Daehee Formula Associated with the q-Extensions of Trigonometric Functions

0.12 MB·English

by  T. Kim

Daehee Formula Associated with the
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q-Extensions of Trigonometric Functions
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Taekyun Kim, EECS, Kyungpook National University,
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J Taegu, 702-701. South Korea

3 e-mail: [email protected]
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] Soonchul Park, EECS, Kyungpook National University,
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Taegu, 702-701. South Korea
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e-mail:[email protected]
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t
a Seog-Hoon Rim, Department of Mathematics Education,
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Kyungpook National University, Taegu, 702-701, S. Korea
[ e-mail: [email protected]

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v February 2, 2008
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Abstract
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InthispaperweintroduceaDaeheeconstantwhichiscalledq-extension
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of Napier constant, and consider Daehee formula associated with the q-
/
h extensionsoftrigonometricfunctions. Thatis,wederivetheq-extensions
t of sine and cosine functions from this Daehee formula. Finally, we give
a
m the q-calculus related to theq-extensions of sine and cosine functions.
: Key words. q-series, q-Euler formula, q-trigonometric function
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i AMS(MOS) subject classifications. 11S80,11B98
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r
a
1 Introduction
When one talks of q-extension, q is variously considered as an indeterminate, a
complex number q C, or a p-adic number q C . The q-basic numbers are
p
∈ ∈
defined [1, 3, 4, 6, 7] by
1 qn
[n]q = − =1+q+q2+ +qn−1,
1 q ···
−
∗This work was partially supported by Jangjeon Mathematical Society(JMS2006-12-
C0007).
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and the q-factorials by [n] ! = [n] [n 1] [1] . Newton’s binomial formula
q q q q
− ···
says:
n
(x+y)n = n yn−kxk, n Z , (1)
k ∈ +
k=0(cid:18) (cid:19)
X
where
n n(n 1) (n k+1) n!
= − ··· − = , (2)
k k! k!(n k)!
(cid:18) (cid:19) −
which is called binomial coefficient.
Throughout this paper, let us assume that q C with q <1. Here we use
∈ | |
the q-deformed binomial coefficient (or Gaussian binomial coefficient) which is
also defined by
n (q :q) [n] !
= n = q , (3)
(cid:18) k (cid:19)q (q :q)k(q :q)n−k [n−k]q![k]q!
while the q-shifted factorial is given by
(a:q) =(1 a)(1 aq) (1 aqk−1), a C, k Z .
k +
− − ··· − ∈ ∈
The recurrencerelationsbelow showthatthe q-binomialcoefficientisa polyno-
mial in q.
x x 1 x 1
=qk − + − .
k k k 1
(cid:18) (cid:19)q (cid:18) (cid:19)q (cid:18) − (cid:19)q
Notethattheq 1yieldstheconventionalnumber[n] =nandtherefore,
q=1
→
the conventionalbinomial coefficient (see [3, 4, 6]);
n n
= .
k k
(cid:18) (cid:19)q=1 (cid:18) (cid:19)
It is well known that
x
1 1
lim 1+ = lim(1+x)x =e, (4)
x→∞ x x→0
(cid:18) (cid:19)
which is called Napier constant (or Euler number). Let i be a complex that is
1
defined by i=√ 1=( 1)2. Then Euler formula is given by
− −
eix =cosx+isinx, (see [5, 2]). (5)
From (5), we derive
1 1
cosx= (eix+e−ix), and sinx= (eix e−ix). (6)
2 2i −
By using Taylor expansion, we easily see ([5]) that
∞ ∞
( 1)n ( 1)n
sinx= − x2n+1, and cosx= − x2n. (7)
(2n+1)! (2n)!
n=0 n=0
X X
2
In the recent paper, M. Schork has studied Ward’s “Calculus of Sequences”
and introduced q-addition x y in [6] by
q
⊕
n
n
(x y)n = xkyn−k. (8)
⊕q k
k=0(cid:18) (cid:19)q
X
This q-addition was already known to Jackson and was generalized later on by
Ward and AI-Salam [6]. The two q-exponentials are defined in [4] by
∞ ∞
zn ((1 q)z)n 1
eq(z)= = − = , (9)
[n]q! (q :q)n (z(1 q):q)∞
n=0 n=0 −
X X
∞ q( n2 )zn ∞ ((1 q)z)nq( n2 )
Eq(z)= = − =( z(1 q):q)∞, (10)
[n] ! (q :q) − −
q n
n=0 n=0
X X
where (a : q)∞ = limk→∞(a : q)k = Π∞i=1(1−aqi−1) (see [4, 6, 2]). From (9)
and (10), we note that e (z) E ( z)=1 for z <1.
q q
· − | |
ThepurposeofthispaperistoconstructDaeheeformulaandtheq-extensions
ofsineandcosine. Fromtheseq-extensions,wederivesomeinterestingformulae
related to Daehee formula and the q-extension of sine and cosine functions.
2 q-extension of Euler formula and trigonomet-
ric functions
Let us consider the Jacksonq-derivative D by
q
f(x) f(qx)
Dqf(x)= − , see [3, 4, 6]. (11)
(1 q)x
−
It satisfies D xn = [n] xn−1 and reduces in the limit q 1 to the ordinary
q q
→
derivative. From the definition of q-exponential function, we derive
D (e (λx))=λe (λx),
q q q
since
∞ λnxn ∞ λnxn−1 ∞ λn+1xn
D = = =λe (λx), see [6].
q q
[n] ! [n 1] ! [n] !
q ! q q
n=0 n=1 − n=0
X X X
The q-integral was defined in [3] by
x ∞
f(t)d t=(1 q) f(qkx)qkx, (12)
q
−
Z0 k=0
X
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where x R, and the right-hand side converges absolutely. In particular, if we
∈
take f(x)=xn, then we have
x ∞ xn+1
tnd t=(1 q) q(n+1)kxn+1 = , see [3].
q
− [n+1]
Z0 k=0 q
X
From (11) and (12), we derive
x ∞
D f(t)d t= (f(qkx) f(qk+1x))=f(x), cf. [3].
q q
−
Z0 k=0
X
Therefore we have the following:
Lemma 1. Let f be a q-integrable function. Then we have
x
D f(t)d t=f(x).
q q
Z0
By the definition of q-integral, we easily see that
D (f(x)g(x))=f(x)D g(x)+g(qx)D f(x),
q q q
x x
f(t)D (g(t))d t=f(x)g(x) g(qt)D (f(t))d t.
q q q q
−
Z0 Z0
Let us consider the q-extensionof Napier constant (or Euler number) which
is called Daehee constaint:
x ∞ k
1 x 1
lim 1 = lim
x→∞(cid:18) ⊕q [x]q(cid:19) x→∞k=0(cid:18) k (cid:19)q(cid:18)[x]q(cid:19)
X
∞ (1 qx) (1 qx−k+1)
= lim − ··· −
x→∞ [k]q!(1 qx)k
k=0 −
X
∞
1
= =e . (13)
q
[k] !
q
k=0
X
Proposition 2. ( Daehee constant)
x
1
e = lim 1 .
q q
x→∞(cid:18) ⊕ [x]q(cid:19)
From the definition of e (x), we consider
q
∞ ∞ ∞
(ix)n ( 1)nx2n ( 1)nx2n+1
eq(ix)= = − +i − (14)
[n] ! [2n] ! [2n+1] !
q q q
n=0 n=0 n=0
X X X
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In the viewpoint of (7), we define the q-extension of sine and cosine function as
follows:
∞ ∞
( 1)nx2n+1 ( 1)nx2n
sinqx= − , and cosqx= − (15)
[2n+1] ! [2n] !
q q
n=0 n=0
X X
From(14)and(15), wederiveDaehee formulawhichis theq-extensionofEuler
formula:
e (ix)=cos x+isin x.
q q q
Note that
lime (ix)=cosx+isinx.
q
q→1
Therefore we obtain the following:
Theorem 3. (Daehee formula)
e (ix)=cos x+isin x.
q q q
By the definition of q-addition, we easily see that
∞ xk ∞ yl ∞ n xkyn−l
e (x)e (y) = =
q q
[k] ! [l] ! [k] ![n k] !
q ! q ! q q !
k=0 l=0 n=0 k=0 −
X X X X
∞ n ∞ n
[n] ! 1 n 1
= q xkyn−l = xkyn−l
[k] ![n k] ! [n] ! k [n] !
n=0 k=0 q − q ! q n=0 k=0(cid:18) (cid:19)q ! q
X X X X
∞
(x y)n
= ⊕q =eq(x qy).
[n] ! ⊕
q
k=0
X
Therefore we obtain the following:
Proposition 4. For x,y R, we have
∈
e (x)e (y)=e (x y).
q q q q
⊕
From proposition 4, we can derive the following:
e (ix)e (iy)=e (i(x y)).
q q q q
⊕
By using Daehee formula, we see that
(cos x+isin x)(cos y+isin y)=cos (x y)+isin (x y).
q q q q q q q q
⊕ ⊕
Thus, we have
(cos xcos y sin xsin y)+i(sin xcos y+cos xsin y)
q q q q q q q q
−
=cos (x y)+isin (x y).
q q q q
⊕ ⊕
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By comparing the coefficients on both sides, we obtain the following:
Theorem 5. For x,y R, q C with q <1, we have
∈ ∈ | |
cos (x y)=cos xcos y sin xsin y,
q q q q q q
⊕ −
sin (x y)=sin xcos y+cos xsin y.
q q q q q q
⊕
By the same motivation of (8), we can also define
n
(x y)n = n ( 1)n−kxkyn−k. (16)
⊖q k −
k=0(cid:18) (cid:19)q
X
By (15), we easily see that
sin ( x)= sin (x) and cos ( x)=cos (x). (17)
q q q q
− − −
From Theorem 5, (16) and (17), we note that
cos2x+sin2x = cos (x(1 1)),
q q q ⊖q
cos (x(1 1)) = cos2x sin2x,
q ⊕q q − q
sin (x(1 1)) = 2sin xcos x.
q q q q
⊕
Therefore we obtain the following:
Corollary 6. For x,y R, we have
∈
cos2x+sin2x = cos (x(1 1)),
q q q ⊖q
cos2x sin2x = cos ((1 1)x),
q − q q ⊕q
2sin xcos x = sin (x(1 1)).
q q q q
⊕
By using Daehee formula, we easily see that
e (ix)=cos x+isin x,
q q q
e ( ix)=cos x isin x.
q q q
− −
Thus, we have the following:
Corollary 7. For x,y R, we have
∈
e (ix)+e ( ix) e (ix) e ( ix)
cosqx= q q − , and sinqx= q − q −
2 2i
We now define q-extension of tanx, secx, cscx and cotx as follows:
sin x 1 1 cos x
tan x= q , sec x= , csc x= , and cot x= q .
q q q q
cos x cos x sin x sin x
q q q q
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Note that
1+tan2x = sin2qx+cos2qx = cos (x(1 1))sec2x,
q cos2qx q ⊖q q (18)
1+cot2x = cos2qx+sin2qx = cos (x(1 1))csc2x.
q sin2qx q ⊖q q
From (11), we can derive
f(x) 1
D = [D (f)g(qx) f(qx)D (g)]. (19)
q q q
g(x) g(x)g(qx) −
(cid:18) (cid:19)
By (11), (15) and (19), we easily see that
D (sin x)=cos x, and D (cos x)= sin x.
q q q q q q
−
Moreover,
D (tan x)=1+tan xtan (qx).
q q q q
Therefore we obtain the following.
Theorem 8. For x R, we have
∈
D (sin x)=cos x, and D (cos x)= sin x.
q q q q q q
−
Moreover,
D (tan x)=1+tan xtan (qx).
q q q q
By Lemma 1 and Theorem 8, we have the following:
Corollary 9. For x R, we have
∈
x x
sin td t= cos x, and cos td t= sin x.
q q q q q q
− −
Z0 Z0
Finally,
x
(1+tan ttan (qt))d t=tan x.
q q q q
Z0
References
[1] T. Kim, Power series and asymtotic series associated with the q-analog
of the two-variable p-adic L-function, Russ. J. Math. Phys., 12(2005), pp.
186-196.
[2] T. Kim, An invariant p-adic integral associated with Daehee numbers, In-
tegral Transforms Spec. Funct., 13(2002), pp. 65-69.
[3] T. Kim, S. H. Rim, A note on the q-integral and q-series, Advan. Stud.
Contemp. Math., 2(2000), pp. 37-45.
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[4] T. H. Koornwinder, Special Functions and q-Commuting Variables,
Field Institute Communications, vol 14(1997), pp. 131-166.
[5] E. Kreyszig, Advanced Engineering Mathematics, 8thEd.,JohnWiley &
sons, 1999.
[6] M.Schork,Wards“CalculusofSequences”q-calculusandthelimitq 1,
→
Advan. Stud. Contemp. Math., 13(2006), pp. 131-141.
[7] Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-
series, J. Math. Anal. Appl., 318(2006), pp. 333-351.
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