# Eikonal Blog

## 2010.10.20

### “Quantum” calculus

Currently all this material is retyped from the reference [1].

Definitions:

• The q-analogue of $n$: $[n]:\equiv \frac{q^n-1}{q-1}$;
• The q-analogue of factorial $n!$: $[n]! :\equiv \prod_{k=1}^{n}[k]$ for $k\in{\Bbb N}$ (and $[0]!=1$);
• $(x-a)^{n}_{q} :\equiv \prod_{k=0}^{n-1} (x-q^k a)$ for $n\in{\Bbb N}$.
• $(x-a)^{-n}_{q} :\equiv \frac1{(x- q^{-n}a)^{n}_{q}}$.

Properties:

• $[-n] = - q^{-n} [n]$.
• $(x-a)^{m+n}_{q} = (x-a)^{m}_{q} (x- q^m a)^{n}_{q}$,
• $(a-x)^{n}_{q} = (-)^n q^{n(n-1)/2} (x - q^{-n+1}a)^{n}_{q}$,

## the “quantum” differentials

Definitions: For an arbitrary function $f:{\Bbb R}\rightarrow {\Bbb R}$ define:

• its q-differential: $d_q f(x) :\equiv f(q x) - f(x)$;
• its h-differential: $d_h f(x) :\equiv f(x+h) - f(x)$;
• its q-derivative: $D_q f(x) :\equiv \frac{d_q f(x)}{d_q x} = \frac{f(q x) - f(x)}{(q-1)x}$;
• its h-derivative: $D_h f(x) :\equiv \frac{d_h f(x)}{d_h x} = \frac{f(x+h) - f(x)}{h}$;

Note that $d_q x = (q-1) x$ and $d_h x = h$.

Basic properties:

• all four operators ($d_q$, $d_h$, $D_q$ and $D_h$) are linear, e.g. $d_q (\alpha f(x) + \beta g(x)) = \alpha d_q f(x) + \beta d_q g(x)$.
• $d_q (f(x)g(x)) = (d_q f(x)) g(x) + f(qx) (d_q g(x))$;
• $d_h (f(x)g(x)) = (d_h f(x)) g(x) + f(x+h) (d_h g(x))$;
• $D_q (f(x)g(x)) = (D_q f(x)) g(x) + f(qx) (D_q g(x))$;
• $D_q \left(\frac{f(x)}{g(x)}\right) = \frac{D_q f(x) \ g(x) - f(x) \ D_q g(x)}{g(x)g(qx)} = \frac{D_q f(x) \ g(q x) - f(q x) \ D_q g(x)}{g(x)g(qx)}$;
• there does not exist a general chain rule for q-derivatives
• such rule exists for monomial changes of variables $x\rightarrow x' = \alpha x^{\beta}$, where $D_q f(x'(x)) = (D_{q^\beta}f)(x') \cdot D_q x'(x)$.

Examples and properties:

• $D_q x^n = [n] x^{n-1}$.
• $(D^n_q f)(0) = \frac{f^{(n)}(0)}{n!} [n]!$.
• $D_q f(x) = \sum_{n=0}^{\infty}\frac{(q-1)^n}{(n+1)!} x^n f^{(n+1)}(x)$.
• $P_n(x) :\equiv \frac{x^n}{[n]!}$ satisfies $D_q P_n(x) = P_{n-1}(x)$.
• $D_q (x-a)^{n}_{q} = [n] (x-a)^{n-1}_{q}$,
• $D_q (a-x)^{n}_{q} = - [n] (a- q x)^{n-1}_{q}$,
• $D_q \frac1{(x-a)^{n}_{q}} = [-n] (x-q^n a)^{-n-1}_{q}$,
• $D_q \frac1{(a-x)^{n}_{q}} = \frac{[n]}{(a-x)^{n+1}_{q}}$,

## q-binomial calculus

Definition: The q-binomial coefficient is defined by $\left[{n \atop j}\right] :\equiv \frac{[n]!}{[j]![n-j]!}$.

