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 ! à ¸ 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 à ¸ 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] LupasĖ§ 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. LupasĖ§ 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