Generating Functions for the q-Bernstein Bases

Generating Functions for the q-Bernstein Bases Ron Goldman Department of Computer Science, Rice University Houston, Texas 77251 USA Plamen Simeonov Department of Computer and Mathematical Sciences University of Houston-Downtown Houston, Texas 77002 USA Yilmaz Simsek Department of Mathematics, Akdeniz University Antalya, Turkey January 19, 2014 Abstract We derive explicit formulas for the generating functions of the q-Bernstein basis functions in terms of q-exponential functions. Using these explicit formulas, we derive a collection of functional equations for these generating functions which we apply to prove a variety of identities–some old and some new–for the q-Bernstein bases.

Keywords and phrases: q-Bernstein bases, generating functions, q-exponential functions, q-Calculus, q-B´ezier curves 2010 Mathematics Subject Classification: 65Qxx, 33Dxx, 65D17, 26Cxx, 30C10. File name: G-S-S-qGF-January-2014.

1

Introduction

The Bernstein bases play a central role in the theory of B´ezier curves and surfaces. These polynomial curves and surfaces are of fundamental importance in Computer Aided Geometric Design (CAGD) for modeling free-form shapes [6]. Recently Phillips and Oru¸c [10], [11], [12] advocated using the q-Bernstein basis functions as a generalization of the classical Bernstein basis functions for CAGD. Building on this work of Phillips and Oru¸c on q-Bernstein bases and q-B´ezier curves as well as on the seminal papers of Ramshaw on blossoming [13], [14], Simeonov et al [15] introduced q-blossoming to investigate the analytic and geometric properties of qBernstein bases and q-B´ezier curves. Simeonov et al also discuss several potential applications of q-B´ezier curves and surfaces to CAGD. In order to investigate the properties of classical Bernstein bases, Simsek [17], [18] defined a novel collection of generating functions to derive many known and some new identities for the 1

2 classical Bernstein basis functions. The goal of this paper is to construct analogous generating functions for the q-Bernstein basis functions. By using an approach similar to that of Simsek as well as some known properties of basic hypergeometric series [2], [5], [8], we shall derive some known and some new identities and formulas for the q-Bernstein basis functions, including the partition of unity property, formulas for representing the monomials, a convolution formula with the monomials, recurrence relations, formulas for q-derivatives, subdivision identities, and a q-Marsden identity. We shall also derive for each fixed index k two new closed formulas for the sum of the q-Bernstein basis functions of degree n scaled by q n . In Section 2 we recall the definition of the classical and the q-Bernstein basis functions along with some basic notation from the q-Calculus. We also introduce the main generating functions discussed in this paper. In Section 3 we briefly review the definition and some basic properties of the q-exponential function, the q-derivative, and the q-integral, and state the main functional equation for the generating functions defined in Section 2. Section 4 contains most of our main results. Here we use the results from Section 3 concerning the q-exponential function and the q-Calculus to establish a collection of functional equations for the generating functions from Section 2, and we apply these functional equations to derive identities for the q-Bernstein bases and for their discrete q-derivatives. We also evaluate two discrete integrals for the q-Bernstein bases. Section 5 contains several auxiliary definitions and results from the theory of basic hypergeometric series, the proofs of the properties of the q-exponential function listed in Section 3, and the proof of the functional equation in Section 3 for the generating functions from Section 2. In Section 6, we derive for each fixed index k two novel closed formulas for the sum of the q-Bernstein basis functions of degree n scaled by q n , using a new generating function introduced in Section 5. We close in Section 7 with a brief review of our work along with a table summarizing the main identities and generating functions derived in this paper.

2

The q-Bernstein Bases and their Generating Functions

The classical Bernstein basis functions of degree n on the interval [0, 1] are defined by   n k n Bk (x) = x (1 − x)n−k , k = 0, . . . , n. k

(2.1)

Simsek and Acikgoz [19] and Acikgoz and Araci [1] introduced the generating functions gk (x, t) =

∞ X

Bkn (x)

n=k

tn xk tk (1−x)t = e . n! k!

(2.2)

Using these generating functions, Simsek proved many identities for the classical Bernstein basis functions [17], [18]. The q-Bernstein basis functions over the interval [0, 1] are defined by Bkn (x; q)

=

hni k

k

q

x

n−k−1 Y j=0

(1 − q j x),

k = 0, . . . , n.

(2.3)

3 Here

n

k q

denotes the q-binomial coefficient defined by hni k

q

=

[n]q ! , [k]q ![n − k]q !

k = 0, . . . , n

(2.4)

where [k]q ! = [1]q [2]q · · · [k]q ,

q 6= 1,

[0]q ! := 1

(2.5)

q 6= 1, q = 1.

(2.6)

and for k ∈ Z+ , [k]q is the q-integer defined by [k]q = 1 + q + · · · + q

k−1

=

(

1−q k 1−q ,

k,

We shall have occasions to consider both [k]q and [k]1/q . It follows from (2.5) and (2.6) that [k]q = q k−1 [k]1/q and

(2.7)

k

[k]q ! = q (2) [k]1/q!.

(2.8)

The q-Bernstein basis functions were introduced by Phillips [11] and Oru¸c [10], who advocated their use in CAGD. q-Bernstein basis functions on arbitrary intervals were introduced in [9], and later in [15] as blending functions for q-B´ezier curves over arbitrary intervals. Other types of q-Bernstein polynomials have been studied, see for example [19] for a type used in interpolation theory. The classical Bernstein polynomials Bkn (x) are obtained as the limiting case q → 1 of the q-Bernstein polynomials in [19]. In analogy with the generating functions gk (x, t) in equation (2.2) for the classical Bernstein basis functions, we define generating functions for the q-Bernstein basis functions by setting Gk (x, t; q) =

∞ X

Bkn (x; q)

n=k

tn . [n]q!

(2.9)

We shall see shortly that the series for Gk (x, t; q) converges for all |t| < 1/|1 − q| if |q| < 1, and for all t ∈ C if |q| > 1 or q = 1. Therefore for any fixed value of q > 0, there is an interval for t in which the series for Gk (x, t; q) converges.

3

The q-Exponential Function and the q-Calculus

Just as the generating functions gk (x, t) in equation (2.2) for the classical Bernstein basis functions are closely related to the classical exponential function ex , so too the generating functions Gk (x, t; q) for the q-Bernstein basis functions defined in equation (2.9) are closely related to the q-exponential function Eq (x). The q-exponential function Eq (x) is defined by [4] Eq (x) =

∞ X xn . [n]q ! n=0

(3.1)

4 This function is closely related to the standard q-exponential functions-see Section 5. By the ratio test this series for Eq (x) converges and therefore Eq (x) is well defined for all |x| < 1/|1 − q| if |q| < 1, and for all x ∈ C if |q| > 1 or q = 1. Therefore for any fixed value of q > 0, there is an interval in which the series for Eq (x) converges. The continuous derivatives of the classical exponential functions are easy to compute. Similarly the discrete q-derivatives of the q-exponential function have an analogous simple form. To understand what we mean by the discrete q-derivatives of the q-exponential function, we need to take a quick detour into the q-Calculus.

3.1

The q-Calculus

The discrete q-derivative of a function f (x) is defined by [2], [5], [8] Dq,xf (x) =

f (qx) − f (x) , (q − 1)x

q 6= 1.

Notice that if f 0 (x) exists, then f 0 (x) = limq→1 Dq,xf (x). The q-derivative is a linear operator. In addition there is a simple product rule and chain rule for the q-derivative [8] Dq,x(u(x)v(x)) = Dq,x(u(x))v(x) + u(qx)Dq,x(v(x))

(3.2)

and for any constant c Dq,x (f (cx)) = c Dq,y f (y)|y=cx .

(3.3)

These two formulas are easy to derive directly from the definition of the q-derivative. For polynomials the q-derivatives are easy to compute. Indeed, it follows easily from the definition of the q-derivative and equation (2.7) that Dq,x(xk ) = [k]q xk−1 , k

D1/q,x(x ) = [k]q (x/q)

(3.4) k−1

.

(3.5)

Just as there is a discrete form of differentiation, there is also a discrete form of integration. The discrete q-integral of a function f (x) is defined by [2], [5], [8] Z t ∞ X f (x) dq x = t(1 − q) f (tq j )q j , (3.6) 0

j=0

provided that the infinite sum converges. Since the q-integral is an infinite Riemann sum with the division points in geometric progression, the q-integral converges to the classical integral as q → 1− for functions that are continuous on [0, t]. The q-integral has many properties analogous to the properties of the classical integral. For example, the q-integral is a linear operator. Moreover if f (x) is continuous at 0, then Z t Dq,xf (x) dq x = f (t) − f (0). (3.7) 0

In addition, by (3.6) Z

t 0

1 f (cx) dq x = c

Z

ct

f (x) dqx. 0

(3.8)

5 For polynomials, q-integrals are easy to compute. Indeed, it follows easily from equations (3.7), (3.4), and (3.5) that Z t tn+1 xn dq x = , (3.9) [n + 1]q 0 Z t tn+1 q n tn+1 xn d1/q x = = . (3.10) [n + 1]1/q [n + 1]q 0

3.2

Properties of the q-Exponential Function

Returning to the q-exponential functions, we now have the following simple formulas for the q- and the 1/q-derivatives: Dq,xEq (x) = Eq (x),

(3.11)

Dq,x(1/Eq (x)) = −1/Eq (qx),

(3.12)

D1/q,x(1/Eq (x)) = −1/Eq (x).

(3.13)

Equation (3.11) follows directly from (3.1) and (3.4). Equations (3.12) and (3.13) will be proved in Section 5. In Section 5 we shall also derive the following identities: E1/q (x) = 1/Eq (−x),

(3.14)

Eq (qx) = (1 − (1 − q)x)Eq (x).

(3.15)

We close this section with an explicit formula for the generating functions Gk (x, t; q). Theorem 3.1. Let max{|q|, |x|} < 1 and |t| < 1/|1 − q|. Then Gk (x, t; q) =

(xt)k Eq (t) . [k]q ! Eq (xt)

(3.16)

Proof. See Section 5.

4

Identities for the q-Bernstein Basis Functions from Functional Equations for their Generating Functions

In this section we will use our generating functions to derive properties of the q-Bernstein basis functions. We shall proceed in the following fashion. In each case we first derive a functional equation satisfied by the generating functions. We then extract an identity for the q-Bernstein basis functions by comparing the coefficients of tn on both sides of this functional equation. All of the identities in this section can also be verified using standard summation formulas for basic hypergeometric series. The formulas for the basis functions are typically valid for all values of q, even though there are constraints on where (for what values of t and q) the functional equations for the generating functions are valid. The reason these identities hold for the q-Bernstein basis functions for all values of q is that the identities for the functional equations hold for all values

6 of x ∈ [0, 1/(1 − q)] and all values of t ∈ [0, 1] for 0 < q < 1. Thus the identities for the q-Bernstein basis functions are valid for all values of x ∈ [0, 1/(1 − q)] for 0 < q < 1. But the q-Bernstein basis functions Bkn (x; q) are also polynomials in q, and polynomial identities that are valid over an interval are valid everywhere. Thus it is enough to derive identities for the functional equations for 0 < q < 1 and x ∈ [0, 1/(1 − q)]. Our functional equations hold for all values of |x| ∈ [0, 1/|1 − q|] and all values of |t| ∈ [0, 1] for |q| < 1. Using Theorem 3.1 and equation (3.1), we immediately obtain the following result. Theorem 4.1. Let max{|q|, |x|, |z|} < 1 and |t| < 1/|1 − q|. Then ∞ X

Gk (x, t; q)z k =

k=0

Eq (t) Eq (zxt). Eq (xt)

(4.1)

In particular, ∞ X

Gk (x, t; q) = Eq (t).

