The main object of this paper is to construct a new generating function of the (q-) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q-) Bernstein-type polynomials. We also give relations between the (q-) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second-kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q-) Bernstein-type polynomials. Moreover, we give some applications and questions
on approximations of (q-) Bernstein-type polynomials, moments of some distributions in
Statistics.

1. Introduction

In [1], Bernstein introduced the Bernstein polynomials. Since that time, many authors have studied these polynomials and other related subjects (cf, [1–25]), and see also the references cited in each of these earlier works. 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. These polynomials have been used not only for approximations of functions in various areas in Mathematics, but also for the other fields such as smoothing in statistics, numerical analysis and constructing Bezier curve which have many interesting applications in computer graphics (cf, [1, 5, 7, 13–20, 25] and see also the references cited in each of these earlier works).

The (q-) Bernstein polynomials have been investigated and studied by many authors without generating function. So far, we have not found any generating function of (q-) Bernstein polynomials in the literature. Therefore, we will consider the following question

How can one construct generating function of (q-) Bernstein-type polynomials?

The aim of this paper is to give answer this question and to construct generating function of the (q-) Bernstein-type polynomials which is given in Section 3. By using this generating function, we not only give recurrence relation and derivative of the (q-) Bernstein-type polynomials, but also find relations between higher-order Bernoulli polynomials, the Stirling numbers of the second-kind, and the Hermite polynomials. In Section 5, by applying Mellin transformation to the generating function of the (q-) Bernstein-type polynomials, we define interpolation function, which interpolates the (q-) Bernstein-type polynomials at negative integers.

2. Preliminary Results Related to the Classical Bernstein, Higher-Order Bernoulli, and Hermit Polynomials as well as the Stirling Numbers of the Second-Kind

The Bernstein polynomials play a crucial role in approximation theory and the other branches of Mathematics and Physics. Thus in this section we give definition and some properties of these polynomials.

Let f be a function on [0,1]. The classical Bernstein polynomials of degree n are defined by

𝔹nf(x)=∑j=0nf(jn)Bj,n(x),0≤x≤1,
where 𝔹nf is called the Bernstein operator and

Bj,n(x)=(nj)xj(1-x)n-j,j=0,1,…,n are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n). There are n+1nth degree Bernstein polynomials. For mathematical convenience, we set Bj,n(x)=0 if j<0 or j>n (cf, [1, 5, 7, 9, 14, 18–20]).

If f:[0,1]→ℂ is a continuous function, then the sequence of Bernstein polynomials 𝔹nf(x) converges uniformly to f on [0,1] (cf, [10]).

A recursive definition of the kth nth Bernstein polynomials can be written as

Bk,n(x)=(1-x)Bk,n-1(x)+xBk-1,n-1(x).

For proof of the above relation see [9].

For 0≤k≤n, derivatives of the nth degree Bernstein polynomials are polynomials of degree n-1:

ddtBk,n(t)=n(Bk-1,n-1(t)-Bk,n-1(t)),
(cf, [1, 5, 7, 9, 14, 18, 19]). On the other hand, in Section 3, using our new generating function, we give the other proof of (2.4).

Observe that the Bernstein polynomial of degree n, 𝔹nf, uses only the sampled values of f at tnj=j/n, j=0, 1,…,n. For j=0, 1,…,n,

βj,n(x)≡(n+1)Bj,n(x),0≤x≤1,
is the density function of beta distribution beta(j+1,n+1-j).

Let yn(x) be a binomial b(n,x) random variable. Then

The classical higher-order Bernoulli polynomials ℬn(v)(z) are defined by means of the following generating function:

F(v)(z,t)=etx(tet-1)v=∑n=0∞ℬn(v)(z)tnn!.
The higher-order Bernoulli polynomials play an important role in the finite differences and in (analytic) number theory. So, the coefficients in all the usual central-difference formulae for interpolation, numerical differentiation, and integration and differences in terms of derivatives can be expressed in terms of these polynomials (cf, [2, 11, 12, 24]). These polynomials are related to the many branches of Mathematics. By substituting v=1 into the above, we have

F(t)=tetxet-1=∑n=1∞Bntnn!,
where Bn is usual Bernoulli polynomials (cf, [22]).

