Construction a new generating function of Bernstein type polynomials

Yilmaz Simsek

Department of Mathematics, Faculty of Art and Science

University of Akdeniz

TR-07058 Antalya, Turkey

E-mail<EMAIL_ADDRESS>

Dedicated to Professor H. M. Srivastava on the occasion of his seventieth
birth anniversary

Abstract

> Main purpose of this paper is to reconstruct generating function of the
> Bernstein type polynomials. Some properties this generating functions are
> given. By applying this generating function, not only derivative of these
> polynomials but also recurrence relations of these polynomials are found.
> Interpolation function of these polynomials is also constructed via Mellin
> Transformation. This function interpolates these polynomials at negative
> integers which are given explicitly. Moreover, relations between these
> polynomials, the generalized Stirling numbers, and Bernoulli polynomials of
> higher order are given. Furthermore some applications associated with
> B´ezier curve are given.

2010 Mathematics Subject Classification. Primary 11B68, 11M06, 33B15 ;
Secondary 33B15, 65D17.

Key Words and Phrases. Generating function, Bernstein polynomials, Bernoulli
polynomials of higher-order, Stirling numbers of second kind, interpolation
function, Mellin transformation, Gamma function, beta function and B´ezier
curve.

##  1\. Introduction, Definitions and Preliminaries

The Bernstein polynomials, recently, have been defined by many different ways,
for examples in $q$-series, by complex function and many algorithms. These
polynomials are used not only approximations of functions in various, but also
in the other fields such as smoothing in statistics, numerical analysis, the
solution of the differential equations, and constructing B´ezier curve and in
Computer Aided Design cf. ([2], [8], [3], [4], [7], [10], [1]), and see also
the references cited in each of these earlier works.

By the same motivation of Ozden’ [6] paper, which is related to the
unification of the Bernoulli, Euler and Genocchi polynomials, we, in this
paper, construct a generating function of the Bernstein polynomials which
unify generating function in [10], [1].

##  2\. Construction generating functions of Bernstein type polynomials

In this section we unify generating function of the Bernstein polynomials. We
define

$\mathcal{F}(t,b,s:x)=\frac{2^{b}x^{bs}\left(\frac{t}{2}\right)^{bs}e^{t(1-x)}}{\left(bs\right)!}$

where $b,s\in\mathbb{Z}^{+}:=\\{1,2,3,\cdots\\}$, $t\in\mathbb{C}$ and
$x\in\left[0,1\right]$. This function is generating function of the
polynomials $\mathfrak{S}_{n}(bs,x)$:

$\mathcal{F}(t,b,s:x)=\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!},$
(2.1)

where $\mathfrak{S}_{0}(bs,x)=\cdots=\mathfrak{S}_{bs-1}(bs,x)=0$.

Remark 1. If we set $s=1$ in (2.1), we obtain

$\frac{\left(xt\right)^{b}e^{t(1-x)}}{b!}=\sum_{n=0}^{\infty}B_{n}(b,x)\frac{t^{n}}{n!},$

and $\mathfrak{S}_{n}(b,x)=B_{n}(b,x)$, which denotes the Bernstein
polynomials cf. ([2], [3], [4], [8], [10], [1]).

By using Taylor expansion of $e^{t}$ in (2.1), we arrive at the following
theorem:

######  Theorem 1.

Let $x,y\in[0,1]$. Let $b,$ $n$ and $s\ $be nonnegative integers. If $n\geq
bs$, then we have

$\mathfrak{S}_{n}(bs,x)=\left(\begin{array}[]{c}n\\\
bs\end{array}\right)\frac{x^{bs}(1-x)^{n-bs}}{2^{b(s-1)}}.$

Remark 1. Setting $s=1$ in Theorem 1, one can see that the polynomials

$\mathfrak{S}_{n}(b,x)=\left(\begin{array}[]{c}n\\\
b\end{array}\right)x^{b}(1-x)^{n-b},$

which give us the Bernstein polynomials cf. ([10], [1]). Consequently, the
polynomials $\mathfrak{S}_{n}(bs,x)$ are unification of the Bernstein
polynomials.

By using Theorem 1, we easily obtain the following results.

######  Corollary 1.

Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$. Then we have

$\left(\begin{array}[]{c}n\\\
bs\end{array}\right)\mathfrak{S}_{n-bs}(bs;x)=\left(\begin{array}[]{c}n+bs\\\
n\end{array}\right)\mathfrak{S}_{n}(bs;x).$

Setting

$\mathfrak{g}_{n}(bs,x)=2^{b(s-1)}\mathfrak{S}_{n}(bs,x),$

where, for $bs=j$,

$\sum_{j=0}^{n}\mathfrak{g}_{n}(j,x)=1.$

Let $f$ be a continuous function on $\left[0,1\right]$. Then we define
unification Bernstein type operator as follows:

$\mathbb{S}_{n}\left(f(x)\right)=\sum_{j=0}^{n}f\left(\frac{j}{n}\right)\mathfrak{g}_{n}(j;x),$
(2.2)

where $x\in[0,1]$, $n$ is positive integer.

Setting $f(x)=x$ in (2.2), then we have

$\mathbb{S}_{n}\left(x\right)=\sum_{j=0}^{n}\frac{j}{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)x^{j}(1-x)^{n-j}.$

From the above, we get

$\mathbb{S}_{n}\left(x\right)=x\sum_{j=0}^{n}\mathfrak{g}_{n-1}(j-1,x).$

##  3\. Fundamental relations of the polynomials $\mathfrak{S}_{n}(bs,x)$

By using generating function of $\mathfrak{S}_{n}(bs,x)$, in this section we
give derivative of $\mathfrak{S}_{n}(bs,x)$ and recurrence relation of
$\mathfrak{S}_{n}(bs,x)$.

######  Theorem 2.

Let $x\in[0,1]$. Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$.
Then we have

$\frac{d}{dx}\mathfrak{S}_{n}(bs,x)=n\left(\mathfrak{S}_{n-1}(bs-1,x)-\mathfrak{S}_{n}(bs,x)\right).$
(3.1)

###### Proof.

By using the partial derivative of a function in (2.1) with respect to the
variable $x$, we have

$\sum_{n=0}^{\infty}\frac{\partial}{\partial
x}\left(\mathfrak{S}_{n}(bs,x)\right)\frac{t^{n}}{n!}=t\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs-1,x)\frac{t^{n}}{n!}-t\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}.$

From the above, we obtain

$\sum_{n=0}^{\infty}\left(\frac{d}{dx}\mathfrak{S}_{n}(bs,x)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}n\mathfrak{S}_{n-1}(bs-1,x)\frac{t^{n}}{n!}-\sum_{n=0}^{\infty}n\mathfrak{S}_{n-1}(bs,x)\frac{t^{n}}{n!}.$

By using the partial derivative of a function in (2.1) with respect to the
variable $t$, we arrive at the following theorem:

######  Theorem 3.

Let $x\in[0,1]$. Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$.
Then we have

$\mathfrak{S}_{n}(bs,x)=x\mathfrak{S}_{n-1}(bs-1,x)+(1-x)\mathfrak{S}_{n-1}(bs,x).$
(3.2)

Remark 3. If setting $s=1$, then (3.2) reduces to a recursive relation of the
Bernstein polynomials

$B_{n}(b,x)=(1-x)B_{n-1}(b,x)+xB_{n-1}(b-1,x)$

and (3.1) reduces to derivative of the Bernstein polynomials

$\frac{d}{dx}B_{n}(j,x)=n\left(B_{n-1}(j-1,x)-B_{n-1}(j,x)\right),$

respectively.

By the umbral calculus convention in (2.1), we get

$\frac{2^{b}x^{bs}\left(\frac{t}{2}\right)^{bs}}{\left(bs\right)!}=e^{\left(\mathfrak{S}(bs,x)-(1-x)\right)t},$

where $\mathfrak{S}^{n}(bs;x)$ is replaced by $\mathfrak{S}_{n}(bs;x)$. After
some elementary calculation, we arrive at the following theorem.

######  Theorem 4.

If $n=bs$, then we have

$2^{b(1-s)}x^{bs}=\sum_{j=0}^{bs}\left(\begin{array}[]{c}bs\\\
j\end{array}\right)(-1)^{bs-j}\left(1-x\right)^{bs-j}\mathfrak{S}_{j}(bs,x).$

If $n>bs$, then we have

$\sum_{j=bs+1}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)(-1)^{n-j}\left(1-x\right)^{n-j}\mathfrak{S}_{j}(bs,x)=0.$

Relations between the polynomials the polynomial $\mathfrak{S}_{n}(bs,x)$,
Bernoulli polynomial of higher order and Stirling numbers of second kind is
given by the following theorem:

######  Theorem 5.

Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$. Then we have

$\mathfrak{S}_{n}(bs,x)=2^{b(1-s)}x^{bs}\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)S(j,bs)B_{n-j}^{(bs)}(1-x),$

where $B_{n}^{(v)}(x)$ and $S(n,j)$ denote Bernoulli polynomial of higher
order and Stirling numbers of second kind, which are given by means of the
following generating function, respectively

$\frac{t^{v}e^{xt}}{\left(e^{t}-1\right)^{v}}=\sum_{n=0}^{\infty}B_{n}^{(v)}(x)\frac{t^{n}}{n!},\text{
}(\left|t\right|<2\pi)$

and

$(-1)^{v}\frac{\left(1-e^{t}\right)^{v}}{v!}=\sum_{n=0}^{\infty}S(n,v)\frac{t^{n}}{n!}.$

###### Proof.

By (2.1), we have

$2^{b(1-s)}x^{bs}\left(\frac{(-1)^{bs}(e^{t}-1)^{bs}}{\left(bs\right)!}\right)\left(\frac{t^{bs}e^{(1-x)t}}{\left(e^{t}-1\right)^{bs}}\right)=\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}.$

From the above, we have

$\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}=2^{b(1-s)}x^{bs}\left(\sum_{n=0}^{\infty}B_{n}^{(bs)}(1-x)\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}S(n,k)\frac{t^{n}}{n!}\right).$

By Cauchy product in the above, after some calculation, we find the desired
result.

By using same method of Lopez and Temme’ [5], we give contour integral
representation of $\mathfrak{S}_{n}(bs,x)$ as follows:

$\mathfrak{S}_{n}(bs,x)=\frac{\Gamma(m+1)}{\Gamma(k+1)}\frac{1}{2\pi
i}\int_{\mathcal{C}}\mathcal{F}(t,b,s:x)\frac{dz}{z^{m+1}},$

where $\mathcal{C}$ is a circle around the origin and the integration is in
positive direction.

##  4\. Interpolation Function of the polynomials $\mathfrak{S}_{n}(bs,x)$

In this section, we construct meromorphic function. This function interpolates
$\mathfrak{S}_{n}(bs;x)$ at negative integers. These values are given
explicitly in Theorem 6.

For $z\in\mathbb{C}$, by applying the Mellin transformation to (2.1), we
obtain

$\mathfrak{B}(z,bs;x)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}t^{z-1}\mathcal{F}(-t,b,s:x)dt,$

where $\Gamma(z)$ is Euler gamma function. From the above, we define the
following interpolation function.

######  Definition 1.

Let $z\in\mathbb{C}$ with $\Re(z)>0$ and $x\neq 1$. Let $b$ and $s$ be
nonnegative integers. Then we define

$\mathfrak{B}(z,bs;x)=(-1)^{bs}\frac{\Gamma(z+bs)}{\Gamma(bs+1)\Gamma(z)}\frac{2^{b(1-s)}x^{bs}}{\left(1-x\right)^{z+bs}},$
(4.1)

Remark 4. By the well-known identity $\Gamma(bs+1)=bs\Gamma(bs)$, for
$\Re(z)>0$ we have

$\mathfrak{B}(z,k;x)=\frac{(-1)^{bs}2^{b(1-s)}x^{bs}}{bsB(z,k)\left(1-x\right)^{z+bs}},$

where $B(z,k)$ denotes the beta function. Observe that if $x=1$, then

$\mathfrak{B}(z,bs,1)=\infty.$

######  Theorem 6.

Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$ and $x\in[0,1]$.
Then we have

$\mathfrak{B}(-n,bs;x)=\mathfrak{S}_{n}(bs,x).$

###### Proof.

Let $n$ and $b$, and $s$ be positive integers with $bs\leq n$. $\Gamma(z)$ has
simple poles at $z=-n=0,-1,-2,-3,\cdots$. The residue of $\Gamma(z)$ is

$Res(\Gamma(z),-n)=\frac{(-1)^{n}}{n!}.$

Taking $z\rightarrow-n$ into (4.1) and using the above relations, the desired
result can be obtained.

Observe that if we set $s=1$ in Theorem 6, we arrive at

$\mathfrak{B}(-n,b;x)=B_{n}(b,x).$

##  5\. Further Remarks on B´ezier curves

The Bernstein polynomials are used to construct B´ezier curves. B´ezier was an
engineer with the Renault car company and set out in the early 1960’s to
develop a curve formulation which would lend itself to shape design. Engineers
may find it most understandable to think of B´ezier curves in terms of the
center of mass of a set of point masses cf. [13], for example, consider the
four masses $m_{0}$, $m_{1}$, $m_{2}$, and $m_{3}$ located at points $P_{0}$,
$P_{1}$, $P_{2}$, $P_{3}$. The center of mass of these four point masses is
given by the equation

$P=\frac{m_{0}P_{0}+m_{1}P_{1}+m_{2}P_{2}+m_{3}P_{3}}{m_{0}+m_{1}+m_{2}+m_{3}}.$

Next, imagine that instead of being fixed, constant values, each mass varies
as a function of some parameter $x$. In specific case, let $m_{0}=(1-x)^{3}$,
$m_{1}=3t(1-x)^{2}$, $m_{2}=3t^{2}(1-x)$ and $m_{3}=x^{3}$. The values of
these masses are a function of $x$. For each value of $x$, the masses assume
different weights and their center of mass changes continuously. As $x$ varies
between $0$ and $1$, a curve is swept out by the center of masses. This curve
is a cubic B´ezier curve. For any value of $x$, this B´ezier curve is

$P=m_{0}P_{0}+m_{1}P_{1}+m_{2}P_{2}+m_{3}P_{3},$

where $m_{0}+m_{1}+m_{2}+m_{3}\equiv 1$. These variable masses $m_{i}$ are
normally called blending functions and their locations $P_{i}$ are known as
control points or B´ezier points. The blending functions, in the case of
B´ezier curves, are known as Bernstein polynomials. This curve is used in
computer graphics and related fields and also in the time domain, particularly
in animation and interface design cf. ([3], [4], [13]).

The B´ezier curve of degree $n$ can be generalized as follows. Given points
$P_{0}$, $P_{1}$, $P_{2}$,$\cdots$, $P_{n}$ the B´ezier curve is

$B(x)=\sum_{k=0}^{n}P_{k}B_{n}(k,x),$ (5.1)

where $x\in[0,1]$ and $B_{n}(k,t)$ denotes Bernstein polynomials.

We now unify the B´ezier curve in (5.1) by the polynomials
$\mathfrak{g}_{n}(bs,x)$ as follows

$\mathbb{B}_{n}(x,y)=\sum_{k=0}^{n}P_{k}\mathfrak{g}_{n}(k;x),$

with control points $P_{k}$.

######  Acknowledgement 1.

The present investigation was supported by the Scientific Research Project
Administration of Akdeniz University.

## References

  * [1] M. Acikgoz and S. Aracı. On generating function of the Bernstein polynomials, Numer. Anal. Appl. AIP Conf. ICNAAM 2010.
  * [2] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow Sér. 2 t. 13, 1-2 (1912-1913).
  * [3] L. Busé, Goldman, R.: Division algorithms for Bernstein polynomials, Computer Aided Geometric Design, 25(9), 850-865 (2008).
  * [4] G. Morin and R. Goldman, On the smooth convergence of subdivision and degree elevation for B´ezier curves. Computer Aided Geometric Design. 18(7), 657-666 (2001).
  * [5] L. Lopez and N.. M. Temme, Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz polynomials, Modelling. Analysis and Simulation (MAS), MAS-R9927 September 30, (1999).
  * [6] H. Ozden, Unification of generating function of the Bernoulli, Euler and Genocchi numbers and polynomials, to appear in Numerical analysis and applied mathematics, AIP Conference Proceedings.
  * [7] S. Ostrovska, The unicity theorems for the limit $q$-Bernstein opera, Appl. Analy. 88(2), 161–167 (2009).
  * [8] G. M. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics/ Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York, (2003).
  * [9] Y. Simsek, Twisted $(h,q)$-Bernoulli numbers and polynomials related to twisted $(h,q)$-zeta function and $L$-function, J. Math. Anal. Appl. 324 (2006), 790-804.
  * [10] Y. Simsek and M. Acikgoz, A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. doi:10.1155/2010/769095.
  * [11] H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), 77–84.
  * [12] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001.
  * [13] Sederberg, T.: BYU B´ezier curves, http://www.tsplines.com/resources/class_notes/B’ezier_curves.pdf.