arXiv:1009.0098v1 [math.NT] 1 Sep 2010
SOME IDENTITIES OF BERNOULLI NUMBERS AND POLYNOMIALS ASSOCIATED WITH BERNSTEIN POLYNOMIALS MIN-SOO KIM, TAEKYUN KIM, BYUNGJE LEE AND CHEON-SEOUNG RYOO Abstract. We investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on Zp .
1. Introduction Let C[0, 1] be the set of continuous functions on [0, 1]. Then the classical Bernstein polynomials of degree n for f ∈ C[0, 1] are defined by n X k (1.1) Bn (f ) = f Bk,n (x), 0 ≤ x ≤ 1 n k=0
where Bn (f ) is called the Bernstein operator and n k (1.2) Bk,n (x) = x (x − 1)n−k k
are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [10]). Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials (see [1, 2]). Their generating function for Bk,n (x) is given by ∞ X tk e(1−x)t xk tn (1.3) F (k) (t, x) = = Bk,n (x) , k! n! n=0 where k = 0, 1, . . . and x ∈ [0, 1]. Note that ( n k n−k k x (1 − x) Bk,n (x) = 0,
if n ≥ k if n < k
for n = 0, 1, . . . (see [1, 2]). The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. Some researchers have studied the Bernstein polynomials in the area of approximation theory (see [1, 2, 3, 7, 9, 10]). In recent years, Acikgoz and Araci [1, 2] have introduced several type Bernstein polynomials. In the present paper, we introduce the Bernstein polynomials on the ring of padic integers Zp . We also investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on the ring of p-adic integers Zp . 2000 Mathematics Subject Classification. 11B68, 11S80. Key words and phrases. Bernstein polynomials, Bosonic p-adic integrals. 1
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MIN-SOO KIM, TAEKYUN KIM, BYUNGJE LEE AND CHEON-SEOUNG RYOO
2. Bernstein polynomials related to the bosonic p-adic integrals on Zp Let p be a fixed prime number. Throughout this paper, Zp , Qp and Cp will denote the ring of p-adic integers, the field of p-adic numbers and the completion of the algebraic closure of Qp , respectively. Let vp be the normalized exponential valuation of Cp with |p|p = p−1 . For N ≥ 1, the bosonic distribution µ1 on Zp µ(a + pN Zp ) =
(2.1)
1 pN
is known as the p-adic Haar distribution µHaar , where a + pN Zp = {x ∈ Qp | |x − a|p ≤ p−N } (cf. [4]). We shall write dµ1 (x) to remind ourselves that x is the variable of integration. Let U D(Zp ) be the space of uniformly differentiable function on Zp . Then µ1 yields the fermionic p-adic q-integral of a function f ∈ U D(Zp ) : N
p −1 1 X f (x)dµ1 (x) = lim N I1 (f ) = f (x) N →∞ p Zp x=0
Z
(2.2)
(cf. [4]). Many interesting properties of (2.2) were studied by many authors (cf. [4, 8] and the references given there). For n ∈ N, write fn (x) = f (x + n). We have (2.3)
I1 (fn ) = I1 (f ) +
n−1 X
f ′ (l).
l=0
This identity is to derives interesting relationships involving Bernoulli numbers and polynomials. Indeed, we note that Z n (x + y)n dµ1 (y) = Bn (x), (2.4) I1 ((x + y) ) = Zp
where Bn (x) are the Bernoulli polynomials (cf. [4]). From (1.2), we have n−k Z n X n−k Bk,n (x)dµ1 (x) = (−1)n−k−j Bn−j (2.5) j k j=0 Zp and Z
Bk,n (x)dµ1 (x) =
Bn−k,n (1 − x)dµ1 (x)
Zp
Zp
(2.6)
Z
X n−j k X n − j k n k−j (−1) = (−1)l Bl . k j=0 j l l=0
By (2.5) and (2.6), we obtain the following proposition. Proposition 2.1. For n ≥ k, n−j k n−k X n − j X X n − k k k−j n−k−j (−1) (−1)l Bl . (−1) Bn−j = j l j j=0 j=0 l=0
From (2.4), we note that (2.7)
Bn (2) = (B(1) + 1)n − n = (B + 1)n = Bn ,
n>1
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with the usual convention of replacing B n by Bn . Thus, we have Z
Z
xn dµ1 (x) =
(x + 2)n dµ1 (x) − n
Zp
Zp
n
= (−1)
(2.8)
Z
(x − 1)n dµ1 (x) − n
Zp
Z
=
(1 − x)n dµ1 (x) − n
Zp
for n > 1, since (−1)n Bn (x) = Bn (1 − x). Therefore we obtain the following theorem. Theorem 2.2. For n > 1, Z
(1 − x)n dµ1 (x) =
Z
xn dµ1 (x) + n.
