(q, γ)-Bernstein Basis Functions on the Interval [a;b]
Abstract: -In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by qand
γand defined over an interval. We give detailed proofs for several key properties of these functions, including
the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce
and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementa-
tion. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein
basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided
Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions,
considering their impact on future research and developments in the domain. By exploring these aspects, we aim
to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.
Key-Words: -(q, γ)-Bernstein polynomials, (q, γ)-Bernstein basis, Marsden's identity, (q, γ)-Bezier curves, De
Casteljau, interval.
Received: April 7, 2024. Revised: September 2, 2024. Accepted: September 22, 2024. Published: October 22, 2024.
1 Introduction
Bernstein's basic polynomials, developed by the Rus-
sian mathematician Sergei Natanovich Bernstein in
the early 20th century, represent a class of polyno-
mials that play a fundamental role in various fields
of applied mathematics, from numerical analysis to
computer-aided geometric design (CAGD). These
polynomials form the basis of the representation of
Bézier curves and surfaces.
The foundations of Bézier theory, developed by
French engineers Pierre Bézier and Paul de Castel-
jau in the 1960s, have found widespread application
in fields such as computer-aided geometric design
(CAGD), surface modelling and graphic animation.
Recently, basic q-Bernstein polynomials have
been develeped in [1], [2], [3] and inspired by clas-
sical Bernstein polynomials on the segment, have
added an extra dimension thanks to the parameter q,
thus opening up new perspectives for the more pre-
cise approximation of functions on the interval [a;b].
More recently, a special case of q-Bernstein polyno-
mials on the triangle has been developed by [4], in-
spired by classical Bernstein polynomials on trian-
gular [5]. Thus, the introduction of the qparameter
offers greater flexibility and more refined modelling
possibilities for q-Bézier curves and surfaces [4], [6],
[7]. Very recently, the theory of h-Bézier curves was
developed by [8] and that of h-Bézier surfaces by [9].
In this article, we add a new real parameter γto
the parameter qparameter in order to introduce new
polynomial functions with base (q, γ)-Bernstein and
(q, γ)-Bézier curves on an interval [a;b]. We de-
fine these new (q, γ)-Bernstein basis polynomials and
(q, γ)-Bézier curves, then formulate and prove sev-
SORO SIONFON SIMON1, HAUDIÉ JEAN STEPHANE INKPÉ2, KOUA BROU JEAN CLAUDE1
1UFR Mathematiques et informatique
Université Felix Houphouët Boigny,
CÔTE D’IVOIRE
2Digital Research and Expertise Unit
Université Virtuelle de Côte d'Ivoire (UVCI)
28 BP 536 Abidjan
CÔTE D’IVOIRE
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DOI: 10.37394/23206.2024.23.69
Soro Sionfon Simon,
Haudié Jean Stephane Inkpé, Koua Brou Jean Claude
E-ISSN: 2224-2880
660
Volume 23, 2024
eral properties of these polynomials and curves, in-
cluding recurrence relations, partition of unity, degree
elevation, linear independence (polynomial basis), De
Casteljau's algorithm and the (q, γ)-Marsden identity.
Finally we conclude and provide some perspectives.
2 Bernstein polynomials
Bernstein's bases functions currently come in
three elementary forms
nis a natural integer.
•on the segment [0; 1]:[6]
1. the classic form:
Bn
k(t) = n
ktk(1 −t)n−k, k = 0,· · · , n,
2. the h-form:
Bn
k(t;h) = n
k
k−1
Y
i=0
(t+ih)
n−k−1
Y
i=0
(1 −t+ih)
n−1
Y
i=0
(1 + ih)
,
k= 0,· · · , n,
3. the q-form:
Bn
k(t;q) = n
kq
tk
n−k−1
Y
i=0
(1 −tqi),
k= 0,· · · , n,
•on an interval [a;b]:[6]
1. the classic form on [a;b]:
Bn
k(t; [a;b]) = n
k(t−a)k(b−t)n−k
(b−a)n,
k= 0,· · · , n,
2. the h-form on [a;b]:
Bn
k(t; [a;b]; h) =
n
k
k−1
Y
i=0
(t−a+ih)
n−k−1
Y
i=0
(b−t+ih)
n−1
Y
i=0
(b−a+ih)
,
k= 0,· · · , n,
3. the q-form on [a;b]:
Bn
k(t; [a;b]; q) =
n
kq
k−1
Y
i=0
(t−aqi)
n−k−1
Y
i=0
(b−tqi)
n−1
Y
i=0
(b−aqi)
,
k= 0,· · · , n,
where [10]: n
kq
=[n]q!
