Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions

The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions.


Introduction
The Bernstein polynomials have many applications: in approximations of functions, in statistics, in numerical analysis, in p-adic analysis and in the solution of differential equations. It is also well-known that in Computer Aided Geometric Design polynomials are often expressed in terms of the Bernstein basis functions. These polynomials are called Bezier curves and surfaces.
Many of the known identities for the Bernstein basis functions are currently derived in an ad hoc fashion, using either the binomial theorem, the binomial distribution, tricky algebraic manipulations or blossoming. The main purpose of this work is to construct novel functional equations for the Bernstein polynomials. Using these functional equations and Laplace transform, we develop a novel approach both to standard and to new identities for the Bernstein polynomials. Thus these polynomial identities are just the residue of a much more powerful system of functional equations.
The remainder of this study is organized as follows: We find several functional equations and differential equations for the Bernstein basis functions using generating functions. From these equations, many properties of the Bernstein basis functions are then derived. For instance, we give a new proof of the recursive definition of the Bernstein basis functions as well as a novel derivation for the two term formula for the derivatives of the nth degree Bernstein basis functions. Using functional equations, we give new derivations for the sum and alternating sum of the the Bernstein basis functions and a formula for the monomials in terms of the Bernstein basis functions. We also derive identities corresponding to the degree elevation and subdivision formulas for Bezier curves. We prove many new identities for the Bernstein basis functions. Finally, we give some applications of the Laplace transform to the generating functions for the Bernstein basis functions. We obtain interesting series representations for the Bernstein basis functions. We also give some remarks and observations related to the Fourier transform and complex generating functions for the Bernstein basis functions.

Generating Functions
The Bernstein polynomials and related polynomials have been studied and defined in many different ways, for examples by q-series, complex functions, p-adic Volkenborn integrals and many algorithms. In this section, we provide novel generating functions for the Bernstein basis functions.
Generating functions for the Bernstein basis functions can be defined as follows: Note that there is one generating function for each value of k.
Proof. By substituting (2.1) into the right hand side of (2.2), we get The right hand side of the above equation is a Taylor series for e (1−x)t , thus we arrive at the desired result.
We give some alternative forms of the generating functions in (2.2) as follows: (2.6) By using the above alternative forms we derive some new identities for the Bernstein basis functions.
where B n k (x, a, b) denotes the generalized Bernstein basis function defined by: cf. [4].
A Bernstein polynomial P(x) is a polynomial represented in the Bernstein basis functions: cf. [4].

Identities for the Bernstein basis functions
In this section, we use the generating functions for the Bernstein basis functions to derive a family of functional equations. Using these equations, we derive a collection of identities for the Bernstein basis functions.
3.1. Sums and Alternating sums. From (2.3), we get the following functional equations: Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.

Proof. By combining (3.2) and (3.3), we obtain
Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.
Substituting (2.1) into the above equation, we arrive at the desired result.
Remark 3. Theorem 4 is a bit tricky to prove with algebraic manipulations. Goldman [6]-[5, Chapter 5, pages 299-306] proved this identity algebraically. He also proved the following related subdivision identities: For additional identities, see cf.
Summing both sides of the above equation over k, we obtain the following functional equation, which is used to derive a formula for the monomials in terms of the Bernstein basis functions: Theorem 5.
Proof. Combining (2.2) and (3.5), we get Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.
3.4. Differentiating the Bernstein basis functions. In this section we give higher order derivatives of the Bernstein basis functions. We begin by observing that Using Leibnitz's formula for the lth derivative, with respect to x, we obtain the following higher order partial differential equation: From this equation, we arrive at the following theorem: Proof. Formula (3.8) follows immediately from (3.7).
Applying Theorem 6, we obtain a new derivation for the higher order derivatives of the Bernstein basis functions.
Proof. By substituting the right hand side of (2.2) into (3.8), we get Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.
Substituting l = 1 into (3.9), we arrive at the following standard corollary: cf.

Recurrence Relation.
In the previous section we computed the derivative of (3.6) with respect to x to derive a derivative formula for the Bernstein basis functions. In this section we are going to differentiate (3.6) with respect to t to derive a recurrence relation for the Bernstein basis functions. Using Leibnitz's formula for the vth derivative, with respect to t, we obtain the following higher order partial differential equation: (3.10) From the above equation, we have the following theorem: Proof. Formula (3.11) follows immediately from (3.10).
Using definition (2.3) and (2.1) in Theorem 8, we obtain a recurrence relation for the Bernstein basis functions: Proof. By substituting the right hand side of (2.2) into (3.11), we get t n n! .

From the above equation, we get
Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result. Therefore Substituting d = 1 into the above equation, we have (3.14) The above relation can also be proved by (2.1) cf. ( [4], [5], [6]). From (2.3), we also get the following functional equation: Therefore Substituting d = 1, we have Adding (3.14) and (3.15), we get the standard degree elevation formula

New identities
In this section, using alternative forms of the generating functions, functional equations and Laplace transform, we give many new identities for the Bernstein basis functions.
Using (2.3), we obtain the following functional equations: and Theorem 10.
Proof. By substituting the right hand side of (2.2) into (4.1), we get Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.

Proof. Combining (2.2) and (4.2), we get
From the above equation, we obtain ∞ n=0 n j=0 n j Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result. Proof. By using (2.4), we obtain Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.
Theorem 13. For all positive integers k and n, we have Proof. By using (2.5), we get Comparing the coefficients of t n n! on the both sides of the above equation, we arrive at the desired result.
Theorem 14. For all positive integers k and n, we have Proof. Proof of Theorem 14 is same as that of Theorem 12. So we omit it.

Applications of the Laplace transform to the generating functions for the Bernstein basis functions
In this section, we give some applications of the Laplace transform to the generating functions for the Bernstein basis functions. We obtain interesting series representations for the Bernstein basis functions.
on the both sides of (5.1), we find that ∞ n=0 B n k (x) = 1 x .
From the above equation, we arrive at the desired result. Proof. Proof of Theorem 16 is same as that of Theorem 15. That is if we replace t by −t in equation (2.4) and integrate by parts with respect to t from 0 to ∞ and using Laplace transform of the function f (t) = t n , then we arrive at the desired result.

Further Remarks and Observations
Fourier series of the Bernstein polynomials has been studied, without generating functions, by Izumi et al. [8]. They investigated many properties of the Fejer mean of the Fourier series of these polynomials. Fourier transform of the Bernstein polynomials has also been given, without generating functions, by Chui at al. [3]. By replacing t by it in (2.4)-(2.6), one may give applications of the Fourier transform to the complex generating functions for the Bernstein basis functions.