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:

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