[k]q![n−k]q!
[k]q! = [k]q[k−1]q· · · [1]q,
[0]q! = 1
[k]q= 1 + q+· · · +qk−1
=1−qk
1−q,with q6= 1.
3(q, γ)−Bernstein basis functions on
an interval [a;b]
Let γbe a real number.
Let τ=b−a;τ1=t−a;τ2=b−t;
˜γ=γ+t−a=γ+τ1.
On the interval [a;b], we define the (q, γ)-Bernstein
polynomials of degree nby:
Bn
k(t; [a;b]; q, γ) =
n
kq
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
,
for n≥1
1,for n= 0
(1)
We have:
•lim
q7→1Bn
k(t; [a;b]; q, γ) = Bn
k(t; [a, b]); where
Bn
k(t; [a, b]) = n
k(t−a)k(b−t)n−k
(b−a)n
•lim
γ7→aBn
k(t; [a;b]; q, γ) = Bn
k(t; [a, b]; q); where
Bn
k(t; [a;b]; q) = n
kq
k−1
Y
i=0
(t−aqi)
n−k−1
Y
i=0
(b−tqi)
n−1
Y
i=0
(b−aqi)
•Note that Bn
k(t; [a;b]; q, γ)≥0,
for −1< q ≤1and γ≥0
•The effect of the γparameter can be seen in
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4 Recurrence relation for the
(q, γ)-Bernstein polynomials
For k= 0,· · · , n. we have [10]:
n
kq
=qn−kn−1
k−1q
+n−1
kq
(2)
and
n
kq
=n−1
k−1q
+qkn−1
kq
(3)
then the (q, γ)-Bernstein polynomials verify the fol-
lowing recurrence relations:
Bn
k(t; [a;b]; q, γ) =
τ2+ ˜γqn−k−1
τ+γqn−1Bn−1
k(t; [a;b]; q, γ)
+qn−kτ1+γqk−1
τ+γqn−1Bn−1
k−1(t; [a;b]; q, γ)
(4)
and
Bn
k(t; [a;b]; q, γ) =
qkτ2+ ˜γqn−k−1
τ+γqn−1Bn−1
k(t; [a;b]; q)
+τ1+γqk−1
τ+γqn−1Bn−1
k−1(t; [a;b]; q)
(5)
Proof:
We fix n
−1q
= 0 et n
n+ 1 q
= 0
By using (2) we have:
for k= 0,· · · , n.
Bn
k(t; [a;b]; q, γ)
=n
kq
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=n−1
kq
τ2+ ˜γ(1 −qn−k−1)
τ+γ(1 −qn−1)n−2
Y
i=0
[τ+γ(1 −qi)]
×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−2
Y
i=0
[τ2+ ˜γ(1 −qi)]
+qn−kn−1
k−1q
τ1+γ(1 −qk−1)
τ+γ(1 −qn−1)×
k−2
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−2
Y
i=0
[τ+γ(1 −qi)]
=τ2+ ˜γ(1 −qn−k−1)
τ+γ(1 −qn−1)Bn−1
k(t; [a;b]; q, γ)+
qn−kτ1+γ(1 −qk−1)
τ+γ(1 −qn−1)Bn−1
k−1(t; [a;b]; q, γ)
Similarly, by using (3) we show that:
Bn
k(t; [a;b]; q, γ) =
qkτ2+ ˜γ(1 −qn−k−1)
τ+γ(1 −qn−1)Bn−1
k(t; [a;b]; q)+
τ1+γ(1 −qk−1)
τ+γ(1 −qn−1)Bn−1
k−1(t; [a;b]; q)
5 Partition of unity
The (q, γ)-Bernstein polynomials verify the partition
of unity property:
n
X
k=0
Bn
k(t; [a;b]; q, γ) = 1 (6)
Proof: to prove (6), we proceed by recurrence
respect n≥0
Figure 1, Figure 2, Figure 3 and Figure 4, in
Appendix, followed by that of the q parameter
Figure 5, Figure 6, Figure 7 and Figure 8, in
Appendix.
