\(q\)-Pochhammers

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q_analogs q_pochhammers q_binomial_coefficients q_vandermonde_chu_identity

By [{gasper-rahman04}], for \(n \in \mathbb Z\) define

\[ (a)_n := (a; q)_n := \begin{cases} (1 - a) (1 - a q) ... (1 - a q^{n - 1}) & n > 0 \\ 1 & n = 0 \\ {1 \over (1 - a q^{-1}) (1 - a q^{-2}) ... (1 - a q^{-n})} & n < 0 \end{cases} \]

Thus for \(n < 0\)

\[ (q)_n = \infty \qquad (1) \]

\(\phi(q) := (q)_\infty\) is called Euler's function.

For \(n \ge 0\) define the \(q\)-binomial coefficients \({n \choose k}_q\) by

\[ {n \choose k}_q = {(n)_q \over (k)_q (n - k)_q} \qquad (2) \]

Thus by (1) when \(k < 0\) or \(k > n\), \({n \choose k}_q = 0\).

Claim 1. \(\sum_{\lambda \subset n^k} q^{|\lambda|} = {n + k \choose k}_q\)

Proof. Denote the LHS by \(F(n, k)\), then

\[ F(n, k) = F(n - 1, k) + q^n F(n, k - 1) \]

with \(F(0, k) = F(n, 0) = 1\). This recursive relation with the boundary conditions agree with those of the binomial coefficients. \(\square\)

Remark. The Young diagrams confined in the \(k \times n\) rectangle correspond to walks from \((0, 0)\) with \(n\) ups and \(k\) downs. Setting \(q = 1\) in (2) this degenerates to the binomial coefficients as enumerations of such walks.

1. \((x; q^{-1})_n = (- 1)^n x^n q^{-{n \choose 2}} (x^{-1}; q)_n\)
2. \((q^{-n}; q)_k = {(n)_q \over (n - k)_q} (-1)^k q^{{k \choose 2} - n k}\)
3. \({n \choose k}_{q^{-1}} = q^{- n (n - k)} {n \choose k}_q\)
4. \((a)_{n - k} = {(a)_n \over (a^{-1} q^{1 - n})_k} (- q a^{-1})^k q^{{k \choose 2} - n k}\)
5. \((a^{-1} q^{1 - n})_n = (a)_n (-a)^{- n} q^{- {n \choose 2}}\)

Theorem.

\[ {}_1\phi_0(a;;q;z) = \sum_{k \ge 0} {(a)_k \over (q)_k} z^k = {(a z)_\infty \over (z)_\infty}. \qquad |q| < 1, |z| < 1. \qquad (3) \]

Proof.

\begin{align} F(a, z) - F(a, q z) &= (1 - a) z F(a q, z)\\ F(a, z) - F(a q, z) &= - a z F(a q, z) \end{align}

Eliminating \(F(a q, z)\) we have

\[ F(a, z) = {1 - a z \over 1 - z} F(a, q z). \]

Applying this recursively we have

\[ F(a, z) = {(a z)_n \over (z)_n} F(a, q^n z). \]

Let \(n \to \infty\) we have

\[ F(a, z) = {(a z)_\infty \over (z)_\infty} F(a, 0) = {(a z)_\infty \over (z)_\infty}. \]

\(\square\)

Let \(x, y\) be the generators of a \(q\)-Grassmann algebra, namely they satisfy

\[ y x = q x y \]

Theorem. \({n \choose k}_q\) satisfies

\[ (x + y)^n = \sum_k {n \choose k}_q x^k y^{n - k} \]

This is also called \(q\)-binomial theorem.

Claim.

\[ {n \choose k}_q = \sum_{S \in {[n] \choose k}} q^{\sum_{a_i \in S} (a_i - i)} = \sum_{S \in {[n] \choose k}} q^{(\sum_{a_i \in S} a_i) - {k + 1 \choose 2}} \]

Pf. We know by the previous thm \({n \choose k}_q\) gathers the products with \(k\) \(x\)'s and \(n - k\) \(y\)'s, and each of such product corresponds to \(S = \{a_1 \le a_2 ... \le a_k\} \in {[n] \choose k}\) in a obvious way, e.g. when \(n = 8\) and \(k = 2\), \(yyxyxyyy \leftrightarrow \{3, 5\}\). Shifting all the \(x\)'s to the left corresponds \(\sum_i (a_i - i)\) shifts. \(\square\)

Another \(q\)-binomial theorem (a.k.a. Gauss binomial theorem) is

\[ \sum_k q^{{k \choose 2}} \alpha^k {n \choose k}_q = (- \alpha; q)_n. \qquad (4) \]

Proof 1. Set \(a := q^{- n}\) and \(z = - \alpha q^n\) in (3) and simplify. \(\square\)

Proof 2. The RHS is

\[ (1 + \alpha) (1 + \alpha q) ... (1 + \alpha q^{n - 1}) \]

The coefficient of \(\alpha^k\) enumerates the partitions of \(k\) distinct parts (with the possibility that the last part is \(0\)) and first part \(\le n - 1\), and that by removing \(0\) from the last part, \(1\) from the second last part, ... \(k - 1\) from the first part, we have partitions with length \(\le k\) and first part \(\le n - k\). Finally applying Claim 1 we arrive at the conclusion:

\begin{align} (1 + \alpha) &(1 + \alpha q) ... (1 + \alpha q^{n - 1}) \\ &= \sum_k \sum_{\lambda: \ell(\lambda) = k, \lambda_1 \le n - 1, \lambda_1 > \lambda_2 > ... > \lambda_k \ge 0} \alpha^k q^{|\lambda|}\\ &= \sum_k \alpha^k q^{{k \choose 2}} \sum_{\mu: \ell(\mu) \le k, \mu_1 \le n - k} \\ &= \sum_k \alpha^k q^{{k \choose 2}} {n \choose k}_q. \end{align}

\(\square\)

Yet a third \(q\)-deformation of the binomial theorem is as follows:

\[ \sum_k b^k (a; q)_k (b; q)_{n - k} {n \choose k}_q = (a b; q)_n \qquad (5) \]

which when \(q \to 1\) reads

\[ \sum_k (b - a b)^k (1 - b)^{n - k} {n \choose k} = (1 - a b)^n. \]

see Exercise 1.3 on pp25 of [{gasper-rahman04}].

Proof. By (3)

\[ \hg{1}{\phi}{0} (a; ; q, z) \hg{1}{\phi}{0} (b; ; q, a z) = \hg{1}{\phi}{0} (a b; ; q, z). \]

Expanding both sides and compare coefficients of \(z^n\) we are done. \(\square\)

This is also called a \(q\)-Vandermonde-Chu identiy. It is used to define the \(\phi_{q, \xi, \eta}\) distribution in q_hahn_process. As in the derivation of (4) using (3), (5) becomes (6) with certain parameters, see binomial_distribution.

Remark. Using Item 4 in Properties the LHS of (5) can be written as

\[ (b)_n \hg{2}{\phi}{1} (a, q^{- n}; b^{-1} q^{1 - n}; q, q) \]

and (5) can be derived using

\[ \hg{2}{\phi}{1}(q^{- n}, b; c; q, q) = {(c / b)_n \over (c)_n} b^n. (5.5) \]

and Item 5 in Properties.

\[ \sum_{k = 0}^n \alpha^k (\alpha; q)_{n - k} {n \choose k}_q = 1 \]

When \(q \to 1\) it reduces to the usual binomial theorem

\[ \alpha^k (1 - \alpha)^{n - k} {n \choose k} = 1 \]

See also binomial_distribution.

\[ \sum_s q^{(m_1 - s) (k - s)} {m_1 \choose s}_q {m_2 \choose k - s}_q = {m_1 + m_2 \choose k}_q. \qquad (6) \]

Proof 1.

\[ (- x)_{m_1 + m_2} = (- x)_{m_1} (- x q^{m_1})_{m_2}. \]

Applying (4) to both sides of the above formula and gather the coefficients of \(x^k\) we are done. \(\square\)

Proof 2. Applying (5.5) with \(n = m_1, b = q^{- k}, c = q^{m_2 - k + 1}\). \(\square\)

In the following let \(q = e^{- \epsilon}\).

For \(t > 0\)

\[ (q^t; q)_\infty = \Gamma(t)^{-1} \exp(A(\epsilon) + (1 - t) \log \epsilon + O(\epsilon)) \]

where \(A(\epsilon)\) is defined as

\[ A(\epsilon) = - {\pi^2 \over 6 \epsilon} - {1 \over 2} \log {\epsilon \over 2\pi} \]

Specifically, when \(t = 0\) we have asymptotics for Euler's function:

\[ \phi(q) = \exp(A(\epsilon) + O(\epsilon)) \]

This can be derived for example from Thm3.2 of [{banerjee-wilkerson16}].

Hence

\[ (n)_q = \Gamma(n + 1)^{-1} \epsilon^{- n} \exp(O(\epsilon)) \]

in Lemma 3.1 in [{gerasimov-lebedev-oblezin12}]:

Lemma 3.1. For \(\alpha \ge 1\),

\[ (\alpha \epsilon^{-1} \log \epsilon^{-1} + \epsilon^{-1} y)_q = \begin{cases} e^{A(\epsilon) + e^{- y} + O(\epsilon)} & \alpha = 1 \\ e^{A(\epsilon) + O(\epsilon)} & \alpha > 1 \end{cases} \]

References

• [banerjee-wilkerson16] Lambert series and q-functions near q= 1, Shubho Banerjee, Blake Wilkerson, arXiv preprint arXiv:1602.01085 2016.
• [gasper-rahman04] Basic hypergeometric series, George Gasper, Mizan Rahman, , Vol. 96 2004.
• [gerasimov-lebedev-oblezin12] On a classical limit of $q$-deformed Whittaker functions, Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin, Letters in Mathematical Physics, Vol. 100, p.279–290 2012.