Properties of q-binomial coefficients:

• $\left[{n \atop n-j}\right] = \left[{n \atop j}\right]$.
• there exist two q-Pascal rules: $\left[{n \atop j}\right] = \left[{n-1 \atop j-1}\right] + q^j \left[{n-1 \atop j}\right]$ and $\left[{n \atop j}\right] = q ^{n-j}\left[{n-1 \atop j-1}\right] + \left[{n-1 \atop j}\right]$.
• $\left[{n \atop 0}\right] = \left[{n \atop n}\right] = 1$.
• $\left[{n \atop j}\right]$ is a polynomial in $q$ of degree $j(n-j)$ with the leading coefficient equal to $1$.
• $\left[{\alpha \atop j}\right] = \frac{[\alpha] [\alpha-1] \cdots [\alpha -j +1]}{[j]!}$ for any number $\alpha$.
• $\left[{m+n \atop k}\right] = \sum_{j=0}^{k} q^{(k-j)(m-j)} \left[{m \atop j}\right] \left[{n \atop k-j}\right]$.
• $x^n = \sum_{j=0}^{n} \left[{n \atop j}\right] (x-1)^j_q$.
• $\sum_{j=0}^{2m} (-)^j \left[{2m \atop j}\right] = (1-q^{2m-1})(1-q^{2m-3})\cdots(1-q)$.
• $\sum_{j=0}^{2m+1} (-)^j \left[{2m+1 \atop j}\right] = 0$.
• The Gauss’s binomial formula: $(x+a)^n_q = \sum_{j=0}^n \left[{n \atop j}\right] q^{j(j-1)/2} a^{j} x^{n-j}$.
• For two non-commutative operators $\hat{A}$ and $\hat{B}$ s/t $\hat{B}\hat{A} = q\hat{A}\hat{B}$ (with $q$ and ordinary number), the non-commutative Gauss’s binomial formula is: $(\hat{A}+\hat{B})^n = \sum_{j=0}^n \left[{n \atop j}\right] \hat{A}^{j} \hat{B}^{n-j}$. Such two operators are $\hat{x}$ and $\hat{M_q}$ defined as $\hat{x} f(x) :\equiv x f(x)$ and $\hat{M_q} f(x) :\equiv f(qx)$.
• The Heine’s binomial formula: $\frac1{(1-x)^n_q} = 1+ \sum_{j=1}^{\infty} \frac{[n][n+1]\cdots[n+j-1]}{[j]!} x^j$.
• $\frac1{(1-x)^{\infty}_q} = \sum_{j=0}^{\infty} \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}$.
• $(1+x)^{\infty}_q = \sum_{j=0}^{\infty} q^{j(j-1)/2} \frac{x^j}{(1-q)(1-q^2)\cdots(1-q^j)}$.

## Generalized Taylor’s formula for polynomials

For given number $a$ and linear operator $D$ on space of polynomials, there exist a unique sequence of polynomials $\{P_0(x), P_1(x), \cdots\}$ such that

• $P_0(a)=1$ and $P_{n>0}(a)=0$;
• $\hbox{deg}P_n = n$;
• $D P_n(x) = P_{n-1}(x)$ ($\forall n\ge 1$) and $D(1)=0$.

Then any polynomial $f(x)$ of degree $n$ has the unique expansion via following generalized Taylor expression: $f(x) = \sum_{j=0}^n (D^j f)(a) P_j(x)$.

Examples:

• If $D$ is $D_q$ we have: $f(x) = \sum_{j=0}^n (D_q^j f)(a) \frac{(x-a)^j_q}{[j]!}$.
• for $f(x)=x^n$ and $a=1$ one gets: $x^n = \sum_{j=0}^n \left[{n \atop j}\right] (x-1)^j_q$.

## Exponentials and trigonometric functions

Definitions: (q-exponentials):

• $e_q^x :\equiv \sum_{k=0}^\infty \frac{x^k}{[k]!} = \frac1{(1-(1-q)x)_q^\infty}$;
• $E_q^x :\equiv \sum_{k=0}^\infty q^{k(k-1)/2} \frac{x^k}{[k]!} = (1+(1-q)x)_q^\infty$.

