Corrections to the book "Quantum Calculus" by Victor Kac and Pokman Cheung
- page 9, lines 3 and 4 from bottom. Should be $n'$ instead of $n$
- page 10, line 11 from bottom. Should be so the formula is true
- page 11, line 2. Should be $(a-q^2x)$ instead of $(a-q^2a)$
- page 11, line 3. Should be $(q^{-2}a-x)$ instead of $(q^{-2}-x)$
- page 11. In the formula above (3.10) instead of $(a-q^{n+1}x)$ should be $(a-q^{n}x)$
- page 12. In the example should be $[n] D^{j-1}_q x^{n-1} = [n] [n-1]D^{j-2}_q x^{n-2}$
- page 19, line 6 from bottom. Should be where $m\geq 2$. Consider
- page 20, lines 8,9. Should be The last line follows from one of the $q$-Pascal rules (6.3). So (6.5) is true for $0\leq j \leq m$.
- page 24, last paragraph. The proof is incorrect, replace it by the following argument: A linear transformation $T$ of rank $j$ is specified by the subspace $K$ of $A$, mapped by $T$ to 0, and by a collection of $j$ linearly independent vectors $b_1$,..., $b_j$ in $B$ such that $T(a_i)=b_j$ for a fixed collection of vectors $a_1$,...,$a_j$, which together with a basis of $K$ form a basis of $A$. This proves the desired formula for the $j$th summand.
- page 27: The first sentence of Theorem 8.1 should be: Suppose $D$ is a linear operator on the space of formal power series and $P_i=a_ix^i, i=0,1,2,...$ is a sequence, such that all the $a_i$ are non-zero numbers and $D(P_i)=P_{i-1}$ for $i\geq 1$.
- page 28, lines 5-7: The beginning of the first sentence of the proof should be: We have
- page 28, lines 5-7.The beginning of the proof of Theorem 8.1 should read: It is easy to see that for any formal power series $f(x)$, we have
- page 37, line 2: should be if we replace $q$ by $q^{3/2}$ and then put $z=$
- page 37, last line: should be $q^{e_n}$th
- page 38, line 11: should be all partitions of $n$
- page 42, in the second line of formulas the symbol $\sum_{m|n}$ should be deleted (twice)
- page 45, line 10 from the bottom: should be $(1-q^n)$ instead of $(1-q^{n-1})$
- page 52, line 11. Instead of $ n \to -\infty$ should be $n \to \infty$
- page 62, line 7 from bottom. Should be $\infty$ line 5 from bottom. Should be parentheses around the two summations
- page 63, line 2. Should be $k,l\geq 1$ in (17.4)
- page 65. A simple proof of Proposition 18.1: Since $\varphi(x)=\varphi(qx)$, we have $\varphi(x)=\varphi(q^nx)$ for any positive integer $n$. Tending $n$ to infinity, we obtain $\varphi(x)=\varphi(0)$ for any $x$.
- page 66, lines 7,8. Should be: This formula means that $F(u(x))$ is a $q^{1/\beta}$-antiderivative of $f(u(x)) D_{q^{1/\beta}} u(x)$.
- page 74. Corollary 20.1 and its proof should be replaced by the following:
Corollary 20.1. If $f(x)$ is continuous at $x=0$ , we have for $a,b \in [0,A]$:
$\hspace{25ex} \int^b_a D_q f(x) d_qx = f (b) - f(a) \hspace{15ex} $ (20.2)
Proof. Apply Theorem 20.1 to the function $D_q f(x)$.
- page 74. Replace lines 7-9 from the bottom by:
Now suppose $f(x)$ and $g(x)$ are two functions, which are continuos at $x=0$, and assume that $x^\alpha D_q f(x)$ is bounded on the interval $[0,b]$. Using the product rule (1.12), we have
Further, replace lines 4 and 5 from the bottom by:
We can apply Corollary 20.1 and Theorem 19.1 to obtain
- page 76. There are divergence problems with the definition (21.6) of the q-gamma function because the function $E_q^{-x}$, surprisingly, blows up along some sequences as x tends to infinity. (It is because the radius of convergence of the series (9.7) for $e_q^{x}$ is $1/(1-q)$, not infinity.) However, if we replace the upper limit of the integral (21.6) by $1/(1-q)$, this difficulty is removed, but all arguments on page 77 still hold with little modifications, given below.
- page 76, line 4 from the bottom. Add: Then we have: $[\infty]=1/(1-q)$.
The upper limit of the integral in the definition (21.6) of the function $\Gamma_q (t)$ should be $[\infty]$ instead of $\infty$
- page 77. The first sentence should read: First we note that by (9.10),
$E_q^0 = 1$ and $E_q^{-[\infty]}=0$.
The upper limit of the integral in lines 3 and 7 should be $[\infty]$ instead of $\infty$.
The sentence after the definition of the q-beta function should read:
By the definition of the $q$-integral (19.7), we have
In the line that follows the letter a should be removed, the next line should be removed, and in line 9 from the bottom the upper limit of the integral should be $1$ instead of $\infty$. The line after that should be removed.
The upper limit of the integral in line 5 from the bottom should be 1 instead of $\infty$.
In line 4 from the bottom should be (19.14) instead of (19.15).
In line 3 from the bottom the upper limit should be $[\infty]$ instead of $\infty$.
- page 79. Instead of the sentence Then both sides are formal power series in $q$. should be
Then both sides are formal power series in two variables $q$ and $v=q^t$.
- page 80. At the end of the first paragraph add:
We shall assume that $h>0$.
- page 81. In Example replace $(x+b)^N$ by $(x+b)_h^N$ (twice).
- page 82, line 2. One $)$ should be removed.
- page 83. In line 6, instead of $D_h(F(x)g(x))$ should be $D_h(f(x)g(x))$
In line 8 from the bottom, instead of $a$ < $b$ should be $0 \leq a$ < $b$
In the subsequent definition of $f(x)$ add that $f(0)=0$
- page 84. In line 13 after $h>0$ write and $x>a$. By (22.17),
In formula (22.19) replace $\frac{}{1n!}|x-a|^{n+1}$ by $\frac{1}{(n+1)!}|(x-a)_h^{n+1}|$
- page 103. It is not true in general that the polynomials $P_n (x)$ have the form (26.20). Therefore one has to use the following generalization of Theorem 2.1.
Theorem 26.2 Let $a,q$ be some numbers, $D$ be a linear operator on the space of polynomials, and $\{ P_0(x) ,P_1 (x), \ldots \}$ be a sequence of polynomials, satisfying three conditions:
- $P_0(a) =1$, $P_n (a)=0$ if $n$ is odd, and $P_n (qa) = P_n (q^{-1}a) =0$ if $n$ is positive even;
- deg $P_n (x) =n$;
- $D P_n (x) = P_{n-1}(x)$ for any $n \geq 1$ and $D(1) = 0$.
Then for any polynomial $f(x)$ one has:
$f(x) = \displaystyle\sum_{n\geq 0 even} (D^nf) (a) (q^{-n}P_n (qx)) + \displaystyle\sum_{n>0 odd} (D^n f) (q^{-1}a) P_n (x) \,.$
The proof of this theorem is the same as that of Theorem 2.1. However, unlike Theorem 2.1, Theorem 26.2 can be applied to the operator $D=\tilde{D}_q$ and the polynomials $P_n (x) = (x-a)^n_{\tilde{q}}/[n]^{\sim} !$.
Moreover, using the same argument as that in the proof of Theorem 20.2, one can derive a similar $q$-analogue of Taylor's formula with the Cauchy remainder in the symmetric $q$-calculus.
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