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1. for n= 0,0
0q
= 1, replacing
Bn
k(t; [a;b]; q, γ)by its expression given in (1)
gives:
B0
0(t; [a;b]; q, γ)=1. Therefore relation (6) in
true for n= 0
2. suppose there exists n≥0, such that (6) is true
and show that (6) is true at rank n+ 1.
Using (4) we have:
Bn+1
k(t; [a;b]; q, γ) =
τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)+
qn+1−kτ1+γ(1 −qk−1)
τ+γ(1 −qn)Bn
k−1(t; [a;b]; q, γ)
hence
n+1
X
k=0
Bn+1
k(t; [a;b]; q, γ) =
n+1
X
k=0τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)+
qn+1−kτ1+γ(1 −qk−1)
τ+γ(1 −qn)Bn
k−1(t; [a;b]; q, γ)
with Bn
n+1 = 0,Bn
−1= 0 and ˜γ=γ+τ1.
This gives
n+1
X
k=0
Bn+1
k(t; [a;b]; q, γ)
=
n
X
k=0 τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)+
n
X
k=0
qn−kτ1+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)+
qn−kτ1+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0τ2+ (γ+τ1)(1 −qn−k)
τ+γ(1 −qn)+
qn−kτ1+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0τ2+ (γ+τ1)(1 −qn−k)
τ+γ(1 −qn)+
qn−kτ1+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0 τ+γ(1 −qn)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0
Bn
k(t; [a;b]; q, γ)
=1
Therefore relation (6) is true for all n≥0.
6 Degree elevation for
(q, γ)-Bernstein polynomials
The (q, γ)-Bernstein polynomials of degree ncan be
written as (q, γ)-Bernstein polynomials of degree n+
1as follows:
Bn
k(t; [a;b]; q, γ) = qn−k[k+ 1]q
[n+ 1]q
Bn+1
k+1 (t; [a;b]; q, γ)+
[n+ 1 −k]q
[n+ 1]q
Bn+1
k(t; [a;b]; q, γ)
(7)
Proof
By writing Bn+1
k+1 (t; [a;b]; q, γ)and
Bn+1
k(t; [a;b]; q, γ)with (1) we have:
qn−k[k+ 1]q
[n+ 1]q
Bn+1
k+1 (t; [a;b]; q, γ)+
[n+ 1 −k]q
[n+ 1]q
Bn+1
k(t; [a;b]; q, γ)
=qn−k[k+ 1]q
[n+ 1]qn+ 1
k+ 1 q
×
k
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n
Y
i=0
[τ+γ(1 −qi)]
+
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[n+ 1 −k]q
[n+ 1]qn+ 1
kq
×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k
Y
i=0
[τ2+ ˜γ(1 −qi)]
n
Y
i=0
[τ+γ(1 −qi)]
=qn−kτ1+γ(1 −qk)
τ+γ(1 −qn) n
kq
×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
+τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn) n
kq
×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ1+γ(1 −qi)]
=qn−kτ1+γ(1 −qk)
τ+γ(1 −qn)+τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)
| {z }
1
×
n
kq
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=n
kq
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=Bn
k(t; [a;b]; q, γ)
7 Polynomial basis
The (q, γ)-Bernstein polynomials form a basis for
Pn([a;b]). Where Pn([a;b]) is the space of polyno-
mials of degree at most non [a;b]
Proof
It suffices to show that ∀n≥0there exist coefficients
Cn,i,kn
k=0 such that:
ti=
n
X
k=0
Cn,i,kBn
k(t; [a;b]; q, γ), i = 0,· · · , n.
(8)
We reason by recurrence
1. For n= 0 we have B0
0(t; [a;b]; q, γ)=1,
C0,0,0= 1
2. Suppose there exists n≥0such that
∀0≤i≤n, (8) is true and using (7)
We have:
ti=
n
X
k=0
Cn,i,kBn
k(t; [a;b]; q, γ)
=
n
X
k=0
Cn,i,kBn
k(t; [a;b]; q, γ)
=
n
X
k=0
Cn,i,kqn−k[k+ 1]q
[n+ 1]q
Bn+1
k+1 (t; [a;b]; q, γ)+
[n+ 1 −k]q
[n+ 1]q
Bn+1
k(t; [a;b]; q, γ)
=
n+1
X
k=0
Cn+1,i,kBn+1
k(t; [a;b]; q, γ)
with
Cn+1,i,k =Cn,i,k−1qn−k−1[k]q
[n+ 1]q
+
Cn,i,k
[n+ 1 −k]q
[n+ 1]q
(9)
3.
tn+1 =t.tn=
n
X
k=0
Cn,n,ktBn
k(t; [a;b]; q, γ)
=
n
X
k=0
Cn,n,k˜
Dn,kBn+1
k+1 (t; [a;b]; q, γ)+
˜
En,kBn+1
k(t; [a;b]; q, γ),
(10)
This last line of (10) is due to the degree ele-
vation for (q, γ)-Bernstein polynomials (7). We
now determine ˜
Dn,k and ˜
En,k, in such a way tn+1
is a linear combination of Bn+1
k(t; [a;b]; q, γ).
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By applying (1) to this last equality, we obtain:
Bn+1
k+1 (t; [a;b]; q, γ)
=n+ 1
k+ 1 q
×
k
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q!
[k+ 1]q![n−k]q!×τ1+γ(1 −qk)
τ+γ(1 −qn)×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q
[k+ 1]q
τ1+γ(1 −qk)
τ+γ(1 −qn)×[n]q!
[k]q![n−k]q!
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q
[k+ 1]qτ1+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
=[n+ 1]q
[k+ 1]qt−a+γ(1 −qk)
τ+γ(1 −qn)Bn
k(t; [a;b]; q, γ)
(11)
and
Bn+1
k(t; [a;b]; q, γ)
=n+ 1
kq
×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k
Y
i=0
[τ2+ ˜γ(1 −qi)]
n
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q!
[n+ 1 −k]q![k]q!×τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q
[n+ 1 −k]q
τ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)
[n]q!
[n−k]q![k]q!×
k−1
Y
i=0
[τ1+γ(1 −qi)]
n−k−1
Y
i=0
[τ2+ ˜γ(1 −qi)]
n−1
Y
i=0
[τ+γ(1 −qi)]
=[n+ 1]q
[n+ 1 −k]qτ2+ ˜γ(1 −qn−k)
τ+γ(1 −qn)
Bn
k(t; [a;b]; q, γ)
Bn+1
k(t; [a;b]; q, γ) = [n+ 1]q
[n+ 1 −k]q
×
b−t+ (γ+t−a)(1 −qn−k)
τ+γ(1 −qn)×
Bn
k(t; [a;b]; q, γ)
(12)
from (10) , (11) and (12); ˜
Dn,k and ˜
En,k verify
the equation:
˜
Dn,k
[n+ 1]q
[k+ 1]qt−a+γ(1 −qk)
τ+γ(1 −qn)+
˜
En,k
[n+ 1]q
[n+ 1 −k]q
×
b−t+ (γ+t−a)(1 −qn−k)
τ+γ(1 −qn)=t
(13)
with
b−t+ (γ+t−a)(1 −qn−k)
=b−t+t−tqn−k+ (γ−a)(1 −qn−k)
=−tqn−k+b+ (γ−a)(1 −qn−k)
by identification (13) is equivalent to:
•for the coefficient of t:
˜
Dn,k
[n+ 1]q
[k+ 1]q
−˜
En,k
[n+ 1]q
[n+ 1 −k]q
qn−k=
τ+γ(1 −qn)
(14)
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•for constant term:
˜
Dn,k
[n+ 1]q
[k+ 1]q−a+γ(1 −qk)+
˜
En,k
[n+ 1]q
[n+ 1 −k]qb+ (γ−a)(1 −qn−k)= 0
(15)
Multiplying (14) by −−a+γ(1 −qk)and
adding the result to (15) gives:
˜
En,k
[n+ 1]q
[n+ 1 −k]q
×
qn−k−a+γ(1 −qk)+
b+ (γ−a)(1 −qn−k)
=−−a+γ(1 −qk)τ+γ(1 −qn),
with τ=b−a
hence
˜
En,k =[n+ 1 −k]q
[n+ 1]q
×
−−a+γ(1 −qk)b−a+γ(1 −qn)
A;
with
A=qn−k−a+γ(1 −qk)+
b+ (γ−a)(1 −qn−k)
=−aqn−k+γqn−k−γqn+
b+γ−a−γqn−k+aqn−k
=−γqn+b+γ−a
A=b−a+γ(1 −qn);
with the last equality of A above, we have :
˜
En,k =[n+ 1 −k]q
[n+ 1]q
×
−−a+γ(1 −qk)b−a+γ(1 −qn)
b−a+γ(1 −qn)
thus
˜
En,k =[n+ 1 −k]q
[n+ 1]qa−γ(1 −qk)(16)
From (14) and (16) we deduce that:
˜
Dn,k =[k+ 1]q
[n+ 1]qτ+γ(1 −qn)+
˜
En,k
[n+ 1]q
[n+ 1 −k]q
qn−k
=[k+ 1]q
[n+ 1]qτ+γ(1 −qn)+
qn−ka−γ(1 −qk)
=[k+ 1]q
[n+ 1]qτ+γ−γqn+aqn−k−
γqn−k+γqn
=[k+ 1]q
[n+ 1]qτ+γ+aqn−k−γqn−k,
avec τ=b−a
=[k+ 1]q
[n+ 1]qb−a+γ+aqn−k−γqn−k
hence
˜
Dn,k =[k+ 1]q
[n+ 1]qb+(γ−a)(1−qn−k)(17)
Therefore Bn+1
k(t; [a;b]; q, γ)n+1
k=0
is a basis
of the space of Pn+1([a;b])
Which completes the demonstration.
8 The De Casteljau algorithm for
(q, γ)-Bezier curves
Let be a polynomial curve of the form
P(t) =
n
X
k=0
PkBn
k(t; [a;b]; q, γ)(18)
where the coefficients Pkn
k=0 are polynomials
called control points.
The (18) form of Pis a (q, γ)-Bezier curve of degree
non the interval [a;b].
P(t) =
n−r
X
k=0
Pr
kBn−r
k(t; [a;b]; q, γ)(19)
Pr
k=Pr
k(t), k = 0,· · · , n −r, are polynomi-
als of degree r. By the recurrence relation (4) (with
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Bn−r−1
−1=Bn−r−1
n−r= 0), (19) becomes
P(t)
=
n−r
X
k=0
Pr
kτ2+ ˜γqn−r−k−1
τ+γqn−r−1Bn−r−1
k(t; [a;b]; q, γ)+
qn−r−kτ1+γqk−1
τ+γqn−r−1Bn−r−1
k−1(t; [a;b]; q, γ)
=
n−r−1
X
k=0
Bn−r−1
k(t; [a;b]; q, γ)τ2+ ˜γqn−r−k−1
τ+γqn−r−1×
Pr
k+qn−r−k−1τ1+γqk
τ+γqn−r−1Pr
k+1
=
n−r−1
X
k=0
Pr+1
kBn−r−1
k(t; [a;b]; q, γ)
with (at rank r+ 1)
Pr+1
k=τ2+ ˜γqn−r−k−1
τ+γqn−r−1Pr
k+
qn−r−k−1τ1+γqk
τ+γqn−r−1Pr
k+1
(20)
At rank nof this algorithm we obtain a single point
Pn
0which gives the value of the Bézier curve at t, i.e.
Pn
0=P(t).
Similarly, using the recurrence relation (5), we ob-
tain
Pr+1
k=qkτ2+ ˜γqn−r−k−1
τ+γqn−r−1Pr
k+
τ1+γqk
τ+γqn−r−1Pr
k+1,
(21)
where k= 0,1,· · · , n −r−1;r= 0,· · · , n −1,
and, at the last level nwe get Pn
0=P(t).
9 The (q, γ)-Marsden identity
Property 1. Let t∈[a;b]and n≥1.
The (q, γ)-Bernstein polynomials on the interval [a;b]
satisfy for any real number:
n−1
Y
i=0 x−t+ ˜γ(1 −qi)=
n
X
k=0
n−1
Y
j=kx−a+γ(1 −qj)×
k−1
Y
j=0 x−bqj+ (γ−a)(1 −qj)Bn
k(t; [a;b]; q, γ).
(22)
Proof: we proceed by recurrence on n
1. for n= 1.
Using (1) we have:
B0
0(t; [a;b]; q, γ) = 1,
B1
0(t; [a;b]; q, γ) = b−t
b−a,
B1
1(t; [a;b]; q, γ) = t−a
b−a.
(23)
So the left-hand side of (22) is (x−t)and the
right-hand side is
(x−a)B1
0+ (x−b)B1
1=
(x−a)b−t
b−a+ (x−b)t−a
b−a= (x−t)
So (22) is true for n= 1.
2. Assume that (22) is true for some n≥1and show
that (22) is true at rank n+ 1.
Let's set
Pn,k(x) =
n−1
Y
j=kx−a+γ(1 −qi)×
k−1
Y
j=0 x−bqi+ (γ−a)(1 −qi),
k= 0,· · · , n
(24)
then (22) is written as
n−1
Y
i=0 x−t+ ˜γ(1 −qi)=
n
X
k=0
Pn,k(x)Bn
k(t; [a;b]; q, γ).
(25)
Let's prove that (25) is true for n+ 1. For n+ 1,
the left-hand side of (25) is
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n
Y
i=0 x−t+ ˜γ(1 −qi)
=x−t+ ˜γ(1 −qn)×
n−1
Y
i=0 x−t+ ˜γ(1 −qi)
=x−t+ ˜γ(1 −qn)×
n
X
k=0
Pn,k(x)Bn
k(t; [a;b]; q, γ)
=
n
X
k=0
Pn,k(x)x−t+ ˜γ(1 −qn)×
Bn
k(t; [a;b]; q, γ)
=
n
X
k=0
Pn,k(x)fn,kBn+1
k(t; [a;b]; q, γ)+
gn,kBn+1
k+1 (t; [a;b]; q, γ)
=
n+1
X
k=0 Pn,k(x)fn,k +Pn,k−1(x)gn,k−1×
Bn+1
k(t; [a;b]; q, γ)
(26)
with Pn,n−1= 0 et Pn,−1= 0.
Let's determine coefficients fn,k =fn,k(x)and
gn,k =gn,k(x)such that
x−t+ ˜γ(1 −qn)Bn
k(t; [a;b]; q, γ) =
fn,kBn+1
k(t; [a;b]; q, γ) + gn,kBn+1
k+1 (t; [a;b]; q, γ).
(27)
Using (27) , (12) et (11), we obtain the equation:
•x−t+ ˜γ(1 −qn)=
˜
fn,kb−t+ (γ+t−a)(1 −qn−k)+
˜gn,kt−a+γ(1 −qk).
•x−tqn+ (γ−a)(1 −qn)=
˜
fn,k−tqn−k+b+ (γ−a)(1 −qn−k)+
˜gn,kt−a+γ(1 −qk)
(28)
with
˜
fn,k =[n+ 1]q
[n+ 1 −k]q
fn,k
τ+γ(1 −qn)
and
˜gn,k =[n+ 1]q
[k+ 1]q
gn,k
τ+γ(1 −qn)
(29)
Using (28) and equalizing the constant terms and
the terms in variable twe obtain the following (30),
(31) system:
−qn=−qn−k˜
fn,k + ˜gn,k,(30)
x+ (γ−a)(1 −qn) =
b+ (γ−a)(1 −qn−k)˜
fn,k+
−a+γ(1 −qk)˜gn,k,
(31)
Multiplying equation (30) by −−a+γ(1 −qk)
and adding it to equation (31) gives:
˜
fn,k
=x+ (γ−a)(1 −qn) + qn−a+γ(1 −qk)
b+ (γ−a)(1 −qn−k)+qn−k−a+γ(1 −qk)
=x−a+γqn(1 −qk) + γ(1 −qn)
τ+γ(1 −qn)
=x−a+γ(1 −qn+k)
τ+γ(1 −qn)
(32)
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using (30), we obtain:
˜gn,k =qn−k˜
fn,k −qn
=qn−kx−a+γ(1 −qn+k)
τ+γ(1 −qn)−qn
=
qn−kx−a+γ(1 −qn+k)
τ+γ(1 −qn)−
qnτ+γ(1 −qn)
τ+γ(1 −qn)
=
qn−kx−a+γ(1 −qn+k)
τ+γ(1 −qn)−
qnb−a+γ(1 −qn)
τ+γ(1 −qn)
=qn−kx−qn−ka+qn−kγ(1 −qn+k)−bqn
τ+γ(1 −qn)−
qn−a+γ(1 −qn)
τ+γ(1 −qn)
=qn−k(x−bqk)−aqn−k+aqn
τ+γ(1 −qn)+
qn−kγ(1 −qn+k)−γqn(1 −qn)
τ+γ(1 −qn)
=qn−k(x−bqk)−aqn−k(1 −qk)
τ+γ(1 −qn)+
+qn−kγ(1 −qn+k)−γqn(1 −qn)
τ+γ(1 −qn)
=qn−k(x−bqk)−aqn−k(1 −qk)
τ+γ(1 −qn)+
γqn−k1−qn+k−qk(1 −qn)
τ+γ(1 −qn)
=qn−k(x−bqk)−aqn−k(1 −qk) + γqn−k1−qk
τ+γ(1 −qn)
=
qn−kx−bqk+ (γ−a)(1 −qk)
τ+γ(1 −qn)
(33)
From (32) and (33) we deduce:
fn,k =[n+ 1 −k]q
[n+ 1]qx−a+γ(1 −qn+k)(34)
gn,k =[k+ 1]q
[n+ 1]q
qn−kx−bqk+ (γ−a)(1 −qk)
(35)
Using (24), the coefficient of Bn+1
kin the last line
of (26) is given by:
Pn,k(x)fn,k +Pn,k−1(x)gn,k−1
=
n−1
Y
j=kx−a+γ(1 −qi)×
k−2
Y
j=0 x−bqi+ (γ−a)(1 −qi)×
x−bqk−1+ (γ−a)(1 −qk−1)fn,k+
x−a+γ(1 −qk−1)gn,k−1
(36)
In this section, we use the following equalities:
[n+ 1 −k]q+ [k]qqn+1−k= [n+ 1]qand
qn[n+ 1 −k]q+ [k]q= [n+ 1]q
Using (34) and (35) we obtain:
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[n+ 1]qx−bqk−1+ (γ−a)(1 −qk−1)fn,k+
x−a+γ(1 −qk−1)gn,k−1
=x−bqk−1+ (γ−a)(1 −qk−1)×[n+ 1 −k]q
x−a+γ(1 −qn+k)+
x−a+γ(1 −qk−1)[k]qqn+1−k×
x−bqk−1+ (γ−a)(1 −qk−1)
=x−bqk−1+ (γ−a)(1 −qk−1)×
[n+ 1 −k]qx−a+γ(1 −qn+k)+
[k]qqn+1−kx−a+γ(1 −qk−1)
(37)
We have:
[n+ 1 −k]qx−a+γ(1 −qn+k)+
[k]qqn+1−kx−a+γ(1 −qk−1)
=[n+ 1 −k]q+ [k]qqn+1−k(x−a)+
[n+ 1 −k]qγ1−qn+k+
[k]qqn+1−kγ1−qk−1
=[n+ 1]qx−a+
[n+ 1 −k]qγ(1 −qn+k)+
[k]qqn+1−kγ(1 −qk−1)
=[n+ 1]q(x−a) + γ[n+ 1 −k]q−
qn+k[n+ 1 −k]q+ [k]qqn+1−k−[k]qqn
= [n+ 1]q(x−a) + γ[n+ 1 −k]q+
[k]qqn+1−k−qn+k[n+ 1 −k]q−[k]qqn
= [n+ 1]q(x−a) + γ[n+ 1 −k]q+
[k]qqn+1−k−qnqk[n+ 1 −k]q)+[k]q
= [n+ 1]q(x−a) + γ[n+ 1]q−qn[n+ 1]q
= [n+ 1]q(x−a) + γ(1 −qn)
(38)
From (37) and (38) we deduce that:
x−bqk−1+ (γ−a)(1 −qk−1)fn,k+
x−a+γ(1 −qk−1)gn,k−1
=x−bqk−1+ (γ−a)(1 −qk−1)×
(x−a) + γ(1 −qn)
(39)
From (24); (36) and (39), we have :
Pn,k(x)fn,k +Pn,k−1(x)gn,k−1=
n
Y
j=kx−a+γ(1 −qi)×
k−1
Y
j=0 x−bqi+ (γ−a)(1 −qi)=Pn+1,k(x)
(40)
Thus from the formula (26) and (40) we have shown
that (25) is true at rank n+1. This completes the proof
of the (q, γ)-Marsden identity.
10 Conclusion
In this paper, we have defined (q, γ)-Bernstein
basis functions depending on the real parameter γ,
and proved several important properties of these
functions using mathematical induction. We also
introduced the Marsden identity and implemented
De Casteljau's algorithm. Finally, we provided
some graphics of (q, γ)-Bernstein polynomials and
(q, γ)-Bezier curve. Note that these proofs are based
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on the work [11].
Our results show that (q, γ)-Bernstein basis functions
are powerful tools for the representation and ma-
nipulation of polynomial curves, offering increased
flexibility thanks to the second γparameter. Marsden
identity provides an essential link between Bernstein
basis functions and Bezier curves, reinforcing their
usefulness in a variety of areas, from computer-aided
design to geometric modelling. In this case, we can
use such a basis to approximate functions in the sense
of least-squares problems, such as isogeometrics
methods. This work is in progress. Looking to the
future, there is still much to explore.
Further research will focus on extending this concept
to multidimensional contexts, as well as developing
more efficient algorithmic techniques for optimal
data manipulation and representation. In addition, in-
tegrating this basis into practical applications would
open up new avenues for technological innovation.
Declaration of Generative AI and AI assisted
Technologies in the Writing Process
During the preparation of this work, the authors used
chatgpt in order to improve the readability and
language of manuscript. Having use this tool, the
authors have reviewed and corrected the content
where necessary and take full responsibility for the
content of the publication.
References:
[1] G. M. Phillips, A generalization of the Bernstein
polynomials based on the q-integers, ANZIAM
J. 42(2000), pages 79-86.
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https://doi.org/10.1017/
S1446181100011615
[2] H. Oruç, G.M. Phillips, A generalization of
Bernstein polynomials, Proc. Edinburgh Math.
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https://doi.org/10.1017/
S0013091500020332
[3] H. Oruç, N. Tuncer, On the convergence and iter-
ates of q-Bernstein polynomials, J. Approx. The-
ory 117 (2002), pages 301–313
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https://doi.org/10.1006/jath.2002.3703
[4] Soro Sionfon. S, Haudié J. S. Inkpé, Koua Brou.
J. C: q-Bernstein basis functions on a triangulated
domain, Vol. 7, N◦1(2023), pages 72-84
[5] G. Farin : Triangular Bernstein-Bézier patches.
Comput. Aided Geom. Des. Volume 3, Issue 2,
August 1986, pages 83–127.
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https://doi.org/10.1016/0167-8396(86)
90016-6
[6] P. Simeonov, V. Zafiris, R. Goldman: q-
Blossoming: A new approach to algorithms and
identities for q-Bernstein bases and q-Bézier
curves. Volume 164, Issue 1, January 2012,
Pages 77-104
DOI:10.1016/j.jat.2011.09.006
https://doi.org/10.1016/j.jat.2011.09.
006
[7] R. Goldman and P. Simeonov, Formulas and
algorithms for quantum differentiation of quan-
tum Bernstein bases and quantum Bézier curves
based on quantum blossoming, Graphical Mod-
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04.004
[8] P. Simeonov, V. Zafiris and R. Goldman, h-
Blossoming: A new approach to algorithms and
identities for h-Bernstein bases and h-Bézier
curves, Computer Aided Geometric Design,
Volume 28, Issue 9, (December 2011), pages
549–565.
DOI:10.1016/j.cagd.2011.09.003
https://doi.org/10.1016/j.cagd.2011.
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[9] P. Lamberti, M. Lamnii, S. Remogna, D. Sbibih:
h-Bernstein basis functions over a triangular
domain Computer Aided Geometric Design,
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[10] V. Kac and P. Cheung: Quantum Calculus,
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[11] I.Jegdić, J.Larson and P.Simeonov: Algorithms
and Identities for (q, h)-Bernstein Polynomials
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Approach Analysis in Theory and Applications
32 (2016), pages 373-386.
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n4.5
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Some graphical examples on the segment [2;5],
highlighting the effect of parameters γand q.
Figure 1: Effect of γ-parameter for q=−0.95
Figure 2: Effect of γ-parameter for q= 0
Figure 3: Effect of γ-parameter for q= 0.8
Figure 4: Effect of γ-parameter for q= 1.05
APPENDIX
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Figure 5: Effect of q-parameter for γ= 0
Figure 6: Effect of q-parameter for γ= 1
Figure 7: Effect of q-parameter for γ= 2
Figure 8: Effect of q-parameter for γ= 3
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Once we had the results of our article, I was asked
(SORO SIONFON SIMON) to write it. I was
helped in this task by Professor KOUA BROU
JEAN CLAUDE. Doctor HAUDIÉ JEAN
STEPHANE INKPÉ wrote the codes to produce the
graphs.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The publication fees and transfer costs were shared
among the three authors, amounting to 233 euros
per author.
Conflict of Interest
The authors state that they have no financial
interests or personal relationships that could
have influenced the work presented in this paper.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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