The usual Stirling numbers of the second-kind with parameters (n,k) are denoted by S(n,k), that is, the number of partitions of the set {1,2,…,n} into k nonempty set. For any t, it is well known that the Stirling numbers of the second-kind are defined by means of the generating function (cf, [3, 21, 23])

FS(t,k)=(-1)kk!(1-et)k=∑n=0∞S(n,k)tnn!.
These numbers play an important role in many branches of Mathematics, for example, combinatorics, number theory, discrete probability distributions for finding higher-order moments. In [8], Joarder and Mahmood demonstrated the application of the Stirling numbers of the second-kind in calculating moments of some discrete distributions, which are binomial distribution, geometric distribution, and negative binomial distribution.

The Hermite polynomials are defined by the following generating function.

For z, t∈ℂ,

e2zt-t2=∑n=0∞Hn(z)tnn!,
which gives the Cauchy-type integral

Hn(z)=n!2πi∫𝒞e2zt-t2dttn+1,
where 𝒞 is a circle around the origin and the integration is in positive direction (cf, [12]). The Hermite polynomials play a crucial role in certain limits of the classical orthogonal polynomials. These polynomials are related to the higher-order Bernoulli polynomials, Gegenbauer polynomials, Laguerre polynomials, the Tricomi-Carlitz polynomials and Buchholz polynomials, (cf, [12]). These polynomials also play a crucial role not only in Mathematics but also in Physics and in the other sciences. In Section 4 we give relation between the Hermite polynomials and (q-) Bernstein-type polynomials.

3. Generating Function of the Bernstein-Type Polynomials

Let {Bk,n(x)}0≤k≤n be a sequence of Bernstein polynomials. The aim of this section is to construct generating function of the sequence {Bk,n(x)}0≤k≤n. It is well known that most of generating functions are obtained from the recurrence formulae. However, we do not use the recurrence formula of the Bernstein polynomials for constructing generating function of them.

We now give the following notation:

[x]=[x:q]={1-qx1-q,q≠1,x,q=1.

If q∈ℂ, then we assume that |q|<1.

We define

Fk,q(t,x)=(-1)ktkexp([1-x]t)×∑m,l=0(k+l-1l)qlS(m,k)(xlogq)mm!,
where |q|<1, exp(x)=ex and S(m,k) denotes the second-kind Stirling numbers and

∑m,l=0f(m)g(l)=∑m=0∞f(m)∑l=0∞g(l).
By (3.2), we define the following new generating function of polynomial Yn(k;x;q) by

Fk,q(t,x)=∑n=k∞Yn(k;x;q)tnn!,
where t∈ℂ.

Observe that if q→1 in (3.4), we have

Yn(k;x;q)→Bk,n(x).
Hence

Fk(t,x)=∑n=k∞Bk,n(x)tnn!.

From (3.4), we obtain the following theorem.

Theorem 3.1.

Let n be a positive integer with k≤n. Then one has
Yn(k;x;q)=(nk)(-1)kk!(1-q)n-k×∑m,l=0∑j=0n-k(k+l-1l)(n-kk)(-1)jql+j(1-x)S(m,k)(xlogq)mm!.

By using (3.2) and (3.4), we obtain

Fk,q(t,x)=([x]t)kk!exp([1-x]t)=∑n=k∞Yn(k;x;q)tnn!.
The generating function Fk,q(t,x) depends on integer parameter k, real variable x, and complex variable q and t. Therefore the properties of this function are closely related to these variables and parameter. By using this function, we give many properties of the (q-) Bernstein-type polynomials and the other well-known special numbers and polynomials. By applying Mellin transformation to this function, in Section 5, we construct interpolation function of the (q-) Bernstein-type polynomials.

By the umbral calculus convention in (3.8), then we obtain

([x]t)kk!exp([1-x]t)=exp(Y(k;x;q)t).
By using the above, we obtain all recurrence formulae of Yn(k;x;q) as follows:

([x]t)kk!=∑n=0∞(Y(k;x;q)-[1-x])ntnn!,
where each occurrence of Yn(k;x;q)are given by Yn(k;x;q) (symbolically Yn(k;x;q)→Yn(k;x;q)).

By (3.9),

[u+v]=[u]+qu[v],[-u]=-qu[u],
we obtain the following corollary.

Corollary 3.2.

Let n be a positive integer with k≤n. Then one has
Yn+k(k;x;q)=(n+kk)∑j=0n(-1)jqj(1-x)[x]j+k.

Remark 3.3.

By Corollary 3.2, for all k with 0≤k≤n, we see that
Yn+k(k;x;q)=(n+kk)∑j=0n(-1)jqj(1-x)[x]j+k.
The polynomials Yn+k(k;x;q) are so-called q-Bernstein-type polynomials. It is easily seen that
limq→1Yn+k(k;x;q)=Bk,n+k(x)=(n+kk)xk(1-x)n,
which give us (2.2).

By using derivative operator

ddx(limq→1Yn+k(k;x;q))
in (3.2), we obtain

∑n=k∞ddx(Yn(k;x;1))tnn!=∑n=k∞nYn-1(k-1;x;1)tnn!-∑n=k∞nYn-1(k;x;1)tnn!.
Consequently, we have

ddx(Yn(k;x;1))=nYn-1(k-1;x;1)-nYn-1(k;x;1),
or

ddx(Bk,n(x))=nBk-1,n-1(x)-nBk,n-1(x).

Observe that by using our generating function we give different proof of (2.4).

Let f be a function on [0,1]. The (q-) Bernstein-type polynomial of degree n is defined by

𝕐nf(x)=∑j=0nf([j][n])Yn(j;x;q),
where 0≤x≤1. 𝕐n is called the (q-) Bernstein-type operator and Yn(j;x;q), j=0,…,n, defined in (3.7), are called the (q-) Bernstein-type (basis) polynomials.

4. New Identities on Bernstein-Type Polynomials, Hermite Polynomials, and the Stirling Numbers of the Second-KindTheorem 4.1.

Let n be a positive integer with k≤n. Then one has
Yn(k;x;q)=[x]k∑j=0n(nj)ℬj(k)([1-x])S(n-j,k),
where ℬj(k)(x) and S(n,k) denote the classical higher-order Bernoulli polynomials and the Stirling numbers of the second-kind, respectively.

Proof.

By using (2.7), (2.9), and (3.4), we obtain
∑n=k∞Yn(k;x;q)tnn!=[x]k∑n=0∞S(n,k)tnn!∑n=0∞ℬj(k)([1-x])tnn!.
By using Cauchy product in the above, we have
∑n=k∞Y(k,n;x;q)tnn!=[x]k∑n=0∞∑j=0nℬj(k)([1-x])S(n-j,k)tnj!(n-j)!.
From the above, we have
∑n=k∞Yn(k;x;q)tnn!=[x]k∑n=0k-1∑j=0nℬj(k)([1-x])S(n-j,k)tnj!(n-j)!+[x]k∑n=k∞∑j=0nℬj(k)([1-x])S(n-j,k)tnj!(n-j)!.
By comparing coefficients of tn in both sides of the above equation, we arrive at the desired result.

Remark 4.2.

In [18–20], Phillips gave many results concerning the q-integers, and an account of the properties of q-Bernstein polynomials. He gave many applications of these polynomials on approximation theory. In [6], Gould gave a different relation between the Bernstein polynomials, generalized Bernoulli polynomials, and the second-kind Stirling numbers. Oruç and Tuncer [15] gave relation between the q-Bernstein polynomials and the second-kind q-Stirling numbers. In [13], Nowak studied approximation properties for generalized q-Bernstein polynomials and also obtained Stancu operators or Phillips polynomials.

From (4.4), we get the following corollary.

Corollary 4.3.

Let n be a positive integer with k≤n. Then one has
[x]k∑n=0k-1∑j=0nℬj(k)([1-x])S(n-j,k)j!(n-j)!=0.

Theorem 4.4.

Let n be a positive integer with k≤n. Then one has
Hn(1-y)=k!yk∑n=0∞Yn+k(k;y;q)2n(n+k)!.

Proof.

By (2.10), we have
e2zt=∑n=0∞t2nn!∑n=0∞Hn(z)tnn!.
By Cauchy product in the above, we obtain
e2zt=∑n=0∞(∑j=0n(nj)Hj(z))t2n-jn!.
By substituting z=1-y into (4.8), we have
∑n=0∞(∑j=0n(nj)Hj(1-y))t2n-jn!=k!yk∑n=0∞(2nYn+k(k;y;q))tn(n+k)!.
By comparing coefficients of tn in the both sides of the above equation, we arrive at the desired result.

5. Interpolation Function of the (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M150"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-) Bernstein-Type Polynomials

The classical Bernoulli numbers interpolate by Riemann’ zeta function, which has a profound effect on number theory and complex analysis. Thus, we construct interpolation function of the (q-) Bernstein-type polynomials.

For z∈ℂ, and x≠1, by applying the Mellin transformation to (3.2), we get

Sq(z,k;x)=1Γ(s)∫0∞tz-k-1Fk,q(-t,x)dt.
By using the above equation, we defined interpolation function of the polynomials Yn(k;x;q) as follows.

Definition 5.1.

Let z∈ℂ and x≠1. We define
Sq(z,k;x)=(1-q)z-k∑m,l=0(z+l-1l)ql(1-x)S(m,k)(xlogq)mm!.

By using (5.2), we obtain

Sq(z,k;x)=(-1)kk![x]k[1-x]-z,
where z∈ℂ and x≠1.

By (5.2), we have Sq(z,k;x)→S(z,k;x) as q→1. Thus one has

S(z,k;x)=(-1)kk!xk(1-x)-z.
By substituting x=1 into the above, we have

S(z,k;1)=∞.
We now evaluate the mth z-derivatives of S(z,k;x) as follows:

∂m∂zmS(z,k;x)=logm(11-x)S(z,k;x),
where x≠1.

By substituting z=-n into (5.2), we obtain

Sq(-n,k;x)=1(1-q)n+k∑m,l=0(-n+l-1l)ql(1-x)S(m,k)(xlogq)mm!.
By substituting (3.7) into the above, we arrive at the following theorem, which relates the polynomials Yn+k(k;x;q) and the function Sq(z,k;x).

Theorem 5.2.

Let n be a positive integer with k≤n and 0<x<1. Then we have
Sq(-n,k;x)=(-1)kn!(n+k)!Yn+k(k;x;q).

Remark 5.3.

Consider the following.
limq→1Sq(-n,k;x)=S(-n,k;x)=(-1)kn!(n+k)!xk(1-x)n=(-1)kn!(n+k)!Bk,n+k(x).
Therefore, for 0<x<1, the function
S(z,k;x)=(-1)kk!xk(1-x)-z
interpolates the classical Bernstein polynomials of degree n at negative integers.

By substituting z=-n into (5.6), we obtain the following corollary.

Corollary 5.4.

Let n be a positive integer with k≤n and 0<x<1. Then one has
∂m∂zmS(-n,k;x)=(-1)kn!(n+k)!Bk,n+k(x)logm(11-x).

6. Further Remarks and Observation

The Bernstein polynomials are used for important applications in many branches of Mathematics and the other sciences, for instance, approximation theory, probability theory, statistic theory, number theory, the solution of the differential equations, numerical analysis, constructing Bezier curve, q-analysis, operator theory, and applications in computer graphics. Thus we look for the applications of our new functions and the (q-) Bernstein-type polynomials.

Due to Oruç and Tuncer [15], the q-Bernstein polynomials share the well-known shape-preserving properties of the classical Bernstein polynomials. When the function f is convex, then

βn-1(f,x)≥βn(f,x)forn>1,0<q≤1,
where

βn(f,x)=∑r=0nfr[nr]xr∏s=0n-r-1(1-qsx),[nr]=[n]⋯[n-r+1][r]!.
As a consequence of this one can show that the approximation to convex function by the q-Bernstein polynomials is one sided, with βnf≥f for all n. βnf behaves in very nice way when one varies the parameter q. In [2], the authors gave some applications on the approximation theory related to Bernoulli and Euler polynomials.

We conclude this section by the following questions.

How can one demonstrate approximation by (q-) Bernstein-type polynomials Yn+k(k;x;q)?

Is it possible to define uniform expansions of the (q-) Bernstein-type polynomials Yn+k(k;x;q)?

Is it possible to give applications of the (q-) Bernstein-type polynomials in calculating moments of some distributions in Statistics Yn+k(k;x;q)?

How can one give relations between the (q-) Bernstein-type polynomials Yn+k(k;x;q) and the Milnor algebras.

Acknowledgments

The first author is supported by the research fund of Akdeniz University. The authors would like to thank the referee for his/her comments. The authors also would like to thank Professor George M. Phillips for his comments and his some references.

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