the q-Euler numbers and polynomials (see [2], [3] and [4] ). There are
numerous recent investigations on this subject by, among many other authors, Cenki et al. ([12], [13], [14]),
Choi et al. ([15] and [16]), Kim et al. ([17]-[24]), Ozden and Simsek [25], Ryoo et al. [28], Simsek ([29], [30]
and [31]), and Luo and Srivastava [11], Srivastava et al. [32].
Wefirstgiveherethedefinitionsoftheq-Bernoulliandtheq-Eulerpolynomialsofhigherorderasfollows.
Definition 1 Let q,α∈ C, 0 <|q| < 1. The q-Bernoulli numbers B(α) and polynomials B(α)(x,y) in x,y
n,q n,q
of order α are defined by means of the generating function functions:
t α ∞ tn
= B(α) , |t|<2π,
e (t)−1 n,q[n] !
(cid:18) q (cid:19) n=0 q
X
t α ∞ tn
e (tx)E (ty)= B(α)(x,y) , |t|<2π.
e (t)−1 q q n,q [n] !
(cid:18) q (cid:19) n=0 q
X
Definition 2 Let q,α ∈ C, 0 < |q| < 1. The q-Euler numbers E(α) and polynomials E(α)(x,y) in x,y of
n,q n,q
order α are defined by means of the generating functions:
2 α ∞ tn
= E(α) , |t|<π,
e (t)+1 n,q[n] !
(cid:18) q (cid:19) n=0 q
X
2 α ∞ tn
e (tx)E (ty)= E(α)(x,y) , |t|<π.
e (t)+1 q q n,q [n] !
(cid:18) q (cid:19) n=0 q
X
It is obvious that
B(α) =B(α)(0,0), lim B(α)(x,y)=B(α)(x+y), lim B(α) =B(α),
n,q n,q q→1− n,q n q→1− n,q n
E(α) =E(α)(0,0), lim E(α)(x,y)=E(α)(x+y), lim E(α) =E(α),
n,q n,q q→1− n,q n q→1− n,q n
lim B(α)(x,0)=B(α)(x), lim B(α)(0,y)=B(α)(y),
q→1− n,q n q→1− n,q n
lim E(α)(x,0)=E(α)(x), lim E(α)(0,y)=E(α)(y).
q→1− n,q n q→1− n,q n
2
HereB(α)(x)andE(α)(x)denotetheclassicalBernoulliandEulerpolynomialsoforderαwhicharedefined
n n
by
t α ∞ tn 2 α ∞ tn
etx = B(α)(x) and etx = E(α)(x) .
et−1 n [n] ! et+1 n [n] !
(cid:18) (cid:19) n=0 q (cid:18) (cid:19) n=0 q
X X
InfactDefinitions1and2definetwodifferenttypeB(α)(x,0)andB(α)(0,y)oftheq-Bernoullipolynomials
n,q n,q
andtwodifferenttypeE(α)(x,0)andE(α)(0,y)oftheq-Eulerpolynomials. BothpolynomialsB(α)(x,0)and
n,q n,q n,q
B(α)(0,y) (E(α)(x,0) and E(α)(0,y)) coincide with the classical highe order Bernoulli polynomilas (Euler
n,q n,q n,q
polynomilas) in the limiting case q →1−.
For the q-Bernoulli numbers B , the q-Euler numbers E of order n, we have
n,q n,q
B =B (0,0)=B(1) (0,0), E =E (0,0)=E(1) (0,0),
n,q n,q n,q n,q n,q n,q
respectively. Note that the q-Bernoulli numbers B are defined and studied in [26].
n,q
The aim of the present paper is to obtain some results for the above defined q-Bernoulli and q-Euler
polynomials. In this paper the q-analogues of well-known results, for example, Srivastava and Pint´er [10],
Cheon [5], etc., will be given. Also the formulas involving the q-Stirling numbers of the second kind, q-
Bernoulli polynomials and Phillips q-Bernstein polynomials are derived.
2 Preliminaries and Lemmas
In this section we shall provide some basic formulas for the q-Bernoulli and q-Euler polynomials in order to
obtain the main results of this paper in the next section. The following result is q-analogue of the addition
theorem for the classical Bernoulli and Euler polynomials.
Lemma 3 (Addition Theorems) For all x,y ∈C we have
n n
n n
B(α)(x,y)= B(α)(x+y)n−k, E(α)(x,y)= E(α)(x+y)n−k,
n,q k k,q q n,q k k,q q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
n n
n n
B(α)(x,y)= q(n−k)(n−k−1)/2B(α)(x,0)yn−k = B(α)(0,y)xn−k, (1)
n,q k k,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
n n
n n
E(α)(x,y)= q(n−k)(n−k−1)/2E(α)(x,0)yn−k = E(α)(0,y)xn−k. (2)
n,q k k,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
In particular, setting x = 0 and y = 0 in (1) and (2), we get the following formulas for q-Bernoulli and
q-Euler polynomials, respectively.
n n
n n
B(α)(x,0)= B(α)xn−k, B(α)(0,y)= q(n−k6 )(n−k−1)/2B(α)yn−k, (3)
n,q k k,q n,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
n n
n n
E(α)(x,0)= E(α)xn−k, E(α)(0,y)= q(n−k6 )(n−k−1)/2E(α)yn−k. (4)
n,q k k,q n,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
Setting y =1 and x=1 in (1) and (2), we get
n n
n n
B(α)(x,1)= q(n−k)(n−k−1)/2B(α)(x,0), B(α)(1,y)= B(α)(0,y), (5)
n,q k k,q n,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
n n
n n
E(α)(x,1)= q(n−k)(n−k−1)/2E(α)(x,0), E(α)(1,y)= E(α)(0,y). (6)
n,q k k,q n,q k k,q
k=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q
X X
3
Clearly (5) and (6) are q-analogues of
n n
n n
B(α)(x+1)= B(α)(x), E(α)(x+1)= E(α)(x),
n k k n k k
k=0(cid:18) (cid:19) k=0(cid:18) (cid:19)
X X
respectively.
Lemma 4 We have
D B(α)(x,y)=[n] B(α) (x,y), D B(α)(x,y)=[n] B(α) (x,qy),
q,x n,q q n−1,q q,y n,q q n−1,q
D E(α)(x,y)=[n] E(α) (x,y), D E(α)(x,y)=[n] E(α) (x,qy).
q,x n,q q n−1,q q,y n,q q n−1,q
Lemma 5 (Difference Equations) We have
B(α)(1,y)−B(α)(0,y)=[n] B(α−1)(0,y), (7)
n,q n,q q n−1,q
E(α)(1,y)+E(α)(0,y)=2E(α−1)(0,y), (8)
n,q n,q n,q
B(α)(x,0)−B(α)(x,−1)=[n] B(α−1)(x,−1),
n,q n,q q n−1,q
E(α)(x,0)+E(α)(x,−1)=2E(α−1)(x,−1).
n,q n,q n,q
From (7) and (3), (8) and (4) we obtain the following formulas.
Lemma 6 We have
n
B(α−1)(0,y)= 1 n+1 B(α)(0,y), (9)
n−1,q [n+1] k k,q
q k=0(cid:20) (cid:21)q
X
n
1 n
E(α−1)(0,y)= E(α)(0,y)+E(α)(0,y) . (10)
n,q 2 k k,q n,q
"k=0(cid:20) (cid:21)q #
X
Putting α=1 in (9) and (10), and noting that
B(0) (0,y)=E(0) (0,y)=qn(n−1)/2yn,
n,q n,q
we arrive at the following expansions:
n
1 n+1
yn = B (0,y),
qn(n−1)/2[n+1] k k,q
qk=0(cid:20) (cid:21)q
X
n
1 n
yn = E (0,y)+E (0,y) ,
2qn(n−1)/2 "k=0(cid:20) k (cid:21)q k,q n,q #
X
which are q-analoques of the following familiar expansions
n n
1 n+1 1 n
yn = B (y), yn = E (y)+E (y) , (11)
n+1 k k 2 k k n
k=0(cid:18) (cid:19) "k=0(cid:18) (cid:19) #
X X
respectively.
4
Lemma 7 (Recurrence Relationships) The polynomials B(α)(x,0) and E(α)(x,0) satisfy the following dif-
n,q n,q
ference relationships:
k k k−1
k mjB(α)(x,0)− k mjB(α)(x,−1)=[k] k−1 mj+1B(α−1)(x,−1), (12)
j j,q j j,q q j j,q
j=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q
X X X
B(α) 1 ,y − k k 1 −1 k−jB(α)(0,y)=[k] k−1 k−1 1 −1 k−j−1B(α−1)(0,y),
k,q m j m j,q q j m j,q
(cid:18) (cid:19) j=0(cid:20) (cid:21)q(cid:18) (cid:19)q j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X X
(13)
k k k
k k k
mjE(α)(x,0)+ mjE(α)(x,−1)=2 mjE(α−1)(x,−1), (14)
j j,q j j,q j j,q
j=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q
X X X
k k−j k k−j
1 k 1 k 1
E(α) ,y + −1 E(α)(0,y)=2 −1 E(α−1)(0,y). (15)
k,q m j m j,q j m j,q
(cid:18) (cid:19) j=0(cid:20) (cid:21)q(cid:18) (cid:19)q j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X X
3 Explicit relationship between the q-Bernoulli and q-Euler poly-
nomials
In this section we shall investigate some explicit relationships between the q-Bernoulli and q-Euler polyno-
mials. Here some q-analogues of known results will be given. We also obtain new formulas and their some
special cases below. These formulas are some extensions of the formulas of Srivastava and A´. Pint´er,Cheon
and others.
We present natural q-extensions of th main results of the papers [10], [8], see Theorems 8 and 13.
Theorem 8 For n∈N , the following relationship
0
n k
1 n k
B(α)(x,y)= mkB(α)(x,0)+ mjB(α)(x,−1)
n,q 2mn k  k,q j j,q
k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q
X X

k−1
+[k] k−1 mj+1B(α−1)(x,−1) E (0,my),
q j j,q  n−k,q
j=0(cid:20) (cid:21)q
X

n k k−j
1 n k 1
B(α)(x,y)= mk B(α)(0,y)+ −1 B(α)(0,y)
n,q 2mn j  k,q j m j,q
k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X X

k−1 k−1−j
+[k] k−1 1 −1 B(α−1)(0,y) E (mx,0)
q j m j,q  n−k,q
j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X

holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Proof. Using the following identity
t α 2 t e t +1 t α
e (tx)E (ty)= ·E my · q m · e (tx)
e (t)−1 q q e t +1 q m 2 e (t)−1 q
(cid:18) q (cid:19) q m (cid:18) (cid:19) (cid:0) (cid:1) (cid:18) q (cid:19)
(cid:0) (cid:1)
5
we have
∞ tn 1 ∞ tn ∞ tn ∞ tn
B(α)(x,y) = E (0,my) B(α)(x,0)
n,q [n] ! 2 n,q mn[n] ! mn[n] ! n,q [n] !
n=0 q n=0 q n=0 q n=0 q
X X X X
1 ∞ tn ∞ tn
+ E (0,my) B(α)(x,0)
2 n,q mn[n] ! n,q [n] !
n=0 q n=0 q
X X
=:I +I .
1 2
It is clear that
1 ∞ tn ∞ tn 1 ∞ n n tn
I = E (0,my) B(α)(x,0) = mk−nB(α)(x,0)E (0,my) .
2 2 n,q mn[n] ! n,q [n] ! 2 j k,q n−k,q [n] !
n=0 q n=0 q n=0k=0(cid:20) (cid:21)q q
X X XX
On the other hand
1 ∞ tn ∞ n n tn
I = B(α)(x,0) m−nE (0,my)
1 2 n,q [n] ! j j,q [n] !
n=0 q n=0j=0(cid:20) (cid:21)q q
X XX
1 ∞ n n n−k n−k tn
= B(α)(x,0) mk−nE (0,my)
2 k k,q j j,q [n] !
n=0k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q q
XX X
1 ∞ n n n−j n−j tn
= m−n E (0,my) mkB(α)(x,0) .
2 j j,q k k,q [n] !
n=0 j=0(cid:20) (cid:21)q k=0(cid:20) (cid:21)q q
X X X
Therefore
∞ tn 1 ∞ n n k k tn
B(α)(x,y) = mk−n B(α)(x,0)+m−k mjB(α)(x,0) E (0,my) .
n,q [n] ! 2 j  k,q j j,q  n−k,q [n] !
n=0 q n=0k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q q
X XX X
 