Properties:

• $e_q^0 = 1$, $E_q^0 =1$.
• $\frac1{(1-x)_0^\infty} = e_q^{x/(1-q)}$.
• $D_q e_q^x = e_q^x$, $D_q E_q^x = E_q^{qx}$.
• $D_q \frac1{(1-(1-q)x)_q^n} = \frac{(1-q)[n]}{(1-(1-q)x)_q^{n+1}}$, $D_q (1+(1-q)x)_q^n = (1-q) [n] (1+q(1-q)x)^{n-1}_q$.
• $e_q^{x}e_q^{y} \ne e_q^{x+y}$; but $e_q^{x}e_q^{y} = e_q^{x+y}$ iff $yx=qxy$.
• $E_q^{-x} = \frac1{e_q^{x}}$, i.e. $E_q^{x} = \frac1{e_q^{-x}}$.
• $e_{\frac1{q}}^{x} = E_q^x$.
• direct consequence of the previous two lines: $e_q^{-x}e_{\frac1{q}}^{x} = 1$.

Definitions: (q-trigonometric functions):

• $sin_q(x) :\equiv \frac{e_q^{ix}-e_q^{-ix}}{2i}$,
• $Sin_q(x) :\equiv \frac{E_q^{ix}-E_q^{-ix}}{2i}$,
• $cos_q(x) :\equiv \frac{e_q^{ix}+e_q^{-ix}}{2i}$,
• $Cos_q(x) :\equiv \frac{E_q^{ix}+E_q^{-ix}}{2i}$.

Properties:

• $cos_q(x) Cos_q(x) + sin_q(x) Sin_q(x) = 1$.
• $D_q sin_q(x) = cos_q(x)$,
• $D_q Sin_q(x) = Cos_q(qx)$,
• $D_q cos_q(x) = - sin_q(x)$,
• $D_q Cos_q(x) = - Sin_q(qx)$.

## Partition functions and product formulas

Definitions:

• The triangular numbers: $\Delta_n :\equiv \frac{n(n+1)}2$,
• the square numbers: $\Box_n :\equiv n^2$,
• the pentagonal numbers: $\Pi_n :\equiv \frac{n(3n-1)}2$,
• the k-gonal numbers: $m^{(k)}_n :\equiv (k-2) \Delta_{n-1} + n = \frac{n(kn -2n -k +4)}2$.

Definition: The classical partition function $p(n):{\Bbb Z}\rightarrow{\Bbb N}$) is defined as

• $p(n) =$ the number of ways to partition an positive integer number $n$ into sum of positive integers (modulo reordering of summands);
• $p(n)=0$ for $n<0$;
• $p(0)=1$.

Properties:

• Examples: $p(1)=1$, $p(2)=2$, $p(3) =3$, $p(4) = 5$, $p(5)=7$.
• Asymptotic behavior: $p(n) \sim \frac1{3\sqrt{3}n} e^{\pi \sqrt{\frac{2n}{3}}}$ \ as \ $n\rightarrow \infty$.
• $\varphi(q)^{-1} = \sum_{n=0}^\infty p(n) q^n$.
• $p(n) = \sum_{n=0}^\infty (-)^{n-1} (p(n-\Pi_n) + p(n-\Pi_{-n}))$.

Definition: the Euler’s product: $\varphi(q) :\equiv \prod_{n=1}^\infty (1-q^{n})$.

Theorem (Jacobi’s triple product identity): For $|q|<1$ following is true:

$\sum_{n\in{\Bbb Z}} q^{n^2}z^n = \prod_{n=1}^\infty (1-q^{2n})(1+q^{2n-1}z)(1+q^{2n-1}z^{-1})$.

Consequences:

• Euler’s product formula: $\sum_{n\in{\Bbb Z}} (-)^n q^{\frac{n(3n-1)}2} = \prod_{n=1}^\infty (1-q^{n})$.
• This can be rephrased as follows: $\varphi(q) = \sum_{n\in{\Bbb Z}} (-)^n q^{\Pi_n}$ where $\Pi_n$ are the pentagonal numbers defined above.
• Following Gauss identities are special cases of the Jacobi’s triple product identity:
• $\sum_{n=0}^{\infty} q^{\Delta_n} = \prod_{n=1}^\infty \frac{1-q^{2n}}{1-q^{2n-1}}$,
• $\sum_{n=0}^{\infty} (-q)^{\Box_n} = \prod_{n=1}^\infty \frac{1-q^{n}}{1+q^{n}}$.

Sources:

• [1] book: “Quantum Calculus” by Victor Kac and Pokman Cheung (Springer) – ISBN 0-387-9534198; QA303.C537 2001

Other references: