General Mathematics Vol. 16, No. 1 (2008), 47-54
Some new properties in q-Calculus1
Daniel Florin Sofonea
Abstract
Our aim is to present new proofs of some results from q - Calculus.
These results occur in many applications as physics, quantum theory,
number theory, statistical mechanics, etc. Our proofs are based only
on interpolation theory ([4]).
2000 Mathematics Subject Classification: 05A30
Key words: q-calculus
1
Introduction
1. In the following, for q ∈ C\{1}, let us denote [n]q =
(1)
[n]q ! =
(2)
·
n
k
¸
q
½
=
1
[1]q [2]q . . . [n]q
if
if
q n −1
,
q−1
and for n ∈ N
n=0
,
n = 1, 2, . . . ,
[n]q !
for k ∈ {0, 1, . . . , n}.
[k]q ![n − k]q !
·
¸
n
The numbers
, 0 ≤ k ≤ n, are called Gaussian - coefficients.
k q
These coefficients satisfy the q - Pascal identities
1
Received 18 August 2007
Accepted for publication (in revised form) 6 December 2007
47
48
(3)
Daniel Florin Sofonea
·
n+1
k
¸
q
=
·
n
k
¸
k
q +
q
·
n
k−1
¸
q
=
·
n
k
¸
q
+
·
n
k−1
¸
q n+1−k
q
where k ∈ {1, 2, . . . , n}, 1 ≤ k ≤ n.
Let q be an arbitrary complex number, q 6= 1, and D = Dq ⊆ C such
that x ∈ D implies qx ∈ D.
Definition 1.1 Any function f : Dq → C is called q - derivable, under
restriction that if 0 ∈ Dq , there is f ′ (0).
Definition 1.2 A function f : Dq → C is q - derivable of order n, iff
0 ∈ Dq , implies that f (n) (0) exists.
For a function f : Dq → C which is q - derivable its q - derivative Dq f
was defined in 1908 by F.H. Jackson [3], in the following way
(4)
(Dq f )(x) =
f (x) − f (qx)
, q ∈ C\{1}.
(1 − q)x
For instance, Dq (xn ) = [n]q xn−1 , and the linear operator f → Dq f satisfies
(see [1])
(5)
(Dq f g)(x) = g(x)(Dq f )(x) + f (qx)(Dq g)(x).
µ
f
Dq
g
¶
(x) =
g(x)(Dq f )(x) − f (x)(Dq g)(x)
, g(x)g(qx) 6= 0.
g(x)g(qx)
In the following by [x0 , x1 , . . . , xn ; f ] we denote the divided differences
at a system of distinct points x0 , x1 , . . . , xn . More precisely
(6)
=
n
X
k=0
[x0 , x1 , . . . , xn ; f ] =
f (xk )
.
(xk − x0 ) · . . . · (xk − xk−1 )(xk − xk+1 ) · . . . · (xk − xn )
49
Some new properties in q-Calculus
2
Main theoretical results
We consider the points xk = q k x, k = 0, 1, . . . , n, ( x0 = x, x1 = qx, . . . , xn =
q n x ). In this case, we have
Theorem 2.1 Let q ∈ C\{1} and f : Dq → C. Then on the knots xk =
q k x, we have the representation
[x, qx, . . . , qn−1 x, ·qn x;¸f ] =
Pn
k(k−1)
n
q 2 f (q n−k x).
(−1)k
= n(n−1) 1
k=0
k q
q 2
[n]q !xn (q−1)n
(7)
Proof. For 0 ≤ j < k we have xk − xj = xq j (q k−j − 1). Therefore xk − xj =
xq j (q−1)[k−j]q , 0 ≤ j ≤ k−1. When k < j ≤ n xk −xj = xq k (1−q)[j −k]q .
Further
n
k−1
n
Y
Y
Y
(xk − xj ) =
(xk − xj ) ·
(xk − xj ) =
j=0
j=0
j6=k
= xn q
k(k−1)
2
j=k+1
(q − 1)k [k]q [k − 1]q · . . . · [1]q · q n−k (1 − q)k(n−k) [1]q · . . . · [n − k]q =
(1)
= (−1)k (1 − q)n xn q k(n−k)+
k(k−1)
2
[k]q ![n − k]q !.
Replacing the product in the equality (6) and using (2) we obtain
¸
n ·
X
f (xq k )
1
n
.
[x0 , x1 , . . . , xn ; f ] =
[n]q !xn (q − 1)n k=0 k q (−1)n−k q k(n−k)− k(k−1)
2
This completes the proof.
If Dq0 = I, Dq1 = Dq , Dqk = Dq Dqk−1 , a representation of operator Dqn is
given by
Theorem 2.2 Let f : Dq → C is q - derivable of order n, then
(8)
(Dqn f )(x)
n
−n −n −( 2 )
= (q − 1)
x
q
¸
n ·
X
k
n
(−1)k q (2) f (q n−k x).
k q
k=0
50
Daniel Florin Sofonea
Proof. Let us remark that
(Dq f )(x) = [x, qx; f ] = [1]q [x, qx; f ],
and
qf (x) − (q + 1)f (qx) + f (q 2 x)
(Dq f )(x) − (Dq f )(qx)
=
=
(1 − q)x
(1 − q)2 qx2
·
¸
(q + 1)
q
1
2
=
f (x) − f (qx) +
f (q x) =
q(1 − q)2 x2 q + 1
q+1
·
¸
f (x)
f (qx)
f (q 2 x)
= (q + 1) 2
−
+
=
x (1 − q)(1 − q 2 ) x2 q(1 − q)2 x2 q(1 − q)(1 − q 2 )
(Dq2 f )(x) =
= [2]q [x, qx, q 2 x; f ].
By induction, let us prove that (Dqn f )(x) = [n]q [x, qx, . . . , q n x; f ]. Assume
that the formula is proved for n = m. Then it is also true for n = m + 1,
because
by (4)
(Dqm+1 f )(x) = (Dq (Dqn f ))(x) =
=
(Dqm f )(x) − (Dqm f )(qx)
=
(1 − q)x
[m]q ![x, qx, . . . , q m x; f ] − [m]q ![qx, q 2 x, . . . , q m+1 x; f ]
=
(1 − q)x
q m+1 x − x
[x, qx, . . . , q m+1 x; f ] = [m + 1]q ![x, qx, . . . , q m+1 x; f ],
(1 − q)x
where we used formula
= [m]q !
[x0 , x1 , . . . , xn ; ·] =
[x0 , . . . , xn−1 ; ·] − [x1 , . . . , xn ; ·]
.
xn − x0
On the other hand
[x, qx, . . . , q n−1 x, q·n x; f¸] =
Pn
k(k−1)
n
q 2 f (q n−k x).
(−1)k
= n(n−1) 1
k=0
k q
[n]q !xn (q−1)n
q 2
Therefore
(9)
(Dqn )(x) = [n]q [x, qx,·. . . ,¸q n x; f ] =
Pn
k
n
= n 1
(−1)k
q (2) f (q n−k x).
k=0
k q
q ( 2 ) xn (q−1)n
51
Some new properties in q-Calculus
Other proof may be performed in the following way. For n = 1 we find
formula (4), that is the definition of q - difference operator. We presume
that formula (8) is true for n = m. Then it is true for n = m + 1 too,
because
!
Ã
¸
m ·
k
m X
by (5)
m
=
(−1)k q (2) f (q m−k x)
(Dqm+1 f )(x) = Dq (q − 1)−m x−m q −( 2 )
k q
k=0
Ã
¸
m ·
k
q m − 1 −m −m−1 X m
(−1)k q (2) f (q m−k x)+
q x
−
= (q − 1) q
k q
q−1
k=0
!
¸
m ·
X
k
by (3)
m
+
1
=
(−1)k q (2) f (q m+1−k x)
+q −m x−m
k
q
−m −(
)
m
2
k=0
−m−1 −m−1 −(m
2)
= (q − 1)
x
q
m+1
X·
k=0
m+1
k
¸
(−1)k q (2) f (q m+1−k x).
k
q
Theorem 2.3 Let q ∈ C\{1} and k be fixed in {2, 3, . . . , p}, If q k = 1 and
f : Dq → C is q - derivable of order p, then for all x, (Dqp f )(x) = 0.
Proof. From (9) we have
(Dqp f )(x) = [p]q ![x0 , x1 , . . . , xp ; f ],
where xj = q j x, j = 0, 1, . . . , p. According to our hypothesis we have q k = 1
for k fixed in {2, 3, . . . , p}. Because
[p]q ! =
(1 − q)(1 − q 2 ) · . . . · (1 − q k ) · . . . · (1 − q p )
(1 − q)p
we have [p]q ! = 0, and the proposition is proved.
Theorem 2.4 Let f : Dq → C is q - derivable of order n. Then we have
the following representation
(10)
n
f (q x) =
n
X
k=0
k
(q − 1) x q (2)
k k
·
n
k
¸
q
(Dqk f )(x).
52
Daniel Florin Sofonea
Proof. Starting from Newton’s formula we have:
f (z) =
n−1
X
(z − x0 )(z − x1 ) . . . (z − xk−1 )[x0 , x1 , . . . , xk ; f ]+
k=0
+(z − x0 )(z − x1 ) . . . (z − xn−1 )[x0 , x1 , . . . , xn−1 , z; f ].
Replacing xk = q k x, k = 0, 1, . . . , n − 1 in the above formula and z = q n x
and using formula (7) we obtain succesively
f (q n x) =
n−1
X
(q n x − x)(q n x − qx) · . . . · (q n x − q k−1 )
k=0
n
n
n
+(q x − x)(q x − qx) · . . . · (q x − q
n−1
(Dqk f )(x)
+
(−1)k [k]q !
(Dqn f )(x)
=
)
(−1)n [n]q !
n
X
(Dqk f )(x)
(q n x − x)(q n x − qx) · . . . · (q n x − q k−1 )
=
=
k [k] !
(−1)
q
k=0
(q n − 1) · . . . · (q − 1)
k
(1 − q)n
xk q ( 2 )
=
(Dqk f )(x) =
n−k
(q
−
1)
·
.
.
.
·
(q
−
1)
k=0
(−1)n−k
[k]q !
(1 − q)n−k (1 − q)k
(−1)n
n
X
=
n
X
(q − 1)k xk q (2)
k
k=0
[n]q !
(Dk f )(x).
[n − q]q ![k]q ! q
Corollary 2.5 Let q ∈ C\{1} and f : Dq → C is q - derivable of order n.
Then the following identity holds
· ¸
n
X
n ¡ k ¢ n−k
k k (k+1
(1 − q) x q 2 )
(11)
f (x) =
Dq f (q x)
k q
k=0
Proof. We start with following equalities (k ∈ {0, 1, . . . , n}):
1
qx − 1
= q 1−x [x]q ,
[x] 1 = 1
q
−1
q
53
Some new properties in q-Calculus
[n] 1 ! = [1] 1 . . . [n] 1 =
q
q
q
= q 1−1 [1]q q 1−2 [2]q . . . q 1−n [n]q = q n−(1+2+...+n) [n]q ! = q −( 2 ) [n]q !,
n
· ¸
· ¸
[n] 1 !
q −( 2 )
n
n
q
,
=
=
n−k
k
−
−
k 1
[k] 1 ![n − k] 1 !
q (2) q ( 2 ) k q
n
q
q
q
On the other hand
µ ¶
x
µ ¶
f
− f (x)
x
q
(D 1 f )(x) =
q = q(Dq f )
,
q
(1 − q)x
q
µ ¶¸
·
µ ¶
x
x
µ ¶
− q(Dq f ) 2
q q(Dq f )
x
q
q
2
2
2
= q (Dq f ) 2 .
(D 1 f )(x) =
q
(q − 1)x
q
By induction, let us prove that
n
(12)
(D 1 f )(x) = q
q
n
(Dqn f )
µ
x
qn
¶
.
Assume that the formula is proved for n = m. Then
µ ¶
x
(D 1 f )(x) − (D 1 f )
q
q
q
µ
¶
f )(x) = (D 1 (Dm
(Dm+1
=
1 f ))(x) =
1
q
q
1
q
1−
x
q
·
µ ¶
µ
¶¸
x
x
q
m
m
m
m
q (Dq f ) m − q (Dq f ) m+1
=
=
(q − 1)x
q
q
µ
¶
x
m+1
m+1
=q
(Dq f ) m+1 .
q
m
In (10) replacing q →
f
µ
x
qn
¶
=
1
q
n µ
X
1
k=0
m
· ¸
¶k
q
1
n
(Dk1 f )(x)
−1
n−k
k
k
−( 2 ) −( 2 )
k
q
(
)
q
2
q
q q
q
54
Daniel Florin Sofonea
if in the above formula x à q n x we obtain
f (x) =
n
X
(1 − q)k
qk
k=0
=
n
X
q x q
k
(1 − q) x q (2)
k k
k=0
=
n
X
·
2
nk k −kn+ k2 + k2
·
n
k
k
(1 − q) x q (2) q k
k k
k=0
·
n
k
¸ ³
´
Dk1 f (q n x) =
q
q
¸ ³
´
by (12)
Dk1 f (q n x) =
q
q
n
k
¸
q
¡
¢
Dqk f (q n−k x).
Therefore we have obtained representation (11).
References
[1] Exton H., q-hypergeometric functions and applications. Ellis. Horwood,
1983.
[2] Gasper G. & Rahman M., Basic hypergeometric functions series. Cambridge (1990).
[3] Jackson F.H., On q-functions and certain difference operator. Trans.
Roy Soc. Edin. 46. (1908), 253-281.
[4] Lupaş A., q-Analogues of Stancu operators., Mathematical Analysis
and Aproximation Theory, Burg Verlag, 2002, 145-154.
[5] Sofonea F., Numerical Analysis and q - Calculus (I), Octogon, Vol. 11,
No.1, April, 2003, 151-156.
[6] Sofonea F., On a q - Analogue of A. Lupaş Operators, Mathematical
Analysis and Approximation Theory, RoGer2004, Mediamira, ISBN
973-713-078-2, 2005, 211-223.
University Lucian Blaga of Sibiu
Department of Mathematics
5-7 Ratiu str, Sibiu, Romania
E-mail: florin.sofonea@ulbsibiu.ro