It remains to use the formula (12).
Next we discuss some special cases of Theorem 8.
Corollary 9 For n∈N , m∈N the following relationship
0
n k
1 n k
B (x,y)= mkB (x,0)+ mjB (x,−1)
n,q 2mn k  k,q j j,q
k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q
X X

k−1
k−1
+ [k] mj+1(x−1)j E (0,my),
q j q n−k,q
j=0(cid:20) (cid:21)q
X

n k k−j
1 n k 1
B (x,y)= mk mkB (0,y)+ −1 B (0,y)
n,q 2mn j  k,q j m j,q
k=0(cid:20) (cid:21)q j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X X

k−1 k−1−j
+[k]q k−j 1 q21j(j−1) m1 −1 yjEn−k,q(mx,0)
j=0(cid:20) (cid:21)q (cid:18) (cid:19)q
X

holds true between the q-Bernoulli polynomials and q-Euler polynomials.
6
Corollary 10 [8] For n∈N , m∈N the following relationship holds true.
0
n
n k
B (x+y)= B (y)+ yk−1 E (x),
n k k 2 n−k
k=0(cid:18) (cid:19)(cid:18) (cid:19)
X
n
1 n 1
B (x+y)= mkB (x)+mkB x−1+ +km(1+m(x−1))k−1 E (my).
n 2mn k k k m n−k,q
k=0(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
X
Corollary 11 For n∈N the following relationship holds true.
0
Bn,q(x,y)= n nk Bk,q(0,y)+q12(k−1)(k−2)[k2]qyk−1 En−k,q(x,0). (16)
k=0(cid:20) (cid:21)q(cid:18) (cid:19)
X
Corollary 12 For n∈N the following relationship holds true.
0
n
n 1
B (x,0)= B E (x,0)+ B + E (x,0), (17)
n,q k k,q n−k,q 1,q 2 n−1,q
k=0 (cid:20) (cid:21) (cid:18) (cid:19)
(kX6=1)
n
n 1
B (0,y)= B E (0,y)+ B + E (0,y). (18)
n,q k k,q n−k,q 1,q 2 n−1,q
k=0 (cid:20) (cid:21)q (cid:18) (cid:19)
(kX6=1)
The formulas(16)-(18) areq-extensionofthe Cheon’s mainresult[5]. Notice thatB =− 1 , see [26],
1,q [2]
q
and the extra term becomes zeo for q →1−.
Theorem 13 For n∈N , the following relationship
0
n k+1 k+1−j
E(α)(x,y)= 1 2 k+1 1 −1 E(α−1)(0,y)
n,q mn−1[k+1]  j m j,q
k=0 q j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X X

k+1 k+1−j
k+1 1
− −1 E(α)(0,y)−E(α) (0,y) B (mx,0),
j m j,q k+1,q  n−k,q
j=0(cid:20) (cid:21)q(cid:18) (cid:19)q
X

n k+1
E(α)(x,y)= n 1 2 k+1 mjE(α−1)(x,−1)
n,q k mn[k+1]  j j,q
k=0(cid:20) (cid:21)q q j=0(cid:20) (cid:21)q
X X

k+1
k+1
− mjE(α)(x,−1)−mk+1E(α) (x,0) B (0,my)
j j,q k+1,q  n−k,q
j=0(cid:20) (cid:21)q
X

holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Proof. The proof is based on the following identities
2 αe (tx)E (ty)= 2 αE (ty)· eq mt −1 · t e t mx ,
e (t)+1 q q e (t)+1 q t e t −1 q m
(cid:18) q (cid:19) (cid:18) q (cid:19) (cid:0) (cid:1) q m (cid:18) (cid:19)
2 α 2 α e t −1 (cid:0) t(cid:1) t
e (tx)E (ty)= e (tx)· q m · E my
e (t)+1 q q e (t)+1 q t e t −1 q m
(cid:18) q (cid:19) (cid:18) q (cid:19) (cid:0) (cid:1) q m (cid:18) (cid:19)
and similar to that of Theorem 8. (cid:0) (cid:1)
Next we discuss some special cases of Theorem 13.
7
Corollary 14 For n∈N , m∈N the following relationship
0
n n m−n k+1 k+1
E (x,y)= 2 mj(x−1)j
n,q k [k+1]  j q
k=0(cid:20) (cid:21)q q j=0(cid:20) (cid:21)q
X X

k+1
k+1
− mjE (x,−1)−mk+1E (x,0) B (0,my)
j j,q k+1,q  n−k,q
j=0(cid:20) (cid:21)q
X

holds true between the q-Bernoulli polynomials and q-Euler polynomials.
Corollary 15 [8] For n∈N , m∈N the following relationship holds true.
0
n
2 n
E (x+y)= yk+1−E (y) B (x),
n k+1 k k+1 n−k
k=0 (cid:18) (cid:19)
X (cid:0) (cid:1)
n n mk−n+1 1−m k+1 1−m
E (x+y)= 2 x+ −E x+ −E (x) B (my).
n k k+1 m k+1 m k+1 n−k
k=0(cid:18) (cid:19) " (cid:18) (cid:19) (cid:18) (cid:19) #
X
Corollary 16 For n∈N the following relationship holds true.
0
n
En,q(x,y)= nk [k+21] q21k(k+1)yk+1−Ek+1,q(0,y) Bn−k,q(x,0).
kX=0(cid:20) (cid:21)q q (cid:16) (cid:17)
Corollary 17 For n∈N the following relationship holds true.
0
n
n 2
E (x,0)=− E B (x,0),
n,q k [k+1] k+1,q n−k,q
k=0(cid:20) (cid:21) q
X
n
n 2
E (0,y)=− E B (0,y).
n,q k [k+1] k+1,q n−k,q
k=0(cid:20) (cid:21)q q
X
These formulas are q-analogues of the formula of Srivastava and A´. Pint´er [10].
4 q-Stirling Numbers and q-Bernoulli Polynomials
In this section, we aim to derive severalformulas involving the q-Bernoullipolynomials,the q-Euler polyno-
mials of order α, the q-Stirling numbers of the second kind and q-Bernstein polynomials.
Theorem 18 Each of the following relationships holds true for the Stirling numbers S (n,k) of the second
2
kind:
n n−j
mx n
B(α)(x,y)= j! mj−nB(α)(0,y)S (n−k,j),
n,q j k k,q 2
j=0(cid:18) (cid:19) k=0(cid:20) (cid:21)q
X X
n n−j
mx n
E(α)(x,y)= j! mj−nE(α)(0,y)S (n−k,j).
n,q j k k,q 2
j=0(cid:18) (cid:19) k=0(cid:20) (cid:21)q
X X
The familiar q-Stirling numbers S(n,k) of the second kind are defined by
(e (t)−1)k ∞ tm
q = S (m,k) ,
[k] ! 2,q [m] !
q m=0 q
X
8
where k ∈ N. Next we give relationship between q-Bernstein basis defined by Phillips [27] and q-Bernoulli
polynomials
b (q;x):=xk(1−x)n−k.
n,k q
Theorem 19 We have
n
n
b (q;x)=xk S (m,k)B(k) (1,−x). (19)
n,k m 2,q n−m,q
m=0(cid:20) (cid:21)q
X
Proof. The proof follows from the following identities.
xktk xktk ∞ (1−x)ntn ∞ n xk(1−x)n−ktn
e (t)E (−xt)= q = q
[k] ! q q [k] ! [n] ! k [n] !
q q n=0 q n=k(cid:20) (cid:21)q q
X X
∞ tn
= b (q;x) .
n,k [n] !
n=k q
X
and
xktk xk(e (t)−1)k tk
e (t)E (−xt)= q e (t)E (−xt)
[k] ! q q [k] ! (e (t)−1)k q q
q q q
∞ tm ∞ tn
=xk S (m,k) B(k)(1,−x)
2,q [m] ! n,q [n] !
m=0 q n=0 q
X X
∞ n n tn
=xk S (m,k)B(k) (1,−x) .
m 2,q n−m,q [n] !
n=0 m=0(cid:20) (cid:21)q ! q
X X
Finally, in their limit case when q →1−, these last result (19) would reduce to the following formula for
the classical Bernoulli polynomials B(k)(x) and the Bernstein basis b (x)=xk(1−x)n−k :
n n,k
n
n
b (x)=xk S (m,k)B(k) (1−x).
n,k m 2 n−m
m=0(cid:20) (cid:21)q
X
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