(4.2)

k=0

From Theorem 4.1, we can easily show that the q-Bernstein bases form a partition of unity. Corollary 4.2.

n X

Bkn (x; q) = 1.

(4.3)

k=0

Proof. Changing the order of summation in (4.2) yields ∞ n ∞ X X tn X n tn Bk (x; q) = Eq (t) = . [n]q ! [n]q !

n=0

n=0

k=0

Now (4.3) follows by equating the coefficients of

tn [n]q !

on both sides of this equation.

A convolution identity between the monomials and the q-Bernstein basis functions can also be derived from the functional equation (3.16) in Theorem 3.1. Corollary 4.3. The q-Bernstein basis functions satisfy the convolution identity n−k Xh l=0

hni n i l n−l x Bk (x; q) = xk . l q k q

(4.4)

Proof. By (3.16) we have Eq (xt)Gk (x, t; q) =

(xt)k Eq (t). [k]q !

Expanding Eq (xt), Gk (x, t; q), and Eq (t) in this equation using (3.1) and (2.9), and then equating the coefficients of tn /[n]q ! on both sides of the resulting equation yields (4.4).

7 Theorem 4.4. Let max{|q|, |x|, |z|} < 1 and |t| < 1/|1 − q|. Then ∞   X k (xt)j Eq (t) Gk (x, t; q)z k−j = Eq (zxt). j q [j]q ! Eq (xt) k=j ∞   X k (xt)j Eq (t). Gk (x, t; q) = j q [j]q !

(4.5) (4.6)

k=j

j Proof. Applying Dq,z to (4.1), then applying (3.4), (3.11), and (3.3), and finally dividing both sides by [j]q !, we obtain (4.5). Setting z = 1 in (4.5) yields (4.6).

Corollary 4.5.

n   X k

j

k=j

Bkn (x; q)

q

  n = xj . j q

(4.7)

Proof. We use (2.9) and (3.1) to expand both sides of (4.6) in powers of t, and then we equate the coefficients of tn /[n]q ! on both sides of the resulting equation to arrive at (4.7). Corollary 4.6. j X

n

(−1)n−k q −( 2 )

n=k

    k j j Bkn (x; q) = q −(2) xj . n 1/q k 1/q

(4.8)

Proof. From (3.16) and (3.14) we get Gk (x, t; q)E1/q(−t) =

(xt)k E (−xt). [k]q ! 1/q

We expand the functions on both sides of this equation using (2.9) and (3.1) to obtain ∞ X

Bkn (x; q)

n=k

∞ ∞ tn X (−t)l (xt)k X (−xt)l = . [n]q! [l]1/q! [k]q ! [l]1/q! l=0

l=0

Equating the coefficients of tj on both sides of this equation yields j X

n=k

Bkn (x; q)

(−1)j−n (−1)j−k xj = . [n]q![j − n]1/q ! [k]q ![j − k]1/q !

The last equation reduces to (4.8) after multiplying both sides by (−1)j−k [j]1/q! and applying (2.8). The partial q-derivatives of the generating functions also contain information about the q-Bernstein basis functions. We begin with two PDE’s for the q-derivatives of the generating functions with respect to x. These PDE’s lead to formulas for the q-derivatives of the qBernstein basis functions. Theorem 4.7. Let max{|q|, |x|} < 1 and |t| < 1/|1 − q|. Then Dq,x Gk (x, t; q) = tq −k+1 Gk−1 (qx, t; q) − tq −k Gk (qx, t; q), D1/q,xGk (x, t; q) = tq

−k+1

Gk−1 (x, t; q) − tq

−k

Gk (x, t; q).

(4.9) (4.10)

8 Proof. From (3.16), (3.2) (with u(x) = 1/Eq (xt) and v(x) = xk ), (3.12), (3.3), and (3.4), it follows that  k    tk Eq (t) tk Eq (t) [k]q xk−1 x xk t Dq,x Gk (x, t; q) = Dq,x = − + [k]q ! Eq (xt) [k]q ! Eq (qxt) Eq (qxt) which proves (4.9). The proof of (4.10) is similar. Again using (3.16), (3.2) (with u(x) = xk and v(x) = 1/Eq (xt)), (3.13), (3.3), and (3.5), we obtain  k    tk Eq (t) tk Eq (t) [k]q (x/q)k−1 (x/q)k t x D1/q,xGk (x, t; q) = D1/q,x = − [k]q ! Eq (xt) [k]q ! Eq (xt) Eq (xt) which yields (4.10). From definition (2.9) of the generating functions Gk (x, t; q) and formulas (4.9) and (4.10), we can derive differentiation formulas for the q-Bernstein basis functions by equating the coefficients of tn on both sides of these equations. Corollary 4.8.
n−1 Dq,x Bkn (x; q) = [n]q q −k qBk−1 (qx; q) − Bkn−1 (qx; q) ,
n−1 D1/q,xBkn (x; q) = [n]q q −k qBk−1 (x; q) − Bkn−1 (x; q) .

(4.11) (4.12)

Formula (4.12) is a special case of formula (2.2) in [7].

As a direct consequence of formulas (4.11)–(4.12), we evaluate two discrete integrals for the q-Bernstein basis functions. Corollary 4.9. If |q| < 1, then Z

1 0

Bkn (x; q) dqx

=

(

q k+1 , [n+1]q 1 [n+1]q ,

k = 0, . . . , n − 1, k = n.

(4.13)

k = 0, . . ., n.

(4.14)

If |q| > 1, then Z

0

1

Bkn (x; q) d1/qx =

qk , [n + 1]q

Proof. First we prove (4.13). To begin, observe that by (2.3) and (3.9) Z 1 Z 1 1 n Bn (x; q) dqx = xn dq x = . [n + 1]q 0 0

(4.15)

Also by q-integrating (4.11) (with n replaced by n + 1) from 0 to 1/q and applying (3.7) on the left hand side and (3.8) on the right hand side, we get Z 1
n Bkn+1 (1/q; q) − Bkn+1 (0; q) = [n + 1]q q −k Bk−1 (x; q) − q −1 Bkn (x; q) dq x. (4.16) 0

Notice that by (2.3)

Bkn+1 (0; q) = 0, Bkn+1 (1/q; q) =

k ≥ 1, 

[n + 1]q q −n (1 − 1/q), 0,

k = n, k < n.

9 Therefore when k = n, (4.16) yields 1

Z

[n + 1]q q −n (1 − 1/q) = [n + 1]q q −n


n Bn−1 (x; q) − q −1 Bnn (x; q) dq x

0

R1

n which together with (4.15) implies 0 Bn−1 (x; q) dqx = q n /[n + 1]q . When 0 < k < n, (4.16) becomes Z 1 Z 1 n Bk−1 (x; q) dqx = q −1 Bkn (x; q) dqx. 0

0

Iterating this relation n − k times yields Z 1 Z n Bk−1 (x; q) dqx = q −(n−k) 0

1

n Bn−1 (x; q) dqx = q k /[n + 1]q .

0

Next, we prove (4.14). To begin, observe that by (2.3) and (3.10) Z

1 0

Bnn (x; q) d1/qx

=

Z

1

0

xn d1/q x =

qn . [n + 1]q

(4.17)

Now we 1/q-integrate (4.12) (with n replaced by n + 1) from 0 to 1 and apply (3.7) and (2.3). When 0 < k < n + 1, we get Z 1

qk n qBk−1 (x; q) − Bkn (x; q) d1/q x = Bkn+1 (1; q) − Bkn+1 (0; q) = 0, [n + 1]q 0 hence

R1 0

n (x; q) d −1 Bk−1 1/qx = q

Z

0

1

R1 0

Bkn (x; q) d1/qx. Iterating this relation n − k + 1 times yields

n (x; q) d1/qx Bk−1

=q

−(n−k+1)

Z

0

1

Bnn (x; q) d1/qx =

q k−1 . [n + 1]q

Next we derive two more PDE’s for the generating functions Gk (x, t; q), this time by taking the partial q-derivatives with respect to t. These new PDE’s for the generating functions Gk (x, t; q) lead to recurrences for the q-Bernstein basis functions. Theorem 4.10. Let max{|q|, |x|} < 1 and |t| < |1 − q|. Then Dq,tGk (x, t; q) = xGk−1 (x, t; q) + q k Gk (x, t; q) − xGk (x, qt; q), Dq,tGk (x, t; q) = xq

−k+1

Gk−1 (x, qt; q) + Gk (x, t; q) − xq

−k

Gk (x, qt; q).

(4.18) (4.19)

Proof. First we prove (4.18). We multiply both sides of equation (3.16) by Eq (xt) and then we apply Dq,t to the resulting equation. Using the product rule (3.2) with u(t) = Gk (x, t; q) and v(t) = Eq (xt) on the left hand side and with u(t) = tk and v(t) = Eq (t) on the right hand side yields Dq,t(Gk (x, t; q))Eq(xt) + Gk (x, qt; q)xEq(xt) =

 xk  [k]q tk−1 Eq (t) + (qt)k Eq (t) [k]q !

10 where we also used (3.3), (3.11), and (3.4). Dividing both sides of this equation by Eq (xt) and applying (3.16), we get (4.18). To establish (4.19), we apply Dq,t to equation (3.16) and use the product rule (3.2) twice on the right hand side, first with u(t) = Eq (t)/Eq (xt) and v(t) = tk , and then with u(t) = 1/Eq (xt) and v(t) = Eq (t). We then get     Eq (t) Eq (qt) xk k k−1 Dq,tGk (x, t; q) = t Dq,t + [k]q t [k]q ! Eq (xt) Eq (qxt)   (xt)k xEq (t) Eq (t) − + + xq −k+1 Gk−1 (x, qt; q) = [k]q ! Eq (qxt) Eq (qxt)   (xt)k x(1 − (1 − q)t)Eq (t) (1 − (1 − q)xt)Eq (t) (4.20) − + + xq −k+1 Gk−1 (x, qt; q) = [k]q ! Eq (qxt) Eq (qxt)   k xEq (qt) Eq (t) (xt) = − + + xq −k+1 Gk−1 (x, qt; q) [k]q ! Eq (qxt) Eq (xt) = −xq −k Gk (x, qt; q) + Gk (x, t; q) + xq −k+1 Gk−1 (x, qt; q),

where we also used (3.4) in the first line, (3.12), (3.11), (3.3), and (3.16) to get the second line, (3.15) to get the fourth line, and (3.16) to get the fifth line. Corollary 4.11. The q-Bernstein basis functions satisfy the recurrence relations n Bkn+1 (x; q) = xBk−1 (x; q) + q k (1 − xq n−k )Bkn (x; q),

(4.21)

n (x; q) + (1 − xq n−k )Bkn (x; q). Bkn+1 (x; q) = xq n−k+1 Bk−1

(4.22)

Proof. We use (2.9) and (3.4) to expand both sides of (4.18) in powers of t, and then we equate the coefficients of tn /[n]q ! on both sides of the resulting equation to obtain (4.21). Similarly, we use (2.9) and (3.4) to expand both sides of (4.19) in powers of t, and then we equate the coefficients of tn /[n]q ! on both sides of the resulting equation to obtain (4.22). Next, we establish another functional equation for the generating functions Gk (x, t; q) and we use this equation to derive a subdivision relation for the q-Bernstein basis functions. Theorem 4.12. Let max{|q|, |x|, |y|} < 1 and |t| < 1/|1 − q|. Then the generating functions Gk (x, t; q) satisfy the functional equations Gk (xy, t; q) =

[l]q![s]q ! k−l−s k−s k−l−s x y t Gl (x, t; q)Gs(y, xt; q) [k]q !

(4.23)

for all l, s. In particular, Gk (xy, t; q) = G0 (x, t; q)Gk(y, xt; q). Proof. Formula (4.23) follows immediately from (3.16) and the identity Eq (t) Eq (t) Eq (xt) = . Eq (xyt) Eq (xt) Eq (yxt) Formula (4.24) follows from (4.23) by letting l = 0 and s = k.

(4.24)

11 Corollary 4.13. The q-Bernstein basis functions satisfy the subdivision relation n−k X

Bkn (xy; q) =

n−j

n Bn−j (x; q)Bk

(y; q).

(4.25)

j=0

Proof. By (2.9) and (2.3) we have G0 (x, t; q) =

∞ X

∞ n−1

B0n (x; q)

n=0

XY tn tn = (1 − xq l ) . [n]q! n=0 [n]q ! l=0

Also by (2.9) we have Gk (xy, t; q) =

∞ X

Bkn (xy; q)

n=k

tn [n]q!

and Gk (y, xt; q) =

∞ X

Bkn (y; q)

n=k

xntn . [n]q!

Substituting the last three expressions into (4.24) and equating the coefficients of tn on both sides of the resulting equation, we obtain n−k

Bkn (xy; q) X = [n]q ! j=0

Qj−1

− xq l ) Bkn−j (y; q)xn−j . [j]q! [n − j]q !

l=0 (1

By (2.3), this equation reduces to (4.25). Formula (4.25) was first established in [3] using a matrix based technique, and later in [15, Proposition 5.4] using a method based on q-blossoming. Corollary 4.14. n X

Bjn (x; q)

j=0

j−1 n−1 Y Y (1 − tq l ) = (1 − txq l ). l=0

(4.26)

l=0

Proof. Formula (4.26) follows immediately from (4.25) and (2.3) by setting k = 0 and y = t in (4.25) and then replacing j by n − j. A more general version of Corollary 4.14 for q-Bernstein basis functions on arbitrary intervals can be derived using the same method or a method based on q-blossoming [15]. The next result is a q-version of Marsden’s identity on the interval [0, 1]. Corollary 4.15. (q-Marsden’s Identity on the Interval [0, 1]). n−1 Y l=0

(t − xq l ) =

n X j=0

n

n

j Bj (x; q)Bn−j (t; 1/q) h i (−1)j q (2) .

n j 1/q

(4.27)

12 Proof. In equation (4.26) we replace t by 1/t and multiply both sides of the resulting equation by tn , thus obtaining j−1 n−1 n Y X Y l n n−j Bj (x; q)t (t − q l ). (t − xq ) = j=0

l=0

l=0

By (2.3) we have n−j

t

j−1 Y

j (t − q ) = (−1) q (2)tn−j

l

j

l=0

j−1 Y

j

(1 − tq −l ) = (−1)j q (2)

l=0

Combining the last two equations yields formula (4.27).

n Bn−j (t; 1/q) h i . n j 1/q

This q-Marsden identity is a special case of a more general q-Marsden identity over arbitrary intervals established in [15, Proposition 5.1] using a method based on q-blossoming. We close this section with a very general subdivision relation. Corollary 4.16. The q-Bernstein basis functions satisfy the subdivision relations Bkn (xy; q) =

n+l−k [l]q![s]q ![n]q ! k−s X Blm (x; q) Bsn+l+s−k−m (y; q)xn−m y [k]q ! [m]q ! [n + l + s − k − m]q !

(4.28)

m=l

for all l, s. tn [n]q ! tl+s−k

Proof. If k ≥ l + s, formula (4.28) is obtained by equating the coefficients of

on both sides

of (4.23). If l + s ≥ k, formula (4.28) follows after multiplying (4.23) by n+l+s−k the coefficients of t [n]q ! on both sides of the resulting equation.

and equating

5

Proofs of Properties of the q-Exponential Functions

We are now going to prove the assertions we made without proof in Section 3 about the qexponential function Eq (x) and about the generating functions Gk (x, t; q). In order to avoid excessive length in some of the formulas in our proofs, here we introduce a much more concise notation. We begin with the q-shifted factorials defined by [2], [5], [8] (a; q)n =

n Y

(1 − aq j−1 ),

(a; q)0 := 1,

j=1

and (a; q)∞ =

∞ Y

(1 − aq j−1 ),

|q| < 1.

j=1

Notice that

(a; q)k (aq k ; q)n−k = (a; q)n, k

(a; q)k (aq ; q)∞ = (a; q)∞.

(5.1) (5.2)

13 The q-Bernstein basis functions can now be rewritten more concisely in terms of the qshifted factorials. First the q-binomial coefficients defined in (2.4) can be written as hni k

q

=

(q; q)n (q k+1 ; q)n−k = , (q; q)k (q; q)n−k (q; q)n−k

k = 0, . . . , n.

(5.3)

Then, by (2.3) and (5.3) the q-Bernstein basis functions defined in (2.3) can be expressed as Bkn (x; q) =

hni k

q

xk (x; q)n−k =

(q k+1 ; q)n−k xk (x; q)n−k , (q; q)n−k

n ≥ k.

(5.4)

We shall also use the identities (A; q)n−k = (q

−n+1

n

(A; q)n , n−k 2 ) (q −n+1 /A; q)

(5.5)

n −(n 2)

(5.6)

(−A)k q (2 )−(

k

/A; q)n = (−1/A) q

(A; q)n.

These identities follow directly from the definition of the q-shifted factorials. In order to analyze the q-exponential function Eq (x) which appears in the explicit formula (3.16) for the generating functions Gk (x, t; q), we now introduce the standard q-exponential functions eq (x) and Eq (x) defined for |q| < 1 by [2], [5], [8] eq (x) =

∞ X n=0

and

xn , (q; q)n

|x| < 1,

n ∞ X q( 2) n Eq (x) = x . (q; q)n

(5.7)

(5.8)

n=0

We shall see shortly (Proposition 5.2) that the three q-exponential functions Eq (x), eq (x), and Eq (x) are closely related. We shall use the basic hypergeometric series notation [2], [5], [8] ∞ s+1−r X n (a1 , . . . , ar ; q)n n  z (−1)n q ( 2 ) , r φs (a1 , . . . , ar ; b1 , . . . , bs; q, z) = (q, b1, . . ., bs ; q)n

(5.9)

n=0

where (a1 , . . . , ar ; q)n =

r Y

(aj ; q)n .

j=1

In order to analyze the q-exponential functions Eq (x), eq (x), and Eq (x), we will need the q-binomial formula [2, Theorem 10.2.1], [8, Theorem 12.2.5]. Let |q| < 1. Then 1 φ0 (a; −; q, z)

=

∞ X (a; q)n

n=0

(q; q)n

zn =

(az; q)∞ , (z; q)∞

(5.10)

where |z| < 1 or a = q −n for some integer n ≥ 0. With this preparation, we are now ready to prove some identities for the q-exponential functions.

14 Lemma 5.1. (Euler’s Theorem, [8, Theorem 12.2.6]) 1 , (x; q)∞ Eq (x) = (−x; q)∞. eq (x) =

(5.11) (5.12)

Proof. Setting a = 0 in (5.10) yields (5.11), and replacing z by −x/a in (5.10) and letting a → ∞ yields (5.12). Proposition 5.2. Eq (x) = eq ((1 − q)x),

|q| < 1,

(5.13)

E1/q (x) = Eq ((1 − q)x),

0 < |q| < 1.

(5.14)

Proof. Equation (5.13) follows from (3.1) and (5.7). To prove equation (5.14), notice that for 0 < |q| < 1, the function E1/q (x) defined by the series in (3.1) is an entire function of x. From (3.1), (2.8), and (5.8) it follows that E1/q (x) =

∞ X n=0

∞ (n) n X xn q 2 x = = Eq ((1 − q)x). [n]1/q ! [n]q ! n=0

Next we prove equations (3.12)–(3.15). Below we recall these four equations: Dq,x(1/Eq (x)) = −1/Eq (qx), D1/q,x(1/Eq (x)) = −1/Eq (x),

(5.15) (5.16)

E1/q (x) = 1/Eq (−x),

(5.17)

Eq (qx) = (1 − (1 − q)x)Eq (x).

(5.18)

Proof of Equations (5.15) and (5.16). By (5.13) and (5.11) we get ((1 − q)x/q; q)∞ − ((1 − q)x; q)∞ (1/q − 1)x [(1 − (1 − q)x/q) − 1] 1 = ((1 − q)x; q)∞ =− . (1 − q)x/q Eq (x)

D1/q,x(1/Eq (x)) =

Equation (5.15) is proved similarly. Proof of Equations (5.17) and (5.18). To prove equation (5.17), observe that from equations (5.14), (5.12), (5.11), and (5.13) it follows that E1/q (x) = Eq ((1 − q)x) = (−(1 − q)x; q)∞ = 1/eq (−(1 − q)x) = 1/Eq (−x). Equation (5.18) is an immediate consequence of (5.13) and (5.11).

15 To derive the explicit formula (3.16) for the generating functions Gk (x, t; q), next we introduce a new family of generating functions defined by Gk (x, t, c; q) =

∞ X

Bkn (x; q)

n=k

tn . (c; q)n

(5.19)

Theorem 5.3. Let max{|q|, |x|, |t|, |c|} < 1. Then Gk (x, t, c; q) =

(xt)k k+1 ; cq k ; q, t). 2 φ1 (x, q (c; q)k

(5.20)

Proof. This formula follows from equations (5.4), (5.1), and (5.9). Now we are ready to derive our explicit formula for the generating functions Gk (x, t; q) that was stated without proof in Theorem 3.1.

Proof of Theorem 3.1. From (2.9), (5.19), (5.20), (5.10), (5.7), (5.11), and (5.13), we derive ((1 − q)xt)k 1 φ0 (x; −; q, (1 − q)t) (q; q)k (xt)k eq ((1 − q)t) (xt)k Eq (t) = = . [k]q ! eq ((1 − q)xt) [k]q ! Eq (xt)

Gk (t, x; q) = Gk (x, (1 − q)t, q; q) = =

6

(xt)k ((1 − q)xt; q)∞ [k]q ! ((1 − q)t; q)∞

Two New Summation Formulas

In this section we are going to derive two q-analogues of the following summation formula for the classical Bernstein polynomials defined in (2.1): ∞ X

Bkn (x) =

n=k

1 x

(6.1)

for 0 < |x| < 1. Equation (6.1) can be verified by direct summation, by integration, or by induction on k. To derive these new identities, we need Heine’s transformation formula [8, Theorem 12.5.1]: 2 φ1 (a, b; c; q, z)

=

(b, az; q)∞ 2 φ1 (c/b, z; az; q, b), (c, z; q)∞

(6.2)

for |z| < 1 and |b| < 1. Corollary 6.1. ∞ X

n=k

for 0 < |x| < 1 and |q| < 1.

qk Bkn (x; q)q n = x

1 − (x; q)∞

k X l=0

xl (q; q)l

!

(6.3)

16 Proof. Applying (5.19), (5.20), Heine’s transformation (6.2) (with a = q k+1 , b = x, c = 0, and z = q), (5.9), (5.2), (5.1), (5.7), and (5.11), we derive ∞ X

Bkn (x; q)q n = Gk (x, q, 0; q) = (qx)k 2 φ1 (q k+1 , x; 0; q, q)

n=k

∞

X (q k+2 , x; q)∞ xl k+2 k (x; q)∞ φ (0, q; q ; q, x) = (qx) 2 1 (q; q)∞ (q; q)k+1 (q k+2 ; q)l l=0 ! ∞ k X q k (x; q)∞ X xl q k (x; q)∞ 1 xl = = − x (q; q)l x (x; q)∞ (q; q)l

= (qx)k

l=k+1

l=0

which yields (6.3). Corollary 6.2. ∞ X

Bkn (x; q)q n

n=k

(−x)k k D = [k]q ! 1/q,x



1 − (x; q)∞ x



(6.4)

for 0 < |x| < 1 and |q| < 1. Proof. The following differentiation formula is easily verified by induction on n: k D1/q,x ((x; q)n) = (−1)k

[n]q ! (x; q)n−k . [n − k]q !

(6.5)

By the first equality in (5.4), (2.4), and (6.5), we get ∞ X

n=k

=

xk [k]q !

∞ X

n=k

∞

Bkn (x; q)q n

xk X n [n]q ! q (x; q)n−k = [k]q ! [n − k]q ! n=k

∞ X (−x)k k k q n (−1)k D1/q,x ((x; q)n) = D1/q,x (x; q)nq n [k]q ! n=0

!

(6.6) .

Notice that in the last expression on the second line we can start the sum from n = 0 instead k of from n = k, since D1/q,x annihilates polynomials of degree less than k. By (5.9) we have ∞ X n=0

(x; q)nq n = 2 φ1 (q, x; 0; q, q) =

(x; q)∞ 2 2 φ1 (0, q; q ; q, x) 1−q

∞ ∞ (x; q)∞ X xl (x; q)∞ X xl+1 = = 1−q (q 2 ; q)l x (q; q)l+1 l=0 l=0   (x; q)∞ 1 = −1 , x (x; q)∞

(6.7)

where to get the second equality we used Heine’s transformation (6.2) (with a = q, b = x, c = 0, and z = q), and to get the last equality we used (5.7) and (5.11). Formula (6.4) now follows from (6.6) and (6.7).

17

7

Summary

We have derived a suite of identities for the q-Bernstein basis functions Bkn (x; q)

=

hni k

q

k

x

n−k−1 Y

(1 − q j x),

k = 0, . . ., n

j=0

from a collection of functional equations for their generating functions Gk (x, t; q) =

∞ X

n=k

Bkn (x; q)

tn . [n]q

Our method in each case is first to derive a functional equation satisfied by the generating functions, and then to extract an identity for the q-Bernstein basis functions by equating the coefficients of tn on both sides of this functional equation. Similar identities and proofs hold for the h-Bernstein basis functions [16] using analogues of the h-exponential functions, h-derivatives, and h-integrals. Since we have derived many identities, we close here with a list of these formulas in Table 1 for easy future reference.

18 Formula

q-Bernstein Identity

Basis Functions

n (x; q) = Bk

Partition of Unity

Pn

Convolution Identity Monomials I

Monomials II

h

Functional Equation (xt)k

i

E (t)

Gk (x, t; q) = [k] ! E q(xt) q q P∞ k=0 Gk (x, t; q) = Eq (t)

n xk (x; q)n−k k q

n k=0 Bk (x; q) = 1 h i Pn−k h n i j n−j k (x; q) = n j=0 j q x Bk k q x h i h i Pn k n n j k=j j q Bk (x; q) = j q x “ ” h i n Pj − j n (x; q) 2 (−1)n−k q Bk n=k n 1/q “ ” h i − k j 2 =q xj k

(xt)k

Eq (xt)Gk (x, t; q) = [k] ! Eq (t) q P∞ h k i (xt)j Gk (x, t; q) = [j] ! Eq (t) k=j j q

q

(xt)k Gk (x, t; q)E1/q (−t) = [k] ! E1/q (−xt) q

1/q

Discrete q-Derivative

= [n]q q −k

Discrete 1/q-Derivative

= [n]q q −k

First Recurrence Second Recurrence Subdivision Formula I Subdivision Formula II

Polynomial Identity q-Marsden’s Identity

“

n (x; q) Dq,xBk n−1

Summation Formula II

(qx; q)

D B n (x; q) “ 1/q,x k ” n−1 n−1 qBk−1 (x; q) − Bk (x; q)

”

n+1 Bk (x; q) n n = xBk−1 (x; q) + q k (1 − xq n−k )Bk (x; q)

n+1 Bk (x; q) n n (x; q) = xq n−k+1 Bk−1 (x; q) + (1 − xq n−k )Bk Pn−k n n−j n Bk (xy; q) = (y; q) j=0 Bn−j (x; q)Bk [l]q ![s]q ![n]q ! k−s y [k]q ! n+l+s−k−j Pn+l−k xn−j Blj (x;q) Bs (y;q) × j=l [j]q ! [n+l+s−k−j]q ! n (xy; q) = Bk

Pn

n k=0 Bk (x; q)(t; q)k = (tx; q)n Qn−1 (t − xq l ) l=0 “ ” n n j B Pn n−j (t;1/q)B j (x;q) j 2 h i = j=0 (−1) q n j

Summation Formula I

n−1

qBk−1 (qx; q) − Bk

P∞ n n „ n=k Bk (x; q)q « P xl 1 − (x; q)∞ k l=0 (q;q)l P∞ n q)q n n=k Bk (x; “ ” (−x)k 1−(x;q)∞ k = [k] ! D1/q,x x qk

q

R1

n 0 Bk (x; q) dq x =

1/q-Integration Formula

R1

q

D1/q,x Gk (x, t; q) = tq −k+1 Gk−1 (x, t; q) − tq −k Gk (x, t; q) Dq,t Gk (x, t; q) = xGk−1 (x, t; q) +q k Gk (x, t; q) − xGk (x, qt; q) Dq,t Gk (x, t; q) = xq −k+1 Gk−1 (x, qt; q) +Gk (x, t; q) − xq −k Gk (x, qt; q) Gk (xy, t; q) = G0 (x, t; q)Gk (y, xt; q) Gk (xy, t; q) =

Gk (xy, t; q) = G0 (x, t; q)Gk (y, xt; q) Gk (xy, t; q) = G0 (x, t; q)Gk (y, xt; q) (xt)k

Gk (x, t, c; q) = (c;q) 2 φ1 (x, q k+1 ; cq k ; q, t) k (xt)k

Gk (x, t, c; q) = (c;q) 2 φ1 (x, q k+1 ; cq k ; q, t) k

(1−δn,k )k+1 [n+1]q

qk n 0 Bk (x; q) d1/q x = [n+1]q

[l]q ![s]q ! k−l−s k−s k−l−s x y t [k]q !

×Gl (x, t; q)Gs (y, xt; q)

1/q

= x

q-Integration Formula

Dq,x Gk (x, t; q) = tq −k+1 Gk−1 (qx, t; q) − tq −k Gk (qx, t; q)

= [n]q q −k = [n]q q −k

n “ Dq,xBk (x; q) ” n−1 n−1 qBk−1 (qx; q) − Bk (qx; q)

D B n (x; q) “ 1/q,x k ” n−1 n−1 qBk−1 (x; q) − Bk (x; q)

Table 1: Identities for the q-Bernstein basis functions and the corresponding functional equations for their generating functions. Acknowlegment. The research of Yilmaz Simsek was supported by the Scientific Research Project Administration of Akdeniz University.

References [1] M. Acikgoz, S. Araci, On the generating function for Bernstein polynomials, American Institute of Physics CP1281, 2010. [2] G. E. Andrews, R. A. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. [3] C. Disibuyuk, H. Oru¸c, A generalization of rational Bernstein-B´ezier curves, BIT 47 (2007) 313-323. [4] H. Exton, q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983.

19 [5] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 2004. [6] R. Goldman, Pyramid Algorithms, A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling, Elsevier Science, 2003. [7] R. Goldman, P. Simeonov, Formulas and algorithms for quantum differentiation of quantum Bernstein bases and quantum B´ezier curves based on quantum blossoming, Graphical Models, 74(6)(2012) 326–334. [8] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in one Variable, Cambridge University Press, Cambridge, 2005. [9] S. Lewanowicz, P. Wo´zny, Generalized Bernstein polynomials, BIT, 44(2004) 63–78. [10] H. Oru¸c, Generalized Bernstein Polynomials and Total Positivity, Ph.D. Thesis, School of Mathematical and Computational Sciences, University of St. Andrews, 1998. [11] G. M. Phillips, Bernstein polynomials based on the q-integers, The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, Annals of Numerical Math. 1-4, 511–518, 1997. [12] G. M. Phillips, A survey of results on the q-Bernstein polynomials, IMA J. Numer. Analysis 30(2010) 277-288. [13] L. Ramshaw, Blossoming: A Connect-the-Dots Approach to Splines, Digital Equipment Corp., Systems Research Center, Technical Report no. 19, 1987. [14] L. Ramshaw, Blossoms are polar forms, Computer Aided Geometric Design, 6(1989) 323-358. [15] P. Simeonov, V. Zafiris, R. Goldman, q-Blossoming: A new approach to algorithms and identities for q-Bernstein bases and q-B´ezier curves, Journal of Approximation Theory 164(2012), 77–104. [16] P. Simeonov, V. Zafiris, R. Goldman, h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-B´ezier curves, Computer Aided Geometric Design, 28(9)(2011), 549-565. [17] Y. Simsek, Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for these polynomials, to appear in Hacettepe Journal of Mathematics and Statistics. [18] Y. Simsek, Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions, Fixed Point Theory and Applications, 1:80(2013), 13 pages. [19] Y. Simsek, M. Acikgoz, A new generating function of (q−) Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal., (2010), 12 pages, Article ID 769095, DOI: 10.1155/2010/769095.

20 Ron Goldman: [email protected] Plamen Simeonov: [email protected] Yilmaz Simsek: [email protected]