Approximation properties of q-Bernoulli polynomials M. Momenzadeh and I. Y. Kakangi Near East University Lefkosa, TRNC, Mersiin 10, Turkey Email:
[email protected] [email protected] August 8, 2017 Abstract We study the q analogue of Euler-Maclurian formula and by introducing a new q-operator we drive to this form. Moreover, approximation properties of q-Bernoulli polynomials is discussed. We estimate the suitable functions as a combination of truncated series of q-Bernoulli polynomials and the error is calculated. This paper can be helpful in two di¤erent branches, …rst solving di¤erential equations by estimating functions and second we may apply these techniques for operator theory.
1
Introduction
In this paper, we investigate the approximation of suitable function f (x) as a linear combination of q Bernoulli polynomials. We reach to the kind of Euler expansion for f (x) by using q operators and in a parallel way, we expand the function in a terms of q Bernoulli polynomials. This expansion gives the appropriate tool to solve q-di¤erence equations or normal di¤erential equation as well. In fact, several approaches for solving di¤erential equations used a kind of approximation of a function in a terms of Bernoulli polynomials.[20] [21] [7] In this paper, we provide a conditions to approximate capable functions as a linear combination of q-Bernoulli polynomials and some example will be given. We introduce the new q operator to reach the q analogue of Euler-Maclaurian formula. We start by introducing some q-calculus concepts. There are several type of q Bernoulli polynomials and numbers that can be generated by di¤erent q exponential functions.q-analogue of the Bernoulli numbers were …rst studied by Carlitz [2], in the middle of the last century when he introduced a new sequence f m gm 0 : m X
k=0
m k
kq
k+1
m
=
1; 0;
m = 1; m > 1:
(1)
Here and in the remainder of the paper, we make the assumption that jqj < 1:Clearly we recover the ordinary forms when q tends to one from a left side Since Carlitz, there have been many distinct q-analogue of Bernoulli numbers arising from varying motivations. In [3], they used improved q exponential functions to introduce a new class of q Bernoulli polynomials. In addition, they investigate some properties of these q-Bernoulli polynomials. Let us introduce a class of q Bernoulli polynomials in a form of generating function as follows 1 X zeq (zt) = eq (z) 1 n=0
n;q (t)
zn ; [n]q !
jzj < 2 :
(2)
1
where [n]q is q number and q-numbers factorial is de…ned by [a]q =
1 qa 1 q
(q 6= 1) ;
[0]q ! = 1;
[n]q ! = [n]q [n
1]q
n 2 N; a 2 C:
The q-shifted factorial and q-polynomial coe¢ cient are de…ned by (a; q)0 = 1;
(a; q)n =
n Y1
qj a ;
1
n 2 N;
j=0
(a; q)1 = n k
1 Y
qj a ;
1
j=0
= q
jqj < 1; a 2 C:
(q; q)n ; (q; q)n k (q; q)k
respectively.q-standard terminology and notation can be found at [1] and [6]. We call n and n;q = n;q (0). In addition, q-analogue of (x a) is de…ned as
(x
n
a)q =
1; (x
a)(x
aq):::(x
aq n
1
);
n = 0; n 6= 0:
n;q
Bernoulli number
;
In the standard approach to the q calculus, two exponential function are used.These q exponentials are de…ned by
eq (z) =
1 1 Y X zn = [n]q ! (1 n=0
Eq (z) = e1=q (z) =
1 q) q k z)
(1
k=0 1 X
;
0 < jqj < 1; jzj
b
(16)
Now in the aid of q Taylor expansion, we may write F (x + h) as follows
F (x + h) =
j
1 X ((x + h)
x)q
[j]q !
j=0
1 X Dqj Hqj
Dqj F (x) =
(F (x)) = eq (Dq Hq ) (F (x))
[j]q !
j=0
(17)
Here, eq is q shifted operator and can be expressed as expansion of Dq and Hq : We may assume that Dh (F (x)) = f (x) and then f (x) =
F (x + h) h
F (x)
=
eq (Dq Hq ) Hq
1
F (x)
(18)
Since the Jackson integral is q antiderivative, we have Hq Dq eq (Dq Hq )
1
Z
f (x)dq x = F (x)
(19)
And in the aid of (2), the left hand side of the equation can be expressed as a q-Bernoulli numbers, therefore
F (x) =
1 X
n
(Hq Dq ) n;q [n]q !
n=0
= =
Z
Z
Z
f (x)dq x
(20)
1 X h n;q f (x) + Dqn [2]q [n] ! q n=1
f (x)dq x
1 n X X h f (x) + [2]q n=1
f (x)dq x
1
f (x) Hqn (x)
n k
k=0
q
k(k 1) 2
(21) n k
(h + x)
k
( x)
q
!
Dqn
1
f (x) [n]q !
n;q
(22)
Therefore we can state the following q analogue of Euler-Maclaurian formula Theorem 6 If the function f (x) is capable of expansion as a convergent power series and f (x) decrease so rapidly with x such that all normal derivatives approach zero as x ! 1, then we can express the series of function f (x) as follow 1 X
f (n) =
Z
1
f (x)dq x +
a
n=a
lim
b!1
1 n X X
n=1
1 (f (a)) [2]q
n k
k=0
q
(23)
k(k 1) 2
n k
(1 + b)
k
( b)
n k
(1 + a)
f (n) =
n=a
Zb
( a)
q
!
hn Dqn
1
f (a) [n]q !
n;q
(24)
Proof. In the aid of theorem (4) and (18), If we suppose that h = 1 and b b 1 X
k
f (x)d1 x = F (b)
a 2 N; we have
F (a)
(25)
a
=
Z
b
1 (f (b) [2]q
f (x)dq x
a
+
1 n X X
n=1
k=0
n k
q
k(k 1) 2
f (a))
(26) n k
(1 + b)
k
( b)
q
n k
(1 + a)
k
( a)
!
Dqn
1
f (b)
Dqn [n]q !
1
f (a) (27)
5
n;q
f (x) x(q 1) ;
Let f (x) decrease so rapidly with x then Dq f (x) can be estimated by
tend x to in…nity and use
0
f (x) (q 1) :Now
L’Hopital to reach If normal derivatives of f (x) is tend to zero, then q derivatives is also tending zero. The same discussion make Dqn 1 f (b) = 0 for big enough b: Therefore 1 X
f (n) =
Z
1
f (x)dq x +
a
n=a
lim
b!1
1 n X X
n=1
1 (f (a)) [2]q
n k
k=0
q
(28)
k(k 1) 2
n k
k
(1 + b)
n k
( b)
(1 + a)
!
k
( a)
q
hn Dqn
1
f (a) [n]q !
n;q
(29)
Example 7 Let f (x) = xs where s is a positive integer, then we can apply this function at (20) where a = 0 and b 1 2 N b 1 X
bs+1 n = [s + 1]q n=0 s
+
s+1 X n X
n=2
k=0
bs [2]q n k
! q
(30) k(k 1) 2
n k
(1 + b)
k
( b)
q
!
bs
s n
1
n+1
n;q
(31)
[n]q
q
This relation shows the sum of power in the combination of q Bernoulli numbers. when q ! 1 from the right side, we have an ordinary form of this relation. For the another forms of sum of power, see [22] Example 8 Let f (x) = eq x = E1x then this function decreases so rapidly with x such that all normal q derivatives approach zero as x ! 1: For com…rming this, let us mention that Eqx ; for some …xed 0 < jqj < 1 1 1 P Q d and j xj < j1 1 qj is increasing rapidly, since dx Eqx = (1 q)q j 1 + (1 q)q k x > 0: j=1
1 X
eq
n
n=0
1 =1+ [2]q
lim
b!1
1 n X X
n=1
k=0
P1 Moreover, this is q-analogue of n=0 e numbers that is generated by e 11 1 :
3
n k n
q
k(k 1) 2
k=0 k6=j
n k
(1 + b)
k
( b)
q
=
1 e
1
1
=
3 2
+
P1
n=1 (2n)! ; 2n
!
n;q
(32)
[n]q !
where
2n
is a normal Bernoulli
approximation by q-Bernoulli polynomial
We know that the class of q-Bernoulli polynomials are not orthogonal polynomials. In addition, we may apply Gram-Schmit algorithm to make these polynomials orthonormal then according to Theorem 8.11 [19] we have the best approximation in the form of Fourier series. Now, Instead of using that algorithm, we apply properR1 ties of lemma (1) to achieve an approximation. Let H = L2q [0; 1] = fq : [0; 1] ! [0; 1] j fq2 (t) dq t < 1 0
be the Hilbert Space[18], then n;q (x) 2 H for n = 0; :::; N and Y = Span f 0;q (x); 1;q (x); :::; N;q (x)g is …nite dimensional vector subspace of H. Unique best approximation for any arbitrary elements of H like h; is b h 2 Y such that for any y 2 Y the inequality h b h kh yk2;q ; where the norm is de…ned 2;q
by kf k2;q :=
R1
1 2
fq2 (t) dq t
: Following proposition determine the coe¢ cient of q Bernoulli polynomials,
0
6
when we estimate any f 2 L2q [0; 1] by truncated q Bernoulli series. In fact, in order to see how well a certain partial sum approximates the actual value, we would like to drive a formula similar to (19), but with the in…nite sum on the right hand side replaced by the N th partial sum SN , plus an additional term RN: Sippose a 2 Z and b = a + 1:Consider the Nth partial sum SN =
N X
k=0
k;q
Let g(x; y) =
SN =
N X
N;q (x;y)
[N ]q !
[k]q !
N X
DqN
k
Hqk Dqk (f ) (a)
; then in the aid of lemma (1) we have Dqn
k;q (0; 0)
k=0
=
Hqk Dqk (f ) (a + 1)
[k]q !
Hqk Dqk
(f ) (a + 1)
k;q (0; 1)
[k]q !
g(0; 0)Hqk Dqk (f ) (a + 1)
DqN
k
Hqk Dqk
k
g(x; y) =
!
(f ) (a)
k;q (x;y)
[k]q !
and we have
Dq (f ) (a)
g(0; 1)Hqk Dqk (f ) (a)
Dq (f ) (a)
k=0
= Dq (f ) (a)
N X
DqN
k
g(0; x)Hqk Dqk (f ) (a + 1
x)jx=1 x=0
k=0
By taking q derivative of the summation, we may rewrite this partial sum by integral presentation, Moreover this relation give a boundary for RN (x): PN Proposition 9 Let f 2 L2q [0; 1] and be estimated by truncated q Bernoulli series n=0 Cn n;q (x): Then R1 (n) Cn coe¢ cients for n = 0; 1; :::; N can be calculated as an integral form Cn = [n]1 ! Dq f (x)dq x. In addition q
0
we can write f (x) in the following form
f (x) =
Z
N X
1
0
h @ 1 f (x) + [2]q [n]q ! n=1
f (x)dq x
0
Z1 0
1
Dq(n) f (x)dq xA
Moreover Rq;n (x) as a reminder part is bounded by
n;q (x)
+ Rq;n (x)
(n) 2n [N ]q ! Supx2[0;1] j n;q (x)jSupx2[0;1] jDq f (x)j
Proof. In fact we approximate f (x) as a linear combinations of q Bernoulli polynomials and we assume PN that f (x) ' n=0 Cn n;q (x); so take the Jackson integral from both sides lead us to Z
1
0
f (x)dq x '
N X
n=0
Cn
Z
1
n;q (x)dq x
0
= C0
Z
1
0;q (x)dq x
+ C1
0
Z
1
1;q (x)dq x
0
+ ::: + CN
Z
1
N;q (x)dq x
0
In the aid of Lemma 1 part (d) all the terms at the right side except the …rst one has to be zero. we R1 calculate 0;q (x) = 1; so C0 = [0]1 ! f (x)dq x:Using part (a) of that lemma for q derivative of q Bernoulli q
0
polynomial gather by taking q-derivative of f (x) leads to Z 1 Z 1 Z N X Dq (f (x)) dq x ' Cn [n]q ! (x)d x = C n 1;q q 1 0
n=0
0
1 0;q (x)dq x
= C1
0
Repeating this procedure n-times, yields the form of Cn as we mention it at theorem. We mention that if x 2 [0; 1] and 0 < jqj < 1, then Hqn (x) for h = 1 is bounded by 2n :In addition, the reminder part can be presented by integral forms and this boundary can be found easily.
7
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