arXiv:1006.4499v1 [math.NT] 23 Jun 2010
Generating functions for q-Bernstein,
q-Meyer-König-Zeller and q-Beta basis
Vijay Gupta
School of Applied Science, Netaji Subhas Institute of Technology
Sector 3 Dwarka 110078 New Delhi, India
vijaygupta2001@hotmail.com
Taekyun Kim, Jongsung Choi and Young-Hee Kim
Division of General Education-Mathematics, Kwangwoon University
Seoul 139-701, S. Korea,
tkkim@kw.ac.kr, jeschoi@kw.ac.kr, yhkim@kw.ac.kr
Abstract. The present paper deals with the q-analogues of Bernstein, Meyer-König-Zeller and Beta
operators. Here we estimate the generating functions for q-Bernstein, q-Meyer-König-Zeller and q-Beta
basis functions.
Keywords: q−integers, q−binomial coefficient, q-exponential, q-Bernstein basis, q-Meyer-KönigZeller basis and q-Beta basis function.
Mathematical subject classification: 41A25, 41A35.
1. Introduction
For each non-negative integer k, the q−integer [k]q and the q−factorial [k]q ! are respectively defined
by
[k]q =
and
(1 − q k ) (1 − q),
k,
q=
6 1
,
q=1
[k]q [k − 1]q · · · [1]q ,
k≥1
.
1,
k=0
For the integers n, k satisfying n ≥ k ≥ 0, the q−binomial coefficients are defined by
[n]q !
n
=
k q
[k]q ![n − k]q !
[k]q ! =
(see e.g. [3]). We consider the q-exponential function in the following form:
∞ X
1
n+k−1
= lim
xk
lim
k
n→∞
n→∞ (1 − x)n
q
q
k=0
∞
X
(1 − q n+k−1 )....(1 − q n ) k
= lim
x
n→∞
(1 − q)(1 − q 2 )...(1 − q k )
k=0
=
∞
X
k=0
xk
= eq (x).
(1 − q)(1 − q 2 )...(1 − q k )
Another form of q-exponential function is given as follows:
lim (1 + x)nq =
n→∞
∞
X
k=0
q k(k−1)/2 xk
= Eq (x).
(1 − q)(1 − q 2 )...(1 − q k )
It is easily observed that eq (x)Eq (−x) = eq (−x)Eq (x) = 1.
Generating functions
2
Based on the q-integers Phillips [1] introduced the q analogue of the well known Bernstein polynomials.
For f ∈ C[0, 1] and 0 < q < 1, the q-Bernstein polynomials are defined as
n
X
[k]q
q
,
(1.1)
Bn,q (f, x) =
bk,n (x)f
[n]q
k=0
where the q-Bernstein basis function is given by
n
q
bk,n (x) =
xk (1 − x)qn−k , x ∈ [0, 1]
k q
n
and (a − b)q =
n−1
Q
(a − q s b),
s=0
a, b ∈ R.
Also Trif [2] proposed the q analogue of well known Meyer-König-Zeller operators. For f ∈ C[0, 1] and
0 < q < 1, the q-Meyer-König-Zeller operators are defined as
∞
X
[k]q
q
Mn,q (f, x) =
mk,n (x)f
,
(1.2)
[n]q
k=0
where the q-MKZ basis function is given by
n+k+1
mqk,n (x) =
xk (1 − x)nq , x ∈ [0, 1].
k
q
For f ∈ C[0, ∞) and 0 < q < 1, the q-Beta operators are defined as
∞
[k]q
1 X q
vk,n (x)f
,
Vn (f, x) =
[n]q
q k−1 [n]q
(1.3)
k=0
where the q-Beta basis function is given by
q
vk,n
(x) =
q k(k−1)/2
xk
, x ∈ [0, ∞)
Bq (k + 1, n) (1 + x)qn+k+1
and Bq (m, n) is q-Beta function.
In the present paper we establish the generating functions for q-Bernstein, q-Meyer-König-Zeller and
q-Beta basis functions.
2. Generating Function for q-Bernstein basis
Theorem 1.
bqk,n (x)
is the coefficient of
tn
[n]q !
in the expansion of
xk tk
eq ((1 − q)(1 − x)q t).
[k]q !
Proof. First consider
xk tk
eq ((1 − q)(1 − x)q t) =
[k]q !
=
=
=
=
∞
xk tk X (1 − x)nq tn
[k]q ! n=0
[n]q !
∞
1 X xk (1 − x)nq tn+k
[k]q ! n=0
[n]q !
∞
X
[n + 1]q [n + 2]q .....[n + k]q xk (1 − x)nq tn+k
[n + k]q ![k]q !
n=0
∞
X n+k
xk (1 − x)nq tn+k
k
[n + k]q !
q
n=0
∞
∞
X
X
xk (1 − x)qn−k tn
tn
n
=
bqk,n (x)
.
k q
[n]q !
[n]q !
n=0
n=k
This completes the proof of generating function for bqk,n (x).
V.Gupta et al.
3
3. Generating Function for q-MKZ basis
Theorem 2. mqk,n (x) is the coefficient of tk in the expansion of
(1−x)n
q
(1−xt)n+2
q
.
Proof. It is easily seen that
(1 − x)nq
(1 − xt)n+2
q
=
∞ ∞
X
X
n+k+1
(1 − x)nq xk tk =
mqk,n (x)tk .
k
q
k=0
k=0
This completes the proof.
4. Generating Function for q-Beta basis
q
Theorem 3. It is observed by us that vk,n
(x) is the coefficient of
tk
[n+k]q !
in the expansion of
1
Eq
(1+x)n+1
q
(1−q)xt
(1+qn+1 x)q
Proof. First using the definition of q-exponential Eq (x), we have
∞
X
tk
1
1
(1 − q)xt
xk
=
E
q k(k−1)/2
q
n+1
n+1
n+1
n+1
k
(1 + q
x)q
(1 + q
x)q [k]q !
(1 + x)q
(1 + x)q
k=0
∞
X
q k(k−1)/2
=
∞
X
q k(k−1)/2
=
∞
X
=
k=0
xk
tk
n+k+1 [k] !
(1 + x)q
q
xk tk
[k + 1]q [k + 2]q ...[n + k]q
n+k+1
[n + k]q !
(1 + x)q
k=0
∞
X
[n]q tk
xk
n+k
=
q k(k−1)/2
n+k+1
n
(1 + x)q
q [n + k]q !
k=0
=
k=0
∞
X
tk
xk
1
q k(k−1)/2
n+k+1
Bq (k + 1, n)
[n + k]q !
(1 + x)q
q
vk,n
(x)
k=0
tk
.
[n + k]q !
This completes the proof of generating function.
Acknowledgement: The present work was done when the first author visited Division of General
Education-Mathematics, Kwangwoon University, Seoul, S. Korea during June 2010.
References
[1] G. M. Phillips, On generalized Bernstein polynomials. In: Griffiths, D. F., Watson, G.A.(eds): Numerical analysis.
Singapore: World Scientific 1996, pp. 263-269.
[2] T. Trif, Meyer-König and Zeller operators based on the q-integers, Rev. Anal. Numer. Theor. Approx. 29 (2000), No.
2, 221-229.
[3] T. Ernst, The history of q−calculus and a new method, U.U.D.M. Report 2000, 16, Uppsala, Departament of Mathematics, Uppsala University (2000).
.