Zp
Zp
And also we obtain Z Z Bn−k,k (x)dµ1 (x) =
xn−k (1 − x)k dµ1 (x)
Zp
Zp
n−k X
Z n−k = (1 − x)l+k dµ1 (x) (−1)l l Z p l=0 (Z ) n−k X n − k l+k l x dµ1 (x) + l + k = (−1) l Zp l=0 n−k X n − k = (−1)l (Bl+k + l + k). l
(2.9)
l=0
Therefore we obtain the following result. Corollary 2.3. For k > 1, Z
Bn−k,k (x)dµ1 (x) =
Zp
n−k X l=0
n−k (−1)l (Bl+k + l + k). l
From the property of the Bernstein polynomials of degree n, we easily see that Z (2.10)
Bk,n (x)Bk,m (x)dµ1 (x) Zp
Z n m x2k (1 − x)n+m−2k dµ1 (x) k k Zp n+m−2k X n + m − 2k n m = (−1)l B2k+l k k l =
l=0
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MIN-SOO KIM, TAEKYUN KIM, BYUNGJE LEE AND CHEON-SEOUNG RYOO
and
Z
Bk,n (x)Bk,m (x)Bk,s (x)dµ1 (x)
Zp
Z n m s x3k (1 − x)n+m−3k dµ1 (x) k k k Zp n+m+s−3k X n m s n + m + s − 3k = (−1)l B3k+l . k k k l =
(2.11)
l=0
Continuing this process, we obtain the following theorem.
Theorem 2.4. The multiplication of the sequence of Bernstein polynomials Bk,n1 (x), Bk,n2 (x), . . . , Bk,ns (x) for s ∈ N with different degree under p-adic integral on Zp can be given as Z Bk,n1 (x)Bk,n2 (x) · · · Bk,ns (x)dµ1 (x) Zp
n1 +n2 +···+n X s −sk n1 + n2 + · · · + ns − sk ns n2 n1 ··· (−1)l Bsk+l . = k k l k l=0
We put
m Bk,n (x) = Bk,n (x) × · · · × Bk,n (x) . {z } | m-times
Theorem 2.5. The multiplication of
m1 m2 ms Bk,n (x), Bk,n (x), . . . , Bk,n (x) 1 2 s
Bernstein polynomials with different degrees n1 , n2 , · · · , ns under p-adic integral on Zp can be given as Z m1 m2 ms Bk,n (x)Bk,n (x) · · · Bk,n (x)dµ1 (x) 1 2 s Zp
ms n1 m1 +n2 m2 +···+n m1 m2 s ms −(m1 +···+ms )k X ns n2 n1 (−1)l ··· k k k l=0 n1 m1 + n2 m2 + · · · + ns ms − (m1 + · · · + ms )k × B(m1 +···+ms )k+l . l Theorem 2.6. The multiplication of =
Bkm11,n1 (x), Bkm22,n2 (x), . . . , Bkmss,ns (x) Bernstein polynomials with different degrees n1 , n2 , · · · , ns with different powers m1 , m2 , · · · , ms under p-adic integral on Zp can be given as Z Bkm11,n1 (x)Bkm22,n2 (x) · · · Bkmss,ns (x)dµ1 (x) Zp
ms n1 m1 +n2 m2 +···+nsX m m2 ms −(k1 m1 +···+ks ms ) ns n1 1 n2 ··· (−1)l = ks k2 k1 l=0 n1 m1 + n2 m2 + · · · + ns ms − (k1 m1 + · · · + ks ms ) × Bk1 m1 +···+ks ms +l . l
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Problem. Find the Witt’s formula for the Bernstein polynomials in p-adic number field. References [1] M. Acikgoz and S. Araci, A study on the integral of the product of several type Bernstein polynomials, IST Transaction of Applied Mathematics-Modelling and Simulation, 2010. [2] M. Acikgoz and S. Araci, On the generating function of the Bernstein polynomials, Accepted to AIP on 24 March 2010 for ICNAAM 2010. [3] S. Bernstein, Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities, Commun. Soc. Math. Kharkow (2) 13 (1912-1913), 1–2. [4] T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320–329. [5] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299. [6] T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Physics A : Math. Theor., 43 (2010), 255201, 11pp. [7] T. Kim, L.-C. Jang and H. Yi, Note on the modified q-Bernstein polynomials, Discrete Dynamics in Nature and Society (in Press). arXiv 1005.4293. [8] T. Kim, J. Choi and Y.-H. Kim Some identities on the q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 335–341. arXiv 1006.2033. [9] G. M. Phillips, Bernstein polynomials based on the q-integer, Annals of Numerical Analysis 4 (1997), 511–518. [10] Y. Simsek and M. Acikgoz, A new generating function of q-Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal. ID 769095 (2010), 12 pp. Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305701, South Korea E-mail address:
[email protected] Division of General Education-Mathematics, Kwangwoon University, Seoul, 139-701, South Korea E-mail address:
[email protected] Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, South Korea E-mail address: bj
[email protected] Department of Mathematics, Hannam University, Daejeon 306-791, South Korea E